MRKISS 2025-09-10
A tiny library with zero dependencies that aims to make it easy to use & experiment with explicit Runge-Kutta methods.
Loading...
Searching...
No Matches
mrkiss_eerk_verner_9_8.f90
Go to the documentation of this file.
1! -*- Mode:F90; Coding:us-ascii-unix; fill-column:129 -*-
2!.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.H.S.!!
3!>
4!! @file mrkiss_eerk_verner_9_8.f90
5!! @author Mitch Richling http://www.mitchr.me/
6!! @brief Butcher tableau for Verner's 16 stage, Order (9,8) 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 Verner's 16 stage, Order (9,8) Runge-Kutta method.
38!!
39!! @image html eerk_verner_9_8-stab.png
40!!
41!! @par Known Aliases
42!! 'Vern9' (OrdinaryDiffEq.jl), 'ARKODE_VERNER_16_8_9' (SUNDIALS).
43!!
44!! @par Stability Image Links
45!! <a href="eerk_verner_9_8-stab.png"> <img src="eerk_verner_9_8-stab.png" width="256px"> </a>
46!! <a href="eerk_verner_9_8-astab.png"> <img src="eerk_verner_9_8-astab.png" width="256px"> </a>
47!! <a href="eerk_verner_9_8-star1.png"> <img src="eerk_verner_9_8-star1.png" width="256px"> </a>
48!! <a href="eerk_verner_9_8-star2.png"> <img src="eerk_verner_9_8-star2.png" width="256px"> </a>
49!!
50!! IMO: This is my go-to O(9,8) method.
51!!
52!! @par References:
53!! - Verner (1978) "Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error"; SIAM J. Num. Anal. 15 (4); p772-790
54!! - Jim Verner's Refuge for Runge-Kutta Pairs
55!! - Retrieved on 2025-07-16 from :
56!! https://www.sfu.ca/~jverner/
57!! - On the above page, in a section annotated with "Added May, 2007", this tableau was labeled as
58!! A "more efficient" Runge--Kutta (9)8 Pair with Interpolants
59!! - The two links with the data were:
60!! - https://www.sfu.ca/~jverner/RKV98.IIa.Efficient.000000349.081209.FLOAT6040OnWeb zotero://select/items/0_UEQBISIW
61!! - https://www.sfu.ca/~jverner/RKV98.IIa.Efficient.000000349.081210.CoeffsOnlyRADandFLOATS
62!! - The first link above is titled:
63!! A 'most efficient" Runge--Kutta (16:9(8)) pair
64!!
65module mrkiss_eerk_verner_9_8
66 use mrkiss_config, only: rk
67 implicit none
68 public
69 !> The order of the overall method
70 integer, parameter :: s = 16
71 !> Number of methods
72 integer, parameter :: m = 2
73 !> The @f$\mathbf{a}@f$ matrix for the Butcher Tableau. @hideinitializer @hideinlinesource
74 real(kind=rk), parameter :: a(s,s) = reshape([ 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
75 & 0.03462000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
76 & -0.0389335438857287327017042687229284478531893268571582235135993884622582_rk, 0.1359578945245091786499878854939346230294986370378387534996881820898099_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
77 & 0.03638413148954266723060635628912731569111599131775519874478329761033190_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.1091523944686280016918190688673819470733479739532655962343498928309957_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
78 & 2.025763914393969636805657604282571047510830037662906582191234862933749_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, -7.638023836496292020387602153091964592951697697070004819490073470713397_rk, 6.173259922102322383581944548809393545440867659407098237298838607779648_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
79 & 0.05112275589406060872792270881648288397196918900059064260163610374675498_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.1770823794555021537929910813839068684086482623355845760735247980480933_rk, 0.00080277624092225014536138698108025283757893616146955804093042761746477_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
80 & 0.1316006357975216279279871693164256985333565813282958398376595254433496_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, -0.2957276252669636417685183174672273730697987768668985202103174231614870_rk, 0.0878137803564295237421124704053886667082860121433388766947895643361975_rk, 0.6213052975225274774321435005639430026099597958976190269617770039696270_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
81 & 0.07166666666666666666666666666666666666666666666666666666666666666666667_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.3305533578915319409260346730051472207727232256950986731344190751632464_rk, 0.2427799754418013924072986603281861125606101076382346601989142581700870_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
82 & 0.07180664062500000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.3294380283228177160744825466257672816400825123306885781893525221733437_rk, 0.1165190029271822839255174533742327183599174876693114218106474778266563_rk, -0.03401367187500000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
83 & 0.04836757646340646986611287718844085773548709477069979718368675437195896_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.03928989925676163974333190042057047002853617703182596319915186452542175_rk, 0.1054740945890344608263649267140088017603726876952699052952491693373838_rk, -0.02143865284648312665982642293830533996213913432454864572927796457527105_rk, -0.1041229174627194437759832813847147895622568251732470199488098236594934_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
84 & -0.02664561487201478635337289243849737340534379474964909082774468076895606_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.03333333333333333333333333333333333333333333333333333333333333333333333_rk, -0.1631072244872467239162704487554706387140651776819187065152511064265169_rk, 0.03396081684127761199487954930015522928243429529037697621819640215248736_rk, 0.1572319413814626097110769806810024118076538798421353850583416985461703_rk, 0.2152267478031879552303534778794770376959874639657221027331243531634820_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
85 & 0.03689009248708622334786359863227633989718899724778444513830232783626695_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, -0.1465181576725542928653609891758501156785280103192803493730073991381010_rk, 0.2242577768172024345345469822625833796000987410341655851576018936697766_rk, 0.02294405717066072637090897902753790803033136466339767092446101317871955_rk, -0.0035850052905728761357394424889330334334357221832889997121382962511547_rk, 0.08669223316444385506869203619044453906050894327941439662464999838202812_rk, 0.4383840651968337846196219974168630120572063359423373354134681522277429_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
86 & -0.4866012215113340846662212357570395295089013330525351287376507318523452_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, -6.304602650282852990657772792012007122987508211314778264330741972821022_rk, -0.281245618289472564778284183790118418111835108541217871686461460254172_rk, -2.679019236219849057687906597489223155566018523604994098255425457656539_rk, 0.518815663924157511565311164615012522024133872547687447699354391991451_rk, 1.365353187603341710683633635235238678626063069147521981032308410050892_rk, 5.885091088503946585721274891680604830711651667380554116568117966230906_rk, 2.802808786272062889819965117517532194812414567437761817710498854310830_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
87 & 0.4185367457753471441471025246471931649630462388911818955699647395240300_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 6.724547581906459363100870806514855026676707813280324750587138792853772_rk, -0.425444280164611790606983409697113064615566787707483413640215700976551_rk, 3.343279153001265577811816947557982637749506391461231978634108426874946_rk, 0.617081663117537759528421117507709784737295565374306282719063266309661_rk, -0.929966123939932833937749523988800852013651005044220965948971297443105_rk, -6.099948804751010722472962837945508844845793704880962268046183409659988_rk, -3.002206187889399044804158084895173690015280585715939212282029589883511_rk, 0.2553202529443445472336424602988558373637360743415609524071247724007435_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
88 & -0.779374086122884664644623040843840506343404534266428077223411005758477_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, -13.93734253810777678786523664804936051202744012820562193800453886770083_rk, 1.252048853379357320949735183924200895140313266258024599241902994425567_rk, -14.69150040801686878191527989293072091588299839622545945926337204097427_rk, -0.494705058533141685655191992136962873575644704089417219926481238395801_rk, 2.242974909146236657906984549543692874753553717467391893981715687718861_rk, 13.36789380382864375813864978592679139880863615933296251993509179449319_rk, 14.39665048665068644512236935340272139004715102859917581786679600071498_rk, -0.7975813331776800379127866056663258667436626421429459750810383427102171_rk, 0.4409353709534277758753793068298041158234962332723178384733350181869991_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
89 & 2.058051337466886442151242368989994043993490280696814143888221854200939_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 22.35793772796803295519317565842520212899391737358275646000873748563820_rk, 0.90949810997556332745009198137971890782877422351389174176170388069944_rk, 35.89110098240264104710550686568482456492761662151620142494472904615022_rk, -3.442515027624453437985000403608480262215499034415596153258119792895499_rk, -4.865481358036368826566013387928704014495932898502565784147427222445975_rk, -18.90980381354342625688427480879773032856963204427713361543927303960153_rk, -34.26354448030451782929251177395134170514856457225123765338258842959230_rk, 1.264756521695642578827783499806516664685830050136869435624016217846503_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk], [s, s])
90 !> The @f$\mathbf{b}@f$ matrix for the Butcher Tableau. @hideinitializer @hideinlinesource
91 real(kind=rk), parameter :: b(s,m) = reshape([ 0.01461197685842315252051541915018784713459249152088687077766479585369427_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, -0.3915211862331339089410228267288242030809705705305073638727375058683944_rk, 0.2310932500289506415909675644868993669908387496622416288355293061073997_rk, 0.1274766769992852382560589467488989175618410613311259890471852985464958_rk, 0.2246434176204157731566981937082069688984319277998004214658763800815054_rk, 0.5684352689748512932705226972873692126743494150750562298469757988708557_rk, 0.05825871557215827200814768021863420902155197903070008580216322149276118_rk, 0.1364317403482215641609022744494239843327246517433106990078145528127465_rk, 0.03057013983082797397721005067920369646664029436738543908952815210293595_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, &
92 & 0.01996996514886773085518508418098868756464331204128217116895433526690098_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 2.191499304949330054530747099310837524864428909141462483430797731597721_rk, 0.08857071848208438030833722031786358862953433866444769213869493736467726_rk, 0.1140560234865965622484956605091432032673904883224793309753983682325393_rk, 0.2533163805345107065564577734569651977347332151654734735546140625349377_rk, -2.056564386240941011158999594595981300492723931865126321129192745149218_rk, 0.3408096799013119935160094894224543812829761001917277610940297281119055_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.04834231373823958314376726739772871714901756833825340876670358204053585_rk], [s, m])
93 !> The @f$\mathbf{c}@f$ matrix for the Butcher Tableau. @hideinitializer @hideinlinesource
94 real(kind=rk), parameter :: c(s) = [ 0.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 0.03462000000000000000000000000000000000000000000000000000000000000000000_rk, 0.09702435063878044594828361677100617517630931018068052998608879362755172_rk, 0.1455365259581706689224254251565092627644639652710207949791331904413275_rk, 0.5610000000000000000000000000000000000000000000000000000000000000000000_rk, 0.2290079115904850126662751771814700052181963874976447767160913294123131_rk, 0.5449920884095149873337248228185299947818036125023552232839086705876869_rk, 0.6450000000000000000000000000000000000000000000000000000000000000000000_rk, 0.4837500000000000000000000000000000000000000000000000000000000000000000_rk, 0.06757000000000000000000000000000000000000000000000000000000000000000000_rk, 0.2500000000000000000000000000000000000000000000000000000000000000000000_rk, 0.6590650618730998549405331618649220295333706496645300841733376899052785_rk, 0.8206000000000000000000000000000000000000000000000000000000000000000000_rk, 0.9012000000000000000000000000000000000000000000000000000000000000000000_rk, 1.00000000000000000000000000000000000000000000000000000000000000000000000_rk, 1.00000000000000000000000000000000000000000000000000000000000000000000000_rk]
95 !> The method orders
96 integer, parameter :: p(m) = [9, 8]
97 !> Number of stages for each method
98 integer, parameter :: se(m) = [15, 16]
99end module mrkiss_eerk_verner_9_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_verner_9_8
Butcher tableau for Verner's 16 stage, Order (9,8) Runge-Kutta method.
Definition mrkiss_eerk_verner_9_8.f90:65
mrkiss_eerk_verner_9_8::c
real(kind=rk), dimension(s), parameter c
The matrix for the Butcher Tableau.
Definition mrkiss_eerk_verner_9_8.f90:94
mrkiss_eerk_verner_9_8::p
integer, dimension(m), parameter p
The method orders.
Definition mrkiss_eerk_verner_9_8.f90:96
mrkiss_eerk_verner_9_8::m
integer, parameter m
Number of methods.
Definition mrkiss_eerk_verner_9_8.f90:72
mrkiss_eerk_verner_9_8::se
integer, dimension(m), parameter se
Number of stages for each method.
Definition mrkiss_eerk_verner_9_8.f90:98
mrkiss_eerk_verner_9_8::b
real(kind=rk), dimension(s, m), parameter b
The matrix for the Butcher Tableau.
Definition mrkiss_eerk_verner_9_8.f90:91
mrkiss_eerk_verner_9_8::s
integer, parameter s
The order of the overall method.
Definition mrkiss_eerk_verner_9_8.f90:70
mrkiss_eerk_verner_9_8::a
real(kind=rk), dimension(s, s), parameter a
The matrix for the Butcher Tableau.
Definition mrkiss_eerk_verner_9_8.f90:74