rklib_module – rklib

Runge-Kutta integration.

Note

The default real kind (wp) can be changed using optional preprocessor flags. This library was built with real kind: real(kind=real64) [8 bytes]

Uses

- root_module
- iso_fortran_env
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Graph Key

Nodes of different colours represent the following:

Solid arrows point from a submodule to the (sub)module which it is descended from. Dashed arrows point from a module or program unit to modules which it uses.

Used by

- Descendants:
- rklib_fixed_properties
- rklib_fixed_steps
- rklib_variable_dense
- rklib_variable_properties
- rklib_variable_steps
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Graph Key

Nodes of different colours represent the following:

Solid arrows point from a submodule to the (sub)module which it is descended from. Dashed arrows point from a module or program unit to modules which it uses.

Variables

Type	Visibility	Attributes		Name		Initial
integer,	public,	parameter	::	rk_module_rk	=	real64	real kind used by this module [8 bytes]
integer,	private,	parameter	::	wp	=	rk_module_rk	local copy of `rk_module_rk` with a shorter name
real(kind=wp),	private,	parameter	::	zero	=	0.0_wp
integer,	private,	parameter	::	max_error_len	=	100	max size of error message strings
integer,	private,	parameter	::	dense_form_polynomial	=	1	$x(t + \theta h) = x(t) + h \sum_i b_i(\theta) f_i$
integer,	private,	parameter	::	dense_form_dopri	=	2	the form used in `DOPRI5` and `DOP853`
integer,	public,	parameter	::	RKLIB_ERROR_DENSE_NOT_AVAILABLE	=	-16
integer,	public,	parameter	::	RKLIB_ERROR_DENSE_STEP_DISABLED	=	-15
integer,	public,	parameter	::	RKLIB_ERROR_DENSE_NOT_ENABLED	=	-14
integer,	public,	parameter	::	RKLIB_ERROR_DENSE_RECURSIVE	=	-13
integer,	public,	parameter	::	RKLIB_ERROR_DENSE_OUT_OF_RANGE	=	-12
integer,	public,	parameter	::	RKLIB_ERROR_NO_DENSE_OUTPUT	=	-11
integer,	public,	parameter	::	RKLIB_ERROR_TOO_MANY_STEPS	=	-10
integer,	public,	parameter	::	RKLIB_ERROR_INVALID_RTOL_SIZE	=	-9
integer,	public,	parameter	::	RKLIB_ERROR_INVALID_ATOL_SIZE	=	-8
integer,	public,	parameter	::	RKLIB_ERROR_INVALID_H	=	-7
integer,	public,	parameter	::	RKLIB_ERROR_USER_STOPPED	=	-6
integer,	public,	parameter	::	RKLIB_ERROR_MIN_STEP_SIZE	=	-5
integer,	public,	parameter	::	RKLIB_ERROR_TOO_MANY_REDUCTIONS	=	-4
integer,	public,	parameter	::	RKLIB_ERROR_INVALID_HINIT_METHOD	=	-3
integer,	public,	parameter	::	RKLIB_ERROR_G_NOT_ASSOCIATED	=	-2
integer,	public,	parameter	::	RKLIB_ERROR_F_NOT_ASSOCIATED	=	-1
integer,	public,	parameter	::	RKLIB_ERROR_NONE	=	0
character(len=max_error_len),	private,	parameter, dimension(RKLIB_ERROR_DENSE_NOT_AVAILABLE:RKLIB_ERROR_NONE)	::	rklib_error_messages	=	['No step is available for the dense output ', 'dense_output_step requires dense output ', 'Dense output was not enabled ', 'Interpolate called from within interpolate ', 'Dense output time is outside the last step ', 'The method does not have a dense output ', 'Too many steps ', 'Invalid size for rtol array ', 'Invalid size for atol array ', 'Step size cannot be zero ', 'User stopped the integration ', 'Minimum step size reached ', 'Too many attempts to reduce step size ', 'Invalid initial step size estimation method ', 'The event procedure g is not associated ', 'The function procedure f is not associated ', 'Success ']	Status message strings that go with the status codes. The index in this array is the `istatus` code.

Interfaces

interface

private pure module function euler_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(euler_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function midpoint_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(midpoint_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function heun_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(heun_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkssp22_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkssp22_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rk3_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rk3_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkssp33_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkssp33_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkssp53_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkssp53_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rk4_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rk4_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rks4_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rks4_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkr4_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkr4_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkls44_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkls44_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkls54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkls54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkssp54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkssp54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rks5_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rks5_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rk5_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rk5_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkc5_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkc5_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkl5_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkl5_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rklk5a_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rklk5a_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rklk5b_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rklk5b_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkb6_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkb6_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rk7_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rk7_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rk8_10_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rk8_10_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkcv8_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkcv8_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rk8_12_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rk8_12_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rks10_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rks10_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkz10_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkz10_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rko10_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rko10_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkh10_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkh10_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkbs32_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkbs32_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkssp43_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkssp43_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkf45_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkf45_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkck54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkck54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkdp54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkdp54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkt54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkt54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rks54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rks54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkpp54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkpp54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkpp54b_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkpp54b_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkbs54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkbs54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkss54_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkss54_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkdp65_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkdp65_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkc65_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkc65_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rktp64_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rktp64_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv65e_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv65e_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv65r_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv65r_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv65_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv65_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function dverk65_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(dverk65_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rktf65_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rktf65_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rktp75_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rktp75_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rktmy7_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rktmy7_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rktmy7s_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rktmy7s_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv76e_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv76e_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv76r_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv76r_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkss76_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkss76_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkf78_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkf78_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv78_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv78_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function dverk78_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(dverk78_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkdp85_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkdp85_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rktp86_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rktp86_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkdp87_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkdp87_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv87e_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv87e_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv87r_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv87r_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkev87_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkev87_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkk87_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkk87_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkf89_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkf89_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv89_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv89_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkt98a_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkt98a_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv98e_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv98e_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkv98r_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv98r_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rks98_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rks98_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkf108_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkf108_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkc108_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkc108_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkb109_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkb109_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rks1110a_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rks1110a_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkf1210_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkf1210_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rko129_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rko129_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private pure module function rkf1412_properties(me) result(p)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkf1412_class),	intent(in)			::	me

Return Value type(rklib_properties)

properties of the method

interface

private module subroutine euler(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(euler_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine midpoint(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(midpoint_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine heun(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(heun_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkssp22(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkssp22_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rk3(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rk3_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkssp33(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkssp33_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkssp53(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkssp53_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rk4(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rk4_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rks4(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rks4_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkr4(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkr4_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkls44(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkls44_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkls54(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkls54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkssp54(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkssp54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rks5(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rks5_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rk5(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rk5_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkc5(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkc5_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkl5(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkl5_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rklk5a(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rklk5a_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rklk5b(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rklk5b_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkb6(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkb6_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rk7(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rk7_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rk8_10(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rk8_10_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkcv8(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkcv8_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rk8_12(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rk8_12_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rks10(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rks10_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkz10(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkz10_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rko10(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rko10_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkh10(me, t, x, h, xf)

Arguments

Type	Intent	Attributes		Name
class(rkh10_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`

interface

private module subroutine rkbs32(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkbs32_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkssp43(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkssp43_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkf45(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkf45_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkck54(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkck54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkdp54(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkdp54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkt54(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkt54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rks54(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rks54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkpp54(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkpp54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkpp54b(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkpp54b_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkbs54(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkbs54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkss54(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkss54_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkdp65(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkdp65_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkc65(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkc65_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rktp64(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rktp64_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv65e(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv65e_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv65r(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv65r_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv65(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv65_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine dverk65(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(dverk65_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rktf65(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rktf65_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rktp75(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rktp75_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rktmy7(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rktmy7_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rktmy7s(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rktmy7s_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv76e(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv76e_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv76r(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv76r_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkss76(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkss76_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkf78(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkf78_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv78(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv78_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine dverk78(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(dverk78_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkdp85(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkdp85_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rktp86(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rktp86_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkdp87(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkdp87_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv87e(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv87e_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv87r(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv87r_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkev87(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkev87_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkk87(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkk87_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkf89(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkf89_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv89(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv89_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkt98a(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkt98a_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv98e(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv98e_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkv98r(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkv98r_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rks98(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rks98_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkf108(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkf108_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkc108(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkc108_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkb109(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkb109_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rks1110a(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rks1110a_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkf1210(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkf1210_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rko129(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rko129_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private module subroutine rkf1412(me, t, x, h, xf, xerr)

Arguments

Type	Intent	Attributes		Name
class(rkf1412_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x	initial state
real(kind=wp),	intent(in)		::	h	time step
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	state at time `t+h`
real(kind=wp),	intent(out),	dimension(me%n)	::	xerr	truncation error estimate for `x`

interface

private pure module subroutine rkdp54_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkdp54_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private pure module subroutine rkv65e_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkv65e_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkv65e_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv65e_class),	intent(inout)			::	me

interface

private pure module subroutine rkv65r_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkv65r_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkv65r_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv65r_class),	intent(inout)			::	me

interface

private pure module subroutine rkv76e_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkv76e_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkv76e_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv76e_class),	intent(inout)			::	me

interface

private pure module subroutine rkv76r_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkv76r_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkv76r_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv76r_class),	intent(inout)			::	me

interface

private pure module subroutine dverk78_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(dverk78_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine dverk78_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(dverk78_class),	intent(inout)			::	me

interface

private pure module subroutine rkdp85_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkdp85_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkdp85_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkdp85_class),	intent(inout)			::	me

interface

private pure module subroutine rkdp87_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkdp87_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkdp87_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkdp87_class),	intent(inout)			::	me

interface

private pure module subroutine rkv87e_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkv87e_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkv87e_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv87e_class),	intent(inout)			::	me

interface

private pure module subroutine rkv87r_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkv87r_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkv87r_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv87r_class),	intent(inout)			::	me

interface

private pure module subroutine rkev87_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkev87_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkev87_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkev87_class),	intent(inout)			::	me

interface

private pure module subroutine rkv98e_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkv98e_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkv98e_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv98e_class),	intent(inout)			::	me

interface

private pure module subroutine rkv98r_dense_coefficients(me, b, form)

Arguments

Type	Intent	Attributes		Name
class(rkv98r_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b`

interface

private module subroutine rkv98r_dense_extra_stages(me)

Arguments

Type	Intent	Optional	Attributes		Name
class(rkv98r_class),	intent(inout)			::	me

Abstract Interfaces

abstract interface

private pure function norm_func(x) result(xmag)

Vector norm function. Must return a value $\ge 0$ .

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=wp),	intent(in),		dimension(:)	::	x	a vector

Return Value real(kind=wp)

the magnitude of the vector

abstract interface

private pure function properties_func(me) result(p)

Returns the properties of the method.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(in)			::	me

Components

Type	Visibility	Attributes		Name		Initial
integer,	public		::	order	=	0	order of the method
integer,	public		::	number_of_stages	=	0	number of stages
logical,	public		::	fsal	=	.false.	if it is a FSAL method
logical,	public		::	low_storage	=	.false.	if it is a LS method
logical,	public		::	strong_stability_preserving	=	.false.	if it is a SSP method
logical,	public		::	dense	=	.false.	if the method has a dense output (continuous extension)
integer,	public		::	number_of_registers	=	0	number of `f` vectors used
real(kind=wp),	public		::	cfl	=	zero	Courant-Friedrichs-Lewy number
character(len=:),	public,	allocatable	::	short_name			short version of the method name
character(len=:),	public,	allocatable	::	long_name			longer description of the method

type, public :: stepsize_class

Algorithms for adjusting the step size for variable-step Runge-Kutta integrators.

Components

Type	Visibility	Attributes		Name		Initial
logical,	private		::	fixed_step_mode	=	.false.	if true, then the method runs in fixed step mode with not error estimation
real(kind=wp),	private		::	hmax	=	1.0e+6_wp	maximum allowed step size
real(kind=wp),	private		::	hmin	=	1.0e-6_wp	minimum allowed step size
real(kind=wp),	private		::	hfactor_reject	=	0.5_wp	minimum allowed factor for decreasing step size after rejected step
real(kind=wp),	private		::	hfactor_accept	=	2.0_wp	maximum allowed factor for increasing step size after accepted step
integer,	private		::	accept_mode	=	2	method to determine if step is accepted [1,2]
integer,	private		::	max_attempts	=	10000	maximum number of attempts to decrease step size before giving up
logical,	private		::	relative_err	=	.false.	if true, use the error per unit step: $\|\|err\|\| \le \|h\| tol$ . otherwise, use the error per step: $\|\|err\|\| \le tol$ . (The scaled error is then proportional to one less power of `h`, so `p_exponent_offset` should be one less than for the error per step.)
real(kind=wp),	private		::	safety_factor	=	0.9_wp	for `hfactor` equation (>0)
integer,	private		::	p_exponent_offset	=	1	the exponent of the `hfactor` equation is `1/(p + p_exponent_offset)` (`-1`, `0`, or `1`; `p + p_exponent_offset` is limited to be at least 1). It should be one over the power of `h` that the scaled error is proportional to. For a `p(p-1)` pair, the error estimate is proportional to $h^{q+1} = h^p$ (where `q=p-1` is the order of the error estimate), so `0` for the error per step, and `-1` for the error per unit step (`relative_err`).
procedure(norm_func),	private,	nopass, pointer	::	norm	=>	maxval_func	routine for computing the norm of the scaled error (e.g., maxval_func, norm2_func, rms_func, or a user function)
integer,	private		::	rtol_scaling	=	1	the state used to scale the relative tolerance: `1`: `tol = rtol\|x(t+h)\| + atol`, `2`: `tol = rtolmax(\|x(t)\|,\|x(t+h)\|) + atol` (as in `DOPRI5` and `DOP853`)
integer,	private		::	error_combination	=	1	for methods with two error estimates that are combined (currently `rkdp85`, which uses the 5th and 3rd order estimates of `DOP853`, see rkdp85): `1`: combined for each component, `2`: combined using the scaled norms of the two estimates (as in `DOP853`)
real(kind=wp),	private		::	pi_beta	=	0.0_wp	the $\beta$ parameter of the PI (Lund) step size control (stabilized step size control, see Hairer & Wanner, Solving ODEs I, section IV.2). `0` (the default) is the standard controller. Values up to `0.04` are recommended for `DOPRI5`.
real(kind=wp),	private		::	pi_beta_mult	=	0.75_wp	the error exponent is reduced by `pi_beta_mult*pi_beta` (`0.75` is used in `DOPRI5`, `0.2` in `DOP853`)
logical,	private		::	no_increase_after_reject	=	.false.	if true, the step size is not increased for the step after a rejected step (as in `DOPRI5` and `DOP853`)
real(kind=wp),	private		::	err_old	=	1.0e-4_wp	scaled error of the last accepted step (for the PI control)
logical,	private		::	last_rejected	=	.false.	if the last step was rejected

Type-Bound Procedures

procedure, public :: initialize => stepsize_class_constructor
procedure, public :: compute_stepsize
procedure, public :: destroy => destroy_stepsize_class
procedure, public :: reset => stepsize_class_reset
procedure, public :: tolerance => stepsize_class_tolerance

type, public, abstract :: rk_class

main integration class

Components

Type	Visibility	Attributes		Name		Initial
integer,	private		::	istatus	=	0	status code
logical,	private		::	stopped	=	.false.	if user has stopped the integration in `f` or `report`.
integer,	private		::	num_steps	=	0	number of accepted steps taken
integer,	private		::	max_number_of_steps	=	huge(1)	maximum number of steps to take
integer,	private		::	report_rate	=	1	how often to call report function: `0` : no reporting (same as not associating `report`), `1` : report every point, `2` : report every other point, etc. The first and last point are always reported.
logical,	private		::	stop_on_errors	=	.false.	if true, then errors will stop the program
integer,	private		::	n	=	0	user specified number of variables
procedure(deriv_func),	private,	pointer	::	f	=>	null()	user-specified derivative function
procedure(report_func),	private,	pointer	::	report	=>	null()	user-specified report function
procedure(event_func),	private,	pointer	::	g	=>	null()	event function (stop when this is zero)
type(root_method),	private		::	solver	=	root_method_brent	the root solver method to use for even finding
real(kind=wp),	private,	dimension(:,:), allocatable	::	funcs			matrix to store the function evaluations in the step function. this will be size (`n` x `number_of_registers`). If the dense output is enabled, and the method needs extra function evaluations for it (see dense_extra_stages), these are stored in additional columns after the ones for the step, so the size is (`n` x the number of rows of `dense_b`).
logical,	private		::	dense_enabled	=	.false.	if the dense output was enabled by the user (see initialize_variable_step)
integer,	private		::	dense_form	=	0	the form of the `dense_b` coefficients (`dense_form_polynomial` or `dense_form_dopri`)
real(kind=wp),	private,	dimension(:,:), allocatable	::	dense_b			dense output coefficients (allocated if the method has a dense output, even if it is not enabled) (`number_of_stages` x `degree`). See dense_coefficients.
logical,	private		::	dense_available	=	.false.	if the dense output data for the last step is valid
logical,	private		::	dense_computed	=	.false.	if `dense_r` has been computed for the last step
logical,	private		::	dense_computing	=	.false.	if `dense_r` is being computed (to detect recursive calls to interpolate from `f`)
real(kind=wp),	private		::	dense_t	=	zero	initial time of the last accepted step
real(kind=wp),	private		::	dense_h	=	zero	size of the last accepted step
real(kind=wp),	private,	dimension(:), allocatable	::	dense_x			initial state of the last accepted step (`n`)
real(kind=wp),	private,	dimension(:), allocatable	::	dense_xf			final state of the last accepted step (`n`)
real(kind=wp),	private,	dimension(:,:), allocatable	::	dense_r			interpolating polynomial coefficients for the last accepted step (`n` x `degree`)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, private :: init => initialize_rk_class
procedure, private :: begin => begin_integration_rk_class
procedure, private :: raise_exception
procedure, private :: clear_exception
procedure, private :: export_point
procedure, private :: dense_coefficients	dense output coefficients for the method
procedure, private :: dense_extra_stages	additional function evaluations for the dense output
procedure, private :: dense_save
procedure, private :: dense_check
procedure, private :: dense_invalidate
procedure(begin_func), private, deferred :: begin_integration
procedure(properties_func), public, deferred :: properties

type, public, abstract, extends(rk_class) :: rk_fixed_step_class

fixed step size class

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure(properties_func), public, deferred :: properties
procedure(step_func_fixed), private, deferred :: step	the step routine for the rk method
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, private :: begin_integration => begin_integration_rk_fixed_step_class

type, public, abstract, extends(rk_class) :: rk_variable_step_class

Main integration class for variable step size Runge-Kutta methods

Components

Type	Visibility	Attributes		Name		Initial
type(stepsize_class),	private		::	stepsize_method			the method for varying the step size
real(kind=wp),	private,	dimension(:), allocatable	::	rtol			relative tolerance (`size(n)`)
real(kind=wp),	private,	dimension(:), allocatable	::	atol			absolute tolerance (`size(n)`)
integer,	private		::	hinit_method	=	1	if automatically computing the inital step size, which method to use. 1 = `hstart`, 2 = `hinit`.
integer,	private		::	num_rejected_steps	=	0	number of rejected steps
real(kind=wp),	private		::	last_accepted_step_size	=	zero	the last accepted step size `dt` from the integration (positive or negative)
real(kind=wp),	private		::	dense_output_step	=	zero	if > 0, the points are reported at this fixed time interval using the dense output (see initialize_variable_step)
logical,	private		::	dense_reporting	=	.false.	if the dense output is used for reporting in the current integration
real(kind=wp),	private		::	dense_output_t0	=	zero	initial time of the current integration
integer(kind=int64),	private		::	dense_output_k	=	0	index of the last dense output point reported: `t = dense_output_t0 + k*dense_output_step` (with the sign for the direction of integration)
real(kind=wp),	private,	allocatable	::	t_saved			cached `t`
real(kind=wp),	private,	dimension(:), allocatable	::	x_saved			cached `x`
real(kind=wp),	private,	dimension(:), allocatable	::	f_saved			cached `f`

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure(properties_func), public, deferred :: properties
procedure(step_func_variable), private, deferred :: step	the step routine for the rk method
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, private :: hstart	for automatically computing the initial step size [this is from DDEABM]
procedure, private :: hinit	for automatically computing the initial step size [this is from DOP853]
procedure, private :: begin_integration => begin_integration_rk_variable_step_class
procedure, private :: compute_initial_step
procedure, private :: export_step
procedure, private :: export_dense_output
procedure, private :: order	returns `p`, the order of the method
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache

type, public, abstract, extends(rk_variable_step_class) :: rk_variable_step_fsal_class

a variable step method with the "first same as last" (FSAL) property: the last function evaluation of a step (at the final t and x) is the first one of the next step. It is reused by means of the function evaluation cache (see set_fsal_cache and check_fsal_cache).

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure(properties_func), public, deferred :: properties
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache

type, public, extends(rk_fixed_step_class) :: euler_class

Euler

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => euler
procedure, public :: properties => euler_properties

type, public, extends(rk_fixed_step_class) :: midpoint_class

Midpoint

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => midpoint
procedure, public :: properties => midpoint_properties

type, public, extends(rk_fixed_step_class) :: heun_class

Heun

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => heun
procedure, public :: properties => heun_properties

type, public, extends(rk_fixed_step_class) :: rkssp22_class

2-stage, 2nd order TVD Runge-Kutta Shu-Osher

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkssp22
procedure, public :: properties => rkssp22_properties

type, public, extends(rk_fixed_step_class) :: rk3_class

3th order Runge-Kutta

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rk3
procedure, public :: properties => rk3_properties

type, public, extends(rk_fixed_step_class) :: rkssp33_class

3-stage, 3rd order TVD Runge-Kutta Shu-Osher

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkssp33
procedure, public :: properties => rkssp33_properties

type, public, extends(rk_fixed_step_class) :: rkssp53_class

5-stage, 3rd order SSP Runge-Kutta Spiteri-Ruuth

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkssp53
procedure, public :: properties => rkssp53_properties

type, public, extends(rk_fixed_step_class) :: rk4_class

Classic 4th order Runge-Kutta

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rk4
procedure, public :: properties => rk4_properties

type, public, extends(rk_fixed_step_class) :: rks4_class

4th order Runge-Kutta Shanks

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rks4
procedure, public :: properties => rks4_properties

type, public, extends(rk_fixed_step_class) :: rkr4_class

4th order Runge-Kutta Ralston

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkr4
procedure, public :: properties => rkr4_properties

type, public, extends(rk_fixed_step_class) :: rkls44_class

4-stage, 4th order low storage non-TVD Runge-Kutta Jiang-Shu

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkls44
procedure, public :: properties => rkls44_properties

type, public, extends(rk_fixed_step_class) :: rkls54_class

5-stage, 4th order low storage Runge-Kutta Carpenter-Kennedy

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkls54
procedure, public :: properties => rkls54_properties

type, public, extends(rk_fixed_step_class) :: rkssp54_class

5-stage, 4th order SSP Runge-Kutta Spiteri-Ruuth

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkssp54
procedure, public :: properties => rkssp54_properties

type, public, extends(rk_fixed_step_class) :: rks5_class

5th order Runge-Kutta Shanks

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rks5
procedure, public :: properties => rks5_properties

type, public, extends(rk_fixed_step_class) :: rk5_class

5th order Runge-Kutta

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rk5
procedure, public :: properties => rk5_properties

type, public, extends(rk_fixed_step_class) :: rkc5_class

5th order Runge-Kutta Cassity

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkc5
procedure, public :: properties => rkc5_properties

type, public, extends(rk_fixed_step_class) :: rkl5_class

5th order Runge-Kutta Lawson

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkl5
procedure, public :: properties => rkl5_properties

type, public, extends(rk_fixed_step_class) :: rklk5a_class

5th order Runge-Kutta Luther-Konen 1

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rklk5a
procedure, public :: properties => rklk5a_properties

type, public, extends(rk_fixed_step_class) :: rklk5b_class

5th order Runge-Kutta Luther-Konen 2

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rklk5b
procedure, public :: properties => rklk5b_properties

type, public, extends(rk_fixed_step_class) :: rkb6_class

6th order Runge-Kutta Butcher

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkb6
procedure, public :: properties => rkb6_properties

type, public, extends(rk_fixed_step_class) :: rk7_class

7th order Runge-Kutta Shanks

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rk7
procedure, public :: properties => rk7_properties

type, public, extends(rk_fixed_step_class) :: rk8_10_class

10-stage, 8th order Runge-Kutta Shanks

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rk8_10
procedure, public :: properties => rk8_10_properties

type, public, extends(rk_fixed_step_class) :: rkcv8_class

11-stage, 8th order Runge-Kutta Cooper-Verner

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkcv8
procedure, public :: properties => rkcv8_properties

type, public, extends(rk_fixed_step_class) :: rk8_12_class

12-stage, 8th order Runge-Kutta Shanks

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rk8_12
procedure, public :: properties => rk8_12_properties

type, public, extends(rk_fixed_step_class) :: rks10_class

10th order Runge-Kutta Stepanov

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rks10
procedure, public :: properties => rks10_properties

type, public, extends(rk_fixed_step_class) :: rkz10_class

10th order Runge-Kutta Zhang

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkz10
procedure, public :: properties => rkz10_properties

type, public, extends(rk_fixed_step_class) :: rko10_class

10th order Runge-Kutta Ono

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rko10
procedure, public :: properties => rko10_properties

type, public, extends(rk_fixed_step_class) :: rkh10_class

10th order Runge-Kutta Hairer

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_fixed_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_fixed_step
procedure, public :: integrate_to_event => integrate_to_event_fixed_step
procedure, public :: step => rkh10
procedure, public :: properties => rkh10_properties

type, public, extends(rk_variable_step_fsal_class) :: rkbs32_class

Bogacki & Shampine 3(2)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkbs32
procedure, public :: properties => rkbs32_properties

type, public, extends(rk_variable_step_class) :: rkssp43_class

4-stage, 3rd order SSP

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkssp43
procedure, public :: properties => rkssp43_properties

type, public, extends(rk_variable_step_class) :: rkf45_class

Fehlberg 4(5)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkf45
procedure, public :: properties => rkf45_properties

type, public, extends(rk_variable_step_class) :: rkck54_class

Cash & Karp 5(4)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkck54
procedure, public :: properties => rkck54_properties

type, public, extends(rk_variable_step_fsal_class) :: rkdp54_class

Dormand-Prince 5(4)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkdp54
procedure, public :: properties => rkdp54_properties
procedure, public :: dense_coefficients => rkdp54_dense_coefficients

type, public, extends(rk_variable_step_fsal_class) :: rkt54_class

Tsitouras 5(4)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkt54
procedure, public :: properties => rkt54_properties

type, public, extends(rk_variable_step_fsal_class) :: rks54_class

Stepanov 5(4)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rks54
procedure, public :: properties => rks54_properties

type, public, extends(rk_variable_step_fsal_class) :: rkpp54_class

Papakostas-PapaGeorgiou 5(4)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkpp54
procedure, public :: properties => rkpp54_properties

type, public, extends(rk_variable_step_fsal_class) :: rkpp54b_class

Papakostas-PapaGeorgiou 5(4) b

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkpp54b
procedure, public :: properties => rkpp54b_properties

type, public, extends(rk_variable_step_class) :: rkbs54_class

Bogacki & Shampine 5(4)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkbs54
procedure, public :: properties => rkbs54_properties

type, public, extends(rk_variable_step_class) :: rkss54_class

Sharp & Smart 5(4)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkss54
procedure, public :: properties => rkss54_properties

type, public, extends(rk_variable_step_class) :: rkdp65_class

Dormand-Prince 6(5)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkdp65
procedure, public :: properties => rkdp65_properties

type, public, extends(rk_variable_step_class) :: rkc65_class

Calvo 6(5)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkc65
procedure, public :: properties => rkc65_properties

type, public, extends(rk_variable_step_class) :: rktp64_class

Tsitouras & Papakostas NEW6(4)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rktp64
procedure, public :: properties => rktp64_properties

type, public, extends(rk_variable_step_fsal_class) :: rkv65e_class

Verner efficient (9,6(5))

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv65e
procedure, public :: properties => rkv65e_properties
procedure, public :: dense_coefficients => rkv65e_dense_coefficients
procedure, public :: dense_extra_stages => rkv65e_dense_extra_stages

type, public, extends(rk_variable_step_fsal_class) :: rkv65r_class

Verner robust (9,6(5))

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv65r
procedure, public :: properties => rkv65r_properties
procedure, public :: dense_coefficients => rkv65r_dense_coefficients
procedure, public :: dense_extra_stages => rkv65r_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkv65_class

Verner 6(5)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv65
procedure, public :: properties => rkv65_properties

type, public, extends(rk_variable_step_class) :: dverk65_class

Verner 6(5) "DVERK"

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => dverk65
procedure, public :: properties => dverk65_properties

type, public, extends(rk_variable_step_fsal_class) :: rktf65_class

Tsitouras & Famelis 6(5)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rktf65
procedure, public :: properties => rktf65_properties

type, public, extends(rk_variable_step_class) :: rktp75_class

Tsitouras & Papakostas NEW7(5)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rktp75
procedure, public :: properties => rktp75_properties

type, public, extends(rk_variable_step_class) :: rktmy7_class

7th order Tanaka-Muramatsu-Yamashita

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rktmy7
procedure, public :: properties => rktmy7_properties

type, public, extends(rk_variable_step_class) :: rktmy7s_class

7th order Stable Tanaka-Muramatsu-Yamashita

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rktmy7s
procedure, public :: properties => rktmy7s_properties

type, public, extends(rk_variable_step_class) :: rkv76e_class

Verner efficient (10:7(6))

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv76e
procedure, public :: properties => rkv76e_properties
procedure, public :: dense_coefficients => rkv76e_dense_coefficients
procedure, public :: dense_extra_stages => rkv76e_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkv76r_class

Verner robust (10:7(6))

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv76r
procedure, public :: properties => rkv76r_properties
procedure, public :: dense_coefficients => rkv76r_dense_coefficients
procedure, public :: dense_extra_stages => rkv76r_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkss76_class

Sharp & Smart 7(6)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkss76
procedure, public :: properties => rkss76_properties

type, public, extends(rk_variable_step_class) :: rkf78_class

Fehlberg 7(8)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkf78
procedure, public :: properties => rkf78_properties

type, public, extends(rk_variable_step_class) :: rkv78_class

Verner 7(8)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv78
procedure, public :: properties => rkv78_properties

type, public, extends(rk_variable_step_class) :: dverk78_class

Verner "Maple" 7(8)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => dverk78
procedure, public :: properties => dverk78_properties
procedure, public :: dense_coefficients => dverk78_dense_coefficients
procedure, public :: dense_extra_stages => dverk78_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkdp85_class

Dormand-Prince 8(5)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkdp85
procedure, public :: properties => rkdp85_properties
procedure, public :: dense_coefficients => rkdp85_dense_coefficients
procedure, public :: dense_extra_stages => rkdp85_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rktp86_class

Tsitouras & Papakostas NEW8(6)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rktp86
procedure, public :: properties => rktp86_properties

type, public, extends(rk_variable_step_class) :: rkdp87_class

Dormand & Prince RK8(7)13M

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkdp87
procedure, public :: properties => rkdp87_properties
procedure, public :: dense_coefficients => rkdp87_dense_coefficients
procedure, public :: dense_extra_stages => rkdp87_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkv87e_class

Verner efficient (8)7

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv87e
procedure, public :: properties => rkv87e_properties
procedure, public :: dense_coefficients => rkv87e_dense_coefficients
procedure, public :: dense_extra_stages => rkv87e_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkv87r_class

Verner robust (8)7

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv87r
procedure, public :: properties => rkv87r_properties
procedure, public :: dense_coefficients => rkv87r_dense_coefficients
procedure, public :: dense_extra_stages => rkv87r_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkev87_class

Enright-Verner (8)7

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkev87
procedure, public :: properties => rkev87_properties
procedure, public :: dense_coefficients => rkev87_dense_coefficients
procedure, public :: dense_extra_stages => rkev87_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkk87_class

Kovalnogov-Fedorov-Karpukhina-Simos-Tsitouras 8(7)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkk87
procedure, public :: properties => rkk87_properties

type, public, extends(rk_variable_step_class) :: rkf89_class

Fehlberg 8(9)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkf89
procedure, public :: properties => rkf89_properties

type, public, extends(rk_variable_step_class) :: rkv89_class

Verner 8(9)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv89
procedure, public :: properties => rkv89_properties

type, public, extends(rk_variable_step_class) :: rkt98a_class

Tsitouras 9(8) A

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkt98a
procedure, public :: properties => rkt98a_properties

type, public, extends(rk_variable_step_class) :: rkv98e_class

Verner efficient (16:9(8))

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv98e
procedure, public :: properties => rkv98e_properties
procedure, public :: dense_coefficients => rkv98e_dense_coefficients
procedure, public :: dense_extra_stages => rkv98e_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rkv98r_class

Verner robust (16:9(8))

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkv98r
procedure, public :: properties => rkv98r_properties
procedure, public :: dense_coefficients => rkv98r_dense_coefficients
procedure, public :: dense_extra_stages => rkv98r_dense_extra_stages

type, public, extends(rk_variable_step_class) :: rks98_class

Sharp 9(8)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rks98
procedure, public :: properties => rks98_properties

type, public, extends(rk_variable_step_class) :: rkf108_class

Feagin 8(10)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkf108
procedure, public :: properties => rkf108_properties

type, public, extends(rk_variable_step_class) :: rkc108_class

Curtis 10(8)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkc108
procedure, public :: properties => rkc108_properties

type, public, extends(rk_variable_step_class) :: rkb109_class

Baker 10(9)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkb109
procedure, public :: properties => rkb109_properties

type, public, extends(rk_variable_step_class) :: rks1110a_class

Stone 11(10)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rks1110a
procedure, public :: properties => rks1110a_properties

type, public, extends(rk_variable_step_class) :: rkf1210_class

Feagin 12(10)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkf1210
procedure, public :: properties => rkf1210_properties

type, public, extends(rk_variable_step_class) :: rko129_class

Ono 12(9)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rko129
procedure, public :: properties => rko129_properties

type, public, extends(rk_variable_step_class) :: rkf1412_class

Feagin 14(12)

Type-Bound Procedures

procedure, public :: destroy	destructor
procedure, public :: stop => rk_class_stop	user-callable method to stop the integration
procedure, public :: status => rk_class_status	get status code and message
procedure, public :: failed
procedure, public :: interpolate	dense output for the last accepted step
procedure, public :: dense_interval	interval of the last accepted step for the dense output
procedure, public :: initialize => initialize_variable_step	initialize the class (set n,f, and report)
procedure, public :: integrate => integrate_variable_step
procedure, public :: integrate_to_event => integrate_to_event_variable_step
procedure, public :: info => info_variable_step
procedure, public :: destroy_fsal_cache
procedure, public :: check_fsal_cache
procedure, public :: set_fsal_cache
procedure, public :: step => rkf1412
procedure, public :: properties => rkf1412_properties

Functions

private pure function order(me) result(p)

Returns the order of the RK method

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_variable_step_class),	intent(in)			::	me

Return Value integer

order of the method

private function failed(me)

Returns true if there was an error. Can use rk_class_status to get more info.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(in)			::	me

Return Value logical

private function dense_check(me)

Check that the dense output can be used: the method has one, and it was enabled. If not, an error is raised.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me

Return Value logical

public pure function norm2_func(x) result(xmag)

Use intrinsic norm2(x) for computing the vector norm.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=wp),	intent(in),		dimension(:)	::	x

Return Value real(kind=wp)

public pure function rms_func(x) result(xmag)

Use the root mean square, $\sqrt{\sum x_i^2 / n}$ , for computing the vector norm. This is the norm used in Hairer's DOPRI5 and DOP853 codes.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=wp),	intent(in),		dimension(:)	::	x

Return Value real(kind=wp)

public pure function maxval_func(x) result(xmag)

Use maxval(abs(x)) for computing the vector norm.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=wp),	intent(in),		dimension(:)	::	x

Return Value real(kind=wp)

private pure function stepsize_class_tolerance(me, rtol, atol, x, xf) result(tol)

The error tolerance vector for a step from x to xf: rtol*s + atol, where s depends on the rtol_scaling option.

Arguments

Type	Intent	Attributes		Name
class(stepsize_class),	intent(in)		::	me
real(kind=wp),	intent(in),	dimension(:)	::	rtol	relative tolerance
real(kind=wp),	intent(in),	dimension(:)	::	atol	absolute tolerance
real(kind=wp),	intent(in),	dimension(:)	::	x	state at the start of the step
real(kind=wp),	intent(in),	dimension(:)	::	xf	state at the end of the step

Return Value real(kind=wp), dimension(size(x))

the error tolerance for each component

private function compute_initial_step(me, t0, tf, x0, h0) result(dt)

Compute the initial step size.

Arguments

Type	Intent	Attributes		Name
class(rk_variable_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t0	initial time
real(kind=wp),	intent(in)		::	tf	final time
real(kind=wp),		dimension(me%n)	::	x0	initial state
real(kind=wp),	intent(in)		::	h0	user-input initial step size (if zero, then one is computed)

Return Value real(kind=wp)

step size to use

private function hinit(me, x, y, posneg, f0, hmax, atol, rtol)

computation of an initial step size guess

Arguments

Type	Intent	Attributes		Name
class(rk_variable_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	x
real(kind=wp),	intent(in),	dimension(:)	::	y	dimension(n)
real(kind=wp),	intent(in)		::	posneg	posneg = sign(1.0_wp,xend-x)
real(kind=wp),	intent(in),	dimension(:)	::	f0	dimension(n)
real(kind=wp),	intent(in)		::	hmax
real(kind=wp),	intent(in),	dimension(:)	::	atol
real(kind=wp),	intent(in),	dimension(:)	::	rtol

Return Value real(kind=wp)

Subroutines

private subroutine clear_exception(me)

Clear any exception.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me

private subroutine raise_exception(me, error_code)

Raise an exception.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me
integer,	intent(in)			::	error_code	the error to raise

private subroutine destroy(me)

Destructor for rk_class.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(out)			::	me

private subroutine rk_class_stop(me)

User-callable method to stop the integration (e.g., from f, g, or report). No further function evaluations are done, and the integrator returns, with the status set to RKLIB_ERROR_USER_STOPPED. This is not treated as an error (i.e., it does not stop the program if stop_on_errors is true).

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me

private subroutine rk_class_status(me, istatus, message)

Get the status of an integration.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(in)			::	me
integer,	intent(out),	optional		::	istatus	status code (`<0` means an error)
character(len=:),	intent(out),	optional,	allocatable	::	message	status message

private subroutine export_point(me, t, x, first_or_last)

Wrapper for exporting points during integration.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me
real(kind=wp),	intent(in)			::	t
real(kind=wp),	intent(in),		dimension(:)	::	x
logical,	intent(in),	optional		::	first_or_last	if this is the first or last point (always reported)

private pure subroutine dense_coefficients(me, b, form)

Returns the dense output coefficients for the method. This is the default for methods that do not have a dense output (b is not allocated).

Arguments

Type	Intent	Attributes		Name
class(rk_class),	intent(in)		::	me
real(kind=wp),	intent(out),	dimension(:,:), allocatable	::	b	dense output coefficients
integer,	intent(out)		::	form	the form of `b` (`dense_form_polynomial` or `dense_form_dopri`)

private subroutine dense_extra_stages(me)

Compute any additional function evaluations needed for the dense output of the last accepted step, and store them in me%funcs (after the ones used by the step function). This is the default for methods that don't need any.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me

private subroutine dense_save(me, t, x, h, xf)

Save the data needed for the dense output of a step that was just accepted. The function evaluations of the step are taken from me%funcs when interpolate is called, so this is only valid until the next step is taken.

Arguments

Type	Intent	Attributes		Name
class(rk_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	initial time of the step
real(kind=wp),	intent(in),	dimension(:)	::	x	initial state of the step
real(kind=wp),	intent(in)		::	h	size of the step
real(kind=wp),	intent(in),	dimension(:)	::	xf	final state of the step

private subroutine dense_invalidate(me)

Mark the dense output data as invalid. This must be called before me%funcs is overwritten by a new step.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me

private recursive subroutine interpolate(me, t, x, istatus)

Dense output: compute the state at a time within the last accepted step, using the continuous extension of the method (if it has one, see rklib_properties dense).

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me
real(kind=wp),	intent(in)			::	t	time at which to compute the state
real(kind=wp),	intent(out),		dimension(me%n)	::	x	state at `t`
integer,	intent(out),	optional		::	istatus	status of this call

private subroutine dense_interval(me, available, t0, tf, x0, xf)

Returns the interval of the last accepted step, which is where interpolate can be used. For example, this can be called from the report function to get the previous time and state, so that points in between can be computed.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me
logical,	intent(out)			::	available	if dense output is available
real(kind=wp),	intent(out),	optional		::	t0	initial time of the step
real(kind=wp),	intent(out),	optional		::	tf	final time of the step
real(kind=wp),	intent(out),	optional,	dimension(me%n)	::	x0	initial state of the step
real(kind=wp),	intent(out),	optional,	dimension(me%n)	::	xf	final state of the step

private subroutine destroy_fsal_cache(me)

Destructor for the function evaluation cache.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_variable_step_class),	intent(inout)			::	me

private subroutine check_fsal_cache(me, t, x, f)

Compute the function at t and x, or get it from the function evaluation cache if it is there (see set_fsal_cache). This is used by all the variable-step methods for the first stage of a step. It may already be in the cache because:

Arguments

Type	Intent	Attributes		Name
class(rk_variable_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t
real(kind=wp),	intent(in),	dimension(:)	::	x
real(kind=wp),	intent(out),	dimension(:)	::	f

private subroutine set_fsal_cache(me, t, x, f)

Compute the function and add it to the function evaluation cache, so that it can be reused by the next step (see check_fsal_cache). This is used for f at the end of a step: by the first-same-as-last (FSAL) methods (the last stage of the step), and by the methods whose dense output needs it as one of its extra stages (see dense_extra_stages).

Arguments

Type	Intent	Attributes		Name
class(rk_variable_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t
real(kind=wp),	intent(in),	dimension(:)	::	x
real(kind=wp),	intent(out),	dimension(:)	::	f

private subroutine initialize_rk_class(me, n, f, report, g, stop_on_errors, max_number_of_steps, report_rate, solver, dense_output)

Initialize the rk_class.

Arguments

Type	Intent	Optional		Name
class(rk_class),	intent(inout)		::	me
integer,	intent(in)		::	n	number of variables
procedure(deriv_func)			::	f	derivative function
procedure(report_func),		optional	::	report	for reporting the steps
procedure(event_func),		optional	::	g	for stopping at an event
logical,	intent(in),	optional	::	stop_on_errors	stop the program for any errors (default is False)
integer,	intent(in),	optional	::	max_number_of_steps	max number of steps allowed
integer,	intent(in),	optional	::	report_rate	how often to call report function: `0` : no reporting (same as not associating `report`), `1` : report every point, `2` : report every other point, etc. The first and last point are always reported.
class(root_method),	intent(in),	optional	::	solver	the root-finding method to use for even finding. if not present, then `brent_solver` is used.
logical,	intent(in),	optional	::	dense_output	to enable the dense output (default is False). This is ignored if the method does not have one.

private subroutine begin_integration_rk_class(me)

Begin an integration.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_class),	intent(inout)			::	me

private subroutine begin_integration_rk_fixed_step_class(me)

Begin a rk_fixed_step_class integration.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_fixed_step_class),	intent(inout)			::	me

private subroutine initialize_fixed_step(me, n, f, report, g, stop_on_errors, max_number_of_steps, report_rate, solver)

Initialize the rk_fixed_step_class.

Arguments

Type	Intent	Optional		Name
class(rk_fixed_step_class),	intent(inout)		::	me
integer,	intent(in)		::	n	number of variables
procedure(deriv_func)			::	f	derivative function
procedure(report_func),		optional	::	report	for reporting the steps
procedure(event_func),		optional	::	g	for stopping at an event
logical,	intent(in),	optional	::	stop_on_errors	stop the program for any errors (default is False)
integer,	intent(in),	optional	::	max_number_of_steps	max number of steps allowed
integer,	intent(in),	optional	::	report_rate	how often to call report function: `0` : no reporting (same as not associating `report`), `1` : report every point, `2` : report every other point, etc. The first and last point are always reported.
class(root_method),	intent(in),	optional	::	solver	the root-finding method to use for even finding. if not present, then `brent_solver` is used.

private subroutine integrate_fixed_step(me, t0, x0, h, tf, xf)

Main integration routine for the rk_class.

Arguments

Type	Intent	Attributes		Name
class(rk_fixed_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t0	initial time
real(kind=wp),	intent(in),	dimension(:)	::	x0	initial state
real(kind=wp),	intent(in)		::	h	abs(time step)
real(kind=wp),	intent(in)		::	tf	final time
real(kind=wp),	intent(out),	dimension(:)	::	xf	final state

private subroutine integrate_to_event_fixed_step(me, t0, x0, h, tmax, tol, tf, xf, gf)

Event-finding integration routine for the rk_class. Integrates until g(t,x)=0, or until t=tf (whichever happens first).

Arguments

Type	Intent	Attributes		Name
class(rk_fixed_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t0	initial time
real(kind=wp),	intent(in),	dimension(:)	::	x0	initial state
real(kind=wp),	intent(in)		::	h	abs(time step)
real(kind=wp),	intent(in)		::	tmax	max final time if event not located
real(kind=wp),	intent(in)		::	tol	function tolerance for root finding
real(kind=wp),	intent(out)		::	tf	actual final time reached
real(kind=wp),	intent(out),	dimension(:)	::	xf	final state (at tf)
real(kind=wp),	intent(out)		::	gf	g value at tf

private subroutine stepsize_class_constructor(me, hmin, hmax, hfactor_reject, hfactor_accept, norm, accept_mode, relative_err, safety_factor, p_exponent_offset, max_attempts, fixed_step_mode, rtol_scaling, error_combination, pi_beta, pi_beta_mult, no_increase_after_reject)

Constructor for a stepsize_class.

Arguments

Type	Intent	Optional		Name
class(stepsize_class),	intent(out)		::	me	(all options not specified are set to their defaults)
real(kind=wp),	intent(in),	optional	::	hmin	minimum allowed step size (>0)
real(kind=wp),	intent(in),	optional	::	hmax	maximum allowed step size (>0)
real(kind=wp),	intent(in),	optional	::	hfactor_reject	minimum allowed factor for decreasing step size after rejected step (>0)
real(kind=wp),	intent(in),	optional	::	hfactor_accept	maximum allowed factor for decreasing step size after accepted step (>0)
procedure(norm_func),		optional	::	norm	the user-specified $\|\|x\|\|$ function
integer,	intent(in),	optional	::	accept_mode	method to determine if step is accepted [1,2]
logical,	intent(in),	optional	::	relative_err	if true, use the error per unit step: $\|\|err\|\| \le \|h\| tol$ . Otherwise (the default), use the error per step: $\|\|err\|\| \le tol$ . Note that, for the error per unit step, the scaled error is proportional to one less power of `h`, so `p_exponent_offset` should be one less than for the error per step (e.g., `-1` rather than `0` for a `p(p-1)` pair).
real(kind=wp),	intent(in),	optional	::	safety_factor	for `hfactor` equation (>0)
integer,	intent(in),	optional	::	p_exponent_offset	the exponent of the step size factor is `1/(p + p_exponent_offset)` (`-1`, `0`, or `1`). See stepsize_class for details.
integer,	intent(in),	optional	::	max_attempts	max step size change attempts after rejected step
logical,	intent(in),	optional	::	fixed_step_mode	if true, then the method runs in fixed step mode with not error estimation. All the other inputs are ignored. Note that this requires a `dt /= 0` input for the integrator.
integer,	intent(in),	optional	::	rtol_scaling	the state used to scale the relative tolerance: `1`: `\|x(t+h)\|` (the default), `2`: `max(\|x(t)\|,\|x(t+h)\|)` (as in `DOPRI5` and `DOP853`).
integer,	intent(in),	optional	::	error_combination	for methods with two error estimates (`rkdp85`): `1`: combined for each component (the default), `2`: combined using their scaled norms (as in `DOP853`).
real(kind=wp),	intent(in),	optional	::	pi_beta	the $\beta$ parameter for the PI (Lund) stabilized step size control (`0` to disable, which is the default). Values up to `0.04` are recommended for `DOPRI5`.
real(kind=wp),	intent(in),	optional	::	pi_beta_mult	the error exponent is reduced by `pi_beta_mult*pi_beta` (`0.75` is the default, and is used in `DOPRI5`, `0.2` is used in `DOP853`).
logical,	intent(in),	optional	::	no_increase_after_reject	if true, the step size is not increased for the step after a rejected step (default is false).

private pure subroutine stepsize_class_reset(me)

Reset the internal state of the step size control. This is called at the start of each integration.

Arguments

Type	Intent	Optional	Attributes		Name
class(stepsize_class),	intent(inout)			::	me

private subroutine destroy_stepsize_class(me)

Destructor for stepsize_class.

Arguments

Type	Intent	Optional	Attributes		Name
class(stepsize_class),	intent(out)			::	me

private subroutine compute_stepsize(me, n, h, tol, err, p, hnew, accept)

Compute the new step size using the specific method.

Arguments

Type	Intent	Attributes		Name
class(stepsize_class),	intent(inout)		::	me	(`inout` since it has some state, see `pi_beta` and `no_increase_after_reject`)
integer,	intent(in)		::	n	number of variables
real(kind=wp),	intent(in)		::	h	current step size (<>0)
real(kind=wp),	intent(in),	dimension(n)	::	tol	abs error tolerance (>0)
real(kind=wp),	intent(in),	dimension(n)	::	err	truncation error estimate (>0)
integer,	intent(in)		::	p	order of the method
real(kind=wp),	intent(out)		::	hnew	new step size (<>0)
logical,	intent(out)		::	accept	if the step is accepted

private subroutine begin_integration_rk_variable_step_class(me)

Begin a rk_variable_step_class integration.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_variable_step_class),	intent(inout)			::	me

private subroutine initialize_variable_step(me, n, f, rtol, atol, stepsize_method, hinit_method, report, g, stop_on_errors, max_number_of_steps, report_rate, solver, dense_output, dense_output_step)

Initialize the rk_variable_step_class.

Arguments

Type	Intent	Optional	Attributes		Name
class(rk_variable_step_class),	intent(inout)			::	me
integer,	intent(in)			::	n	number of equations
procedure(deriv_func)				::	f	derivative function
real(kind=wp),	intent(in),	optional,	dimension(:)	::	rtol	relative tolerance (if size=1, then same tol used for all equations). If not present, a default of `100*eps` is used
real(kind=wp),	intent(in),	optional,	dimension(:)	::	atol	absolute tolerance (if size=1, then same tol used for all equations). If not present, a default of `100*eps` is used
type(stepsize_class),	intent(in),	optional		::	stepsize_method	method for varying the step size
integer,	intent(in),	optional		::	hinit_method	which method (1 or 2) to use for automatic initial step size computation. 1 = use `hstart`, 2 = use `hinit`.
procedure(report_func),		optional		::	report	for reporting the steps
procedure(event_func),		optional		::	g	for stopping at an event
logical,	intent(in),	optional		::	stop_on_errors	stop the program for any errors (default is False)
integer,	intent(in),	optional		::	max_number_of_steps	max number of steps allowed
integer,	intent(in),	optional		::	report_rate	how often to call report function: `0` : no reporting (same as not associating `report`), `1` : report every point, `2` : report every other point, etc. The first and last point are always reported.
class(root_method),	intent(in),	optional		::	solver	the root-finding method to use for even finding. if not present, then `brent_solver` is used.
logical,	intent(in),	optional		::	dense_output	to enable the dense output (default is False, unless `dense_output_step` is present and not zero). If enabled, interpolate can be used (e.g., from the `report` function), and the dense output is also used for event location in integrate_to_event_variable_step. The method must have a dense output. When not enabled, there is no overhead for the dense output.
real(kind=wp),	intent(in),	optional		::	dense_output_step	if present and not zero, the points are reported at this fixed time interval (starting from the initial time), using the dense output of the method, rather than at the steps taken by the integrator. The first and last point are also reported (so the last interval will generally be shorter). This is only used if `report` is present and `report_rate>0` (`report_rate` is otherwise ignored). The method must have a dense output. This automatically enables the dense output (it is an error if `dense_output=.false.` is also specified).

private subroutine info_variable_step(me, num_steps, num_rejected_steps, last_accepted_step_size)

Return some info about the integration.

Arguments

Type	Intent	Optional		Name
class(rk_variable_step_class),	intent(in)		::	me
integer,	intent(out),	optional	::	num_steps	number of steps taken
integer,	intent(out),	optional	::	num_rejected_steps	number of rejected steps
real(kind=wp),	intent(out),	optional	::	last_accepted_step_size	the last accepted step size `dt` from the integration (positive or negative)

private subroutine export_step(me, t, x, ok)

Export the point(s) after a step has been accepted (other than the last step). Either the end of the step, or the dense output points within the step (see export_dense_output).

Arguments

Type	Intent	Attributes		Name
class(rk_variable_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t	time at the end of the step
real(kind=wp),	intent(in),	dimension(:)	::	x	state at the end of the step
logical,	intent(out)		::	ok	false if there was an error (the integration should not continue)

private subroutine export_dense_output(me, t_stop, ok)

Report the dense output points (at the fixed dense_output_step interval) within the last accepted step, that are before t_stop.

Arguments

Type	Intent		Name
class(rk_variable_step_class),	intent(inout)	::	me
real(kind=wp),	intent(in)	::	t_stop	report points up to (but not including) this time
logical,	intent(out)	::	ok	false if there was an error (the integration should not continue)

private subroutine integrate_variable_step(me, t0, x0, h, tf, xf)

Main integration routine for the rk_variable_step_class.

Arguments

Type	Intent	Attributes		Name
class(rk_variable_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t0	initial time
real(kind=wp),	intent(in),	dimension(:)	::	x0	initial state
real(kind=wp),	intent(in)		::	h	initial abs(time step)
real(kind=wp),	intent(in)		::	tf	final time
real(kind=wp),	intent(out),	dimension(:)	::	xf	final state

private subroutine integrate_to_event_variable_step(me, t0, x0, h, tmax, tol, tf, xf, gf)

Event-finding integration routine for the rk_variable_step_class. Integrates until g(t,x)=0, or until t=tf (whichever happens first).

Arguments

Type	Intent	Attributes		Name
class(rk_variable_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	t0	initial time
real(kind=wp),	intent(in),	dimension(me%n)	::	x0	initial state
real(kind=wp),	intent(in)		::	h	abs(time step)
real(kind=wp),	intent(in)		::	tmax	max final time if event not located
real(kind=wp),	intent(in)		::	tol	function tolerance for root finding
real(kind=wp),	intent(out)		::	tf	actual final time reached
real(kind=wp),	intent(out),	dimension(me%n)	::	xf	final state (at tf)
real(kind=wp),	intent(out)		::	gf	g value at tf

private subroutine hstart(me, a, b, y, yprime, etol, h)

Computes a starting step size to be used in solving initial value problems in ordinary differential equations.

Arguments

Type	Intent	Attributes		Name
class(rk_variable_step_class),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	a	the initial point of integration.
real(kind=wp),	intent(in)		::	b	a value of the independent variable used to define the direction of integration. a reasonable choice is to set `b` to the first point at which a solution is desired. you can also use `b`, if necessary, to restrict the length of the first integration step because the algorithm will not compute a starting step length which is bigger than `abs(b-a)`, unless `b` has been chosen too close to `a`. (it is presumed that hstart has been called with `b` different from `a` on the machine being used. also see the discussion about the parameter `small`.)
real(kind=wp),	intent(in),	dimension(me%n)	::	y	the vector of initial values of the `neq` solution components at the initial point `a`.
real(kind=wp),	intent(in),	dimension(me%n)	::	yprime	the vector of derivatives of the `neq` solution components at the initial point `a`. (defined by the differential equations in subroutine `me%f`)
real(kind=wp),	intent(in),	dimension(me%n)	::	etol	the vector of error tolerances corresponding to the `neq` solution components. it is assumed that all elements are positive. following the first integration step, the tolerances are expected to be used by the integrator in an error test which roughly requires that `abs(local error) <= etol` for each vector component.
real(kind=wp),	intent(out)		::	h	appropriate starting step size to be attempted by the differential equation method.

rklib_module Module

Contents

Variables

Interfaces

Abstract Interfaces

Derived Types

Functions

Subroutines

Uses

Used by

Descendants:

Variables

Interfaces

interface

private pure module function euler_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function midpoint_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function heun_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkssp22_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rk3_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkssp33_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkssp53_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rk4_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rks4_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkr4_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkls44_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkls54_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkssp54_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rks5_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rk5_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkc5_properties(me) result(p) Implementation →

Arguments

Return Value type(rklib_properties)

interface

private pure module function rkl5_properties(me) result(p) Implementation →

Arguments

private pure module function euler_properties(me) result(p)

private pure module function midpoint_properties(me) result(p)

private pure module function heun_properties(me) result(p)

private pure module function rkssp22_properties(me) result(p)

private pure module function rk3_properties(me) result(p)

private pure module function rkssp33_properties(me) result(p)

private pure module function rkssp53_properties(me) result(p)

private pure module function rk4_properties(me) result(p)

private pure module function rks4_properties(me) result(p)

private pure module function rkr4_properties(me) result(p)

private pure module function rkls44_properties(me) result(p)

private pure module function rkls54_properties(me) result(p)

private pure module function rkssp54_properties(me) result(p)

private pure module function rks5_properties(me) result(p)

private pure module function rk5_properties(me) result(p)

private pure module function rkc5_properties(me) result(p)

private pure module function rkl5_properties(me) result(p)

private pure module function rklk5a_properties(me) result(p)

private pure module function rklk5b_properties(me) result(p)

private pure module function rkb6_properties(me) result(p)

private pure module function rk7_properties(me) result(p)

private pure module function rk8_10_properties(me) result(p)

private pure module function rkcv8_properties(me) result(p)

private pure module function rk8_12_properties(me) result(p)

private pure module function rks10_properties(me) result(p)

private pure module function rkz10_properties(me) result(p)

private pure module function rko10_properties(me) result(p)

private pure module function rkh10_properties(me) result(p)

private pure module function rkbs32_properties(me) result(p)

private pure module function rkssp43_properties(me) result(p)

private pure module function rkf45_properties(me) result(p)

private pure module function rkck54_properties(me) result(p)

private pure module function rkdp54_properties(me) result(p)

private pure module function rkt54_properties(me) result(p)