mrkiss_eerk_dormand_prince_5_4 Module Reference

Butcher tableau for Dormand & Prince's 7 stage, Order (5,4) Runge-Kutta method. More...

Variables
integer, parameter	s = 7
	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 = [5, 4]
	The method orders.
integer, dimension(m), parameter	se = [6, 7]
	Number of stages for each method.

Detailed Description

Butcher tableau for Dormand & Prince's 7 stage, Order (5,4) Runge-Kutta method.

IMO

I include this method for historical reasons and for unit tests. It's a good general use method with excellent principle truncation error coefficients. It has been very widely used, frequently as the default method, across many software packages with good results. It's a standard method to use for comparisons in the literature. Today it has largely been replaced by newer methods.

Known Aliases

'The Dormand-Prince Method', 'RKDP', 'DOPRI', 'DP5' (OrdinaryDiffEq.jl), 'ode45' (MATLAB & Octave), 'RK5(4)7M' (Dormand & Prince), & DOPRI5 (Hairer), 'ARKODE_DORMAND_PRINCE_7_4_5' (SUNDIALS)

Stability Image Links

References:

• Dormand & Prince (1980); A family of embedded Runge-Kutta formulae; J. Comput. Appl. Math. 6, no. 1
• Hairer, Norsett & Wanner (2009). Solving Ordinary Differential Equations. I: Nonstiff Problems. p178; zotero://select/items/0_VLZWN2CT
• Butcher (2016); Numerical Methods for Ordinary Differential Equations. 3ed; p224; zotero://select/items/0_V7UTIRPT

Variable Documentation

s

integer, parameter mrkiss_eerk_dormand_prince_5_4::s = 7

The order of the overall method.

Definition at line 68 of file mrkiss_eerk_dormand_prince_5_4.f90.

m

integer, parameter mrkiss_eerk_dormand_prince_5_4::m = 2

Number of methods.

Definition at line 70 of file mrkiss_eerk_dormand_prince_5_4.f90.

a

real(kind=rk), dimension(s,s), parameter mrkiss_eerk_dormand_prince_5_4::a

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

Definition at line 72 of file mrkiss_eerk_dormand_prince_5_4.f90.

b

real(kind=rk), dimension(s,m), parameter mrkiss_eerk_dormand_prince_5_4::b

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

Definition at line 80 of file mrkiss_eerk_dormand_prince_5_4.f90.

c

real(kind=rk), dimension(s), parameter mrkiss_eerk_dormand_prince_5_4::c

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

Definition at line 83 of file mrkiss_eerk_dormand_prince_5_4.f90.

p

integer, dimension(m), parameter mrkiss_eerk_dormand_prince_5_4::p = [5, 4]

The method orders.

Definition at line 85 of file mrkiss_eerk_dormand_prince_5_4.f90.

se

integer, dimension(m), parameter mrkiss_eerk_dormand_prince_5_4::se = [6, 7]

Number of stages for each method.

Definition at line 87 of file mrkiss_eerk_dormand_prince_5_4.f90.