Butcher tableau for Fehlberg's 6 stage, Order (4,5) Runge-Kutta method. More...

Variables
integer, parameter	s = 6
	The order of the overall method.
integer, parameter	m = 2
	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 = [ 4, 5 ]
	The method orders.
integer, dimension(m), parameter	se = [ 5, 6 ]
	Number of stages for each method.

Detailed Description

Butcher tableau for Fehlberg's 6 stage, Order (4,5) Runge-Kutta method.

IMO: Included for historical reasons and for unit tests. Performs poorly in local extrapolation mode. This method was broadly implemented, and widely used. As such it is a standard used for comparison frequently in older literature. It was largely replaced by mrkiss_eerk_dormand_prince_5_4.

Known Aliases: 'RKF78'', 'ode78' (OrdinaryDiffEq.jl), 'ARKODE_FEHLBERG_6_4_5' (SUNDIALS).

Stability Image Links

References:

Erwin Fehlberg (1972); Classical eight and lower-order Runge-Kutta-Nystroem formulas with stepsize control for special second-order differential equations; NASA Tech. Rep. M-533
Butcher (2008); Numerical Methods for Ordinary Differential Equations. 2ed; p209; zotero://select/items/0_8V2GY73E
Butcher (2016); Numerical Methods for Ordinary Differential Equations. 3ed; p222; zotero://select/items/0_V7UTIRPT
Hairer, Norsett & Wanner (2009). Solving Ordinary Differential Equations. I: Nonstiff Problems. p180; zotero://select/items/0_VLZWN2CT

Variable Documentation

◆ s

integer, parameter mrkiss_eerk_fehlberg_4_5::s = 6

The order of the overall method.

Definition at line 68 of file mrkiss_eerk_fehlberg_4_5.f90.

◆ m

integer, parameter mrkiss_eerk_fehlberg_4_5::m = 2

Number of methods.

Definition at line 70 of file mrkiss_eerk_fehlberg_4_5.f90.

◆ a

real(kind=rk), dimension(s,s), parameter mrkiss_eerk_fehlberg_4_5::a

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

Definition at line 72 of file mrkiss_eerk_fehlberg_4_5.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, &
       &                                             45082440.0_rk,           0.0_rk,          0.0_rk,         0.0_rk,         0.0_rk,         0.0_rk, &
       &                                             16905915.0_rk,    50717745.0_rk,          0.0_rk,         0.0_rk,         0.0_rk,         0.0_rk, &
       &                                            158578560.0_rk,  -590976000.0_rk,  598855680.0_rk,         0.0_rk,         0.0_rk,         0.0_rk, &
       &                                            366503540.0_rk, -1442638080.0_rk, 1293593600.0_rk, -37129300.0_rk,         0.0_rk,         0.0_rk, &
       &                                            -53431040.0_rk,   360659520.0_rk, -249157376.0_rk,  81684460.0_rk, -49590684.0_rk,         0.0_rk], [s, s]) / 180329760.0_rk