mrkiss_erk_ralston_4 Module Reference

Butcher tableau for ralston's 4 stage order (4) Runge-Kutta method. More...

Variables
integer, parameter	s = 4
	The order of the overall method.
integer, parameter	m = 1
	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]
	The method orders.
integer, dimension(m), parameter	se = [4]
	Number of stages for each method.

Detailed Description

Butcher tableau for ralston's 4 stage order (4) Runge-Kutta method.

IMO

This method was designed to minimize truncation error over every other consideration includeing stability, roundoff error, performance, and storage.

Stability Image Links

References:

• Ralston (1962); Runge-Kutta Methods with Minimum Error Bounds; Math. of Comp. 16 (80); p431-437; zotero://select/items/0_2FK7F4J3

Variable Documentation

s

integer, parameter mrkiss_erk_ralston_4::s = 4

The order of the overall method.

Definition at line 60 of file mrkiss_erk_ralston_4.f90.

m

integer, parameter mrkiss_erk_ralston_4::m = 1

Number of methods.

Definition at line 62 of file mrkiss_erk_ralston_4.f90.

a

real(kind=rk), dimension(s,s), parameter mrkiss_erk_ralston_4::a

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

Definition at line 64 of file mrkiss_erk_ralston_4.f90.

  real(kind=rk),    parameter :: a(s,s) = reshape([         0.0_rk,         0.0_rk,         0.0_rk,       0.0_rk,                             &
       &                                             98650112.0_rk,         0.0_rk,         0.0_rk,       0.0_rk,                             &
       &                                           -695801205.0_rk, 911598325.0_rk,         0.0_rk,       0.0_rk,                             &
       &                                           -137399680.0_rk, -94224000.0_rk, 478248960.0_rk,       0.0_rk], [s, s]) / 246625280.0_rk + &
       &                                  reshape([         0.0_rk,         0.0_rk,         0.0_rk,       0.0_rk,                             &
       &                                                    0.0_rk,         0.0_rk,         0.0_rk,       0.0_rk,                             &
       &                                             85981665.0_rk, -97542225.0_rk,         0.0_rk,       0.0_rk,                             &
       &                                             21375552.0_rk, -73591360.0_rk,  52215808.0_rk,       0.0_rk], [s, s]) /  61656320.0_rk * sqrt(5.0_rk)

b

real(kind=rk), dimension(s,m), parameter mrkiss_erk_ralston_4::b

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

Definition at line 73 of file mrkiss_erk_ralston_4.f90.

  real(kind=rk),    parameter :: b(s,m) = reshape([   3439777.0_rk,    773875.0_rk,  13705216.0_rk, 5780280.0_rk], [s, m]) /  23699148.0_rk + &
       &                                  reshape([     78474.0_rk,  -1547750.0_rk,   1661952.0_rk, -192676.0_rk], [s, m]) /   5924787.0_rk * sqrt(5.0_rk)

c

real(kind=rk), dimension(s), parameter mrkiss_erk_ralston_4::c

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

Definition at line 76 of file mrkiss_erk_ralston_4.f90.

  real(kind=rk),    parameter :: c(s)   = [                 0.0_rk,        16.0_rk,        35.0_rk,      40.0_rk ]         /        40.0_rk + &
       &                                  [                 0.0_rk,         0.0_rk,        -3.0_rk,       0.0_rk ]         /        16.0_rk * sqrt(5.0_rk)

p

integer, dimension(m), parameter mrkiss_erk_ralston_4::p = [4]

The method orders.

Definition at line 79 of file mrkiss_erk_ralston_4.f90.

se

integer, dimension(m), parameter mrkiss_erk_ralston_4::se = [4]

Number of stages for each method.

Definition at line 81 of file mrkiss_erk_ralston_4.f90.