MRKISS 2025-09-10
A tiny library with zero dependencies that aims to make it easy to use & experiment with explicit Runge-Kutta methods.
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mrkiss_eerk_fehlberg_7_8.f90
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1! -*- Mode:F90; Coding:us-ascii-unix; fill-column:129 -*-
2!.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.H.S.!!
3!>
4!! @file mrkiss_eerk_fehlberg_7_8.f90
5!! @author Mitch Richling http://www.mitchr.me/
6!! @brief Butcher tableau for Fehlberg's 13 stage, Order (7,8) Runge-Kutta method.@EOL
7!! @keywords ode ivp differential equation initial value problem rk
8!! @std F2023
9!! @see https://github.com/richmit/MRKISS
10!! @copyright
11!! @parblock
12!! Copyright (c) 2025, Mitchell Jay Richling <http://www.mitchr.me/> All rights reserved.
13!!
14!! Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following
15!! conditions are met:
16!!
17!! 1. Redistributions of source code must retain the above copyright notice, this list of conditions, and the following
18!! disclaimer.
19!!
20!! 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions, and the following
21!! disclaimer in the documentation and/or other materials provided with the distribution.
22!!
23!! 3. Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products
24!! derived from this software without specific prior written permission.
25!!
26!! THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES,
27!! INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
28!! DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
29!! EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
30!! USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
31!! LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
32!! OF THE POSSIBILITY OF SUCH DAMAGE.
33!! @endparblock
34!.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.H.E.!!
35
36!----------------------------------------------------------------------------------------------------------------------------------
37!> Butcher tableau for Fehlberg's 13 stage, Order (7,8) Runge-Kutta method
38!!
39!! @image html eerk_fehlberg_7_8-stab.png
40!!
41!! @par IMO
42!! Included for historical reasons. Performs poorly in local extrapolation mode. This one was pretty broadly used, but not
43!! nearly as much as mrkiss_eerk_fehlberg_4_5. It was largely replaced by mrkiss_eerk_dormand_prince_7_8.
44!!
45!! @par Known Aliases
46!! 'RKF78'', 'ode78' (OrdinaryDiffEq.jl), 'ARKODE_FEHLBERG_13_7_8' (SUNDIALS).
47!!
48!! @par Stability Image Links
49!! <a href="eerk_fehlberg_7_8-stab.png"> <img src="eerk_fehlberg_7_8-stab.png" width="256px"> </a>
50!! <a href="eerk_fehlberg_7_8-astab.png"> <img src="eerk_fehlberg_7_8-astab.png" width="256px"> </a>
51!! <a href="eerk_fehlberg_7_8-star1.png"> <img src="eerk_fehlberg_7_8-star1.png" width="256px"> </a>
52!! <a href="eerk_fehlberg_7_8-star2.png"> <img src="eerk_fehlberg_7_8-star2.png" width="256px"> </a>
53!!
54!! @par References:
55!! - Erwin Fehlberg (1972); Classical eight- and lower-order Runge-Kutta-Nystroem formulas with stepsize control for
56!! special second-order differential equations; NASA Technical Report M-533
57!! - Butcher (2008); Numerical Methods for Ordinary Differential Equations. 2ed; p209; zotero://select/items/0_8V2GY73E
58!! - Butcher (2016); Numerical Methods for Ordinary Differential Equations. 3ed; p222; zotero://select/items/0_V7UTIRPT
59!! - Hairer, Norsett & Wanner (2009). Solving Ordinary Differential Equations. I: Nonstiff Problems. p180;
60!!
61module mrkiss_eerk_fehlberg_7_8
62 use mrkiss_config, only: rk
63 implicit none
64 public
65 !> The order of the overall method
66 integer, parameter :: s = 13
67 !> Number of methods
68 integer, parameter :: m = 2
69 !> The @f$\mathbf{a}@f$ matrix for the Butcher Tableau. @hideinitializer @hideinlinesource
70 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, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
71 & 32800.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
72 & 12300.0_rk, 36900.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
73 & 18450.0_rk, 0.0_rk, 55350.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
74 & 184500.0_rk, 0.0_rk, -691875.0_rk, 691875.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
75 & 22140.0_rk, 0.0_rk, 0.0_rk, 110700.0_rk, 88560.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
76 & -102500.0_rk, 0.0_rk, 0.0_rk, 512500.0_rk, -1066000.0_rk, 1025000.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
77 & 45756.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 120048.0_rk, -98400.0_rk, 6396.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
78 & 885600.0_rk, 0.0_rk, 0.0_rk, -3911400.0_rk, 6927360.0_rk, -5264400.0_rk, 329640.0_rk, 1328400.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
79 & -373100.0_rk, 0.0_rk, 0.0_rk, 94300.0_rk, -3201280.0_rk, 2550200.0_rk, -140220.0_rk, 1254600.0_rk, -36900.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
80 & 257364.0_rk, 0.0_rk, 0.0_rk, -920700.0_rk, 1942272.0_rk, -1625400.0_rk, 230364.0_rk, 243000.0_rk, 121500.0_rk, 194400.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
81 & 6480.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, -64800.0_rk, -6480.0_rk, -32400.0_rk, 32400.0_rk, 64800.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, &
82 & -191916.0_rk, 0.0_rk, 0.0_rk, -920700.0_rk, 1942272.0_rk, -1560600.0_rk, 236844.0_rk, 275400.0_rk, 89100.0_rk, 129600.0_rk, 0.0_rk, 442800.0_rk, 0.0_rk], [s, s]) / 442800.0_rk
83 !> The @f$\mathbf{b}@f$ matrix for the Butcher Tableau. @hideinitializer @hideinlinesource
84 real(kind=rk), parameter :: b(s,m) = reshape([ 41.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 272.0_rk, 216.0_rk, 216.0_rk, 27.0_rk, 27.0_rk, 41.0_rk, 0.0_rk, 0.0_rk, &
85 & 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 0.0_rk, 272.0_rk, 216.0_rk, 216.0_rk, 27.0_rk, 27.0_rk, 0.0_rk, 41.0_rk, 41.0_rk], [s, m]) / 840.0_rk
86 !> The @f$\mathbf{c}@f$ matrix for the Butcher Tableau. @hideinitializer @hideinlinesource
87 real(kind=rk), parameter :: c(s) = [ 0.0_rk, 8.0_rk, 12.0_rk, 18.0_rk, 45.0_rk, 54.0_rk, 90.0_rk, 18.0_rk, 72.0_rk, 36.0_rk, 108.0_rk, 0.0_rk, 108.0_rk] / 108.0_rk
88 !> The method orders
89 integer, parameter :: p(m) = [7, 8]
90 !> Number of stages for each method
91 integer, parameter :: se(m) = [11, 13]
92
93
94end module mrkiss_eerk_fehlberg_7_8
mrkiss_config
Configuration for MRKISS == MR RK KISS == Mitch Richling's Runge-Kutta Keep It Simple Stupid.
Definition mrkiss_config.f90:38
mrkiss_config::rk
integer, parameter, public rk
Real kind used across the library.
Definition mrkiss_config.f90:46
mrkiss_eerk_fehlberg_7_8
Butcher tableau for Fehlberg's 13 stage, Order (7,8) Runge-Kutta method.
Definition mrkiss_eerk_fehlberg_7_8.f90:61
mrkiss_eerk_fehlberg_7_8::p
integer, dimension(m), parameter p
The method orders.
Definition mrkiss_eerk_fehlberg_7_8.f90:89
mrkiss_eerk_fehlberg_7_8::a
real(kind=rk), dimension(s, s), parameter a
The matrix for the Butcher Tableau.
Definition mrkiss_eerk_fehlberg_7_8.f90:70
mrkiss_eerk_fehlberg_7_8::se
integer, dimension(m), parameter se
Number of stages for each method.
Definition mrkiss_eerk_fehlberg_7_8.f90:91
mrkiss_eerk_fehlberg_7_8::b
real(kind=rk), dimension(s, m), parameter b
The matrix for the Butcher Tableau.
Definition mrkiss_eerk_fehlberg_7_8.f90:84
mrkiss_eerk_fehlberg_7_8::c
real(kind=rk), dimension(s), parameter c
The matrix for the Butcher Tableau.
Definition mrkiss_eerk_fehlberg_7_8.f90:87
mrkiss_eerk_fehlberg_7_8::s
integer, parameter s
The order of the overall method.
Definition mrkiss_eerk_fehlberg_7_8.f90:66
mrkiss_eerk_fehlberg_7_8::m
integer, parameter m
Number of methods.
Definition mrkiss_eerk_fehlberg_7_8.f90:68