Butcher tableau for Cash & Karp's 4 step, order (5,4) Runge-Kutta method. More...

Variables
integer, parameter	s = 6
	The order of the overall method.
integer, parameter	m = 5
	Number of methods.
real(kind=rk), dimension(s, s), parameter	a
	The \(\mathbf{a}\) matrix for the Butcher Tableau.
real(kind=rk), dimension(s, m), parameter	b
	The \(\mathbf{b}\) matrix for the Butcher Tableau.
real(kind=rk), dimension(s), parameter	c
	The \(\mathbf{c}\) matrix for the Butcher Tableau.
integer, dimension(m), parameter	p = [5, 4, 3, 2, 1]
	The method orders.
integer, dimension(m), parameter	se = [6, 6, 4, 2, 1]
	Number of stages for each method.

Detailed Description

Butcher tableau for Cash & Karp's 4 step, order (5,4) Runge-Kutta method.

IMO: I don't recommend this method as a 2 step embedded method for general use. It was designed with five embedded methods for use with stiff applications. It's an interesting method from both a historical and research perspective. Note this module has all five embedded methods defined (b1/p1-b5/p5).

Known Aliases: 'ARKODE_CASH_KARP_6_4_5' (SUNDIALS)

Stability Image Links

References:

Cash & Karp(1990); A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides; TOMS 16; zotero://select/items/0_2YSGGWSD

Variable Documentation

◆ s

integer, parameter mrkiss_eerk_cash_karp_5_4::s = 6

The order of the overall method.

Definition at line 67 of file mrkiss_eerk_cash_karp_5_4.f90.

◆ m

integer, parameter mrkiss_eerk_cash_karp_5_4::m = 5

Number of methods.

Definition at line 69 of file mrkiss_eerk_cash_karp_5_4.f90.

◆ a

real(kind=rk), dimension(s,s), parameter mrkiss_eerk_cash_karp_5_4::a

The \(\mathbf{a}\) matrix for the Butcher Tableau.

Definition at line 71 of file mrkiss_eerk_cash_karp_5_4.f90.

  real(kind=rk),    parameter :: a(s,s) = reshape([        0.0_rk,         0.0_rk,         0.0_rk,        0.0_rk,       0.0_rk,        0.0_rk,  &
                                                      110592.0_rk,         0.0_rk,         0.0_rk,        0.0_rk,       0.0_rk,        0.0_rk,  &
                                                       41472.0_rk,    124416.0_rk,         0.0_rk,        0.0_rk,       0.0_rk,        0.0_rk,  &
                                                      165888.0_rk,   -497664.0_rk,    663552.0_rk,        0.0_rk,       0.0_rk,        0.0_rk,  &
                                                     -112640.0_rk,   1382400.0_rk,  -1433600.0_rk,   716800.0_rk,       0.0_rk,        0.0_rk,  &
                                                       16310.0_rk,    189000.0_rk,     23000.0_rk,   221375.0_rk,   34155.0_rk,        0.0_rk], [s, s]) /   552960.0_rk