mrkiss_eerk_fehlberg_4_5.f90

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! -*- Mode:F90; Coding:us-ascii-unix; fill-column:129 -*-
!.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.H.S.!!
!>
!! @file      mrkiss_eerk_fehlberg_4_5.f90
!! @author    Mitch Richling http://www.mitchr.me/
!! @brief     Butcher tableau for Fehlberg's 6 stage, Order (4,5) Runge-Kutta method.@EOL
!! @keywords  ode ivp differential equation initial value problem rk
!! @std       F2023
!! @see       https://github.com/richmit/MRKISS
!! @copyright
!!  @parblock
!!  Copyright (c) 2025, Mitchell Jay Richling <http://www.mitchr.me/> All rights reserved.
!!
!!  Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following
!!  conditions are met:
!!
!!  1. Redistributions of source code must retain the above copyright notice, this list of conditions, and the following
!!     disclaimer.
!!
!!  2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions, and the following
!!     disclaimer in the documentation and/or other materials provided with the distribution.
!!
!!  3. Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products
!!     derived from this software without specific prior written permission.
!!
!!  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES,
!!  INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
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!!  @endparblock
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!----------------------------------------------------------------------------------------------------------------------------------
!> Butcher tableau for Fehlberg's 6 stage, Order (4,5) Runge-Kutta method
!!
!! @image html eerk_fehlberg_4_5-stab.png
!!
!! @par 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.
!!
!! @par Known Aliases
!! 'RKF78'', 'ode78' (OrdinaryDiffEq.jl), 'ARKODE_FEHLBERG_6_4_5' (SUNDIALS).
!!
!! @par Stability Image Links
!! <a href="eerk_fehlberg_4_5-stab.png">  <img src="eerk_fehlberg_4_5-stab.png"  width="256px"> </a>
!! <a href="eerk_fehlberg_4_5-astab.png"> <img src="eerk_fehlberg_4_5-astab.png" width="256px"> </a>
!! <a href="eerk_fehlberg_4_5-star1.png"> <img src="eerk_fehlberg_4_5-star1.png" width="256px"> </a>
!! <a href="eerk_fehlberg_4_5-star2.png"> <img src="eerk_fehlberg_4_5-star2.png" width="256px"> </a>
!!
!! @par 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
!!
module mrkiss_eerk_fehlberg_4_5
  use mrkiss_config, only: rk
  implicit none
  public
  !> The order of the overall method
  integer,          parameter :: s      = 6
  !> Number of methods
  integer,          parameter :: m      = 2
  !> The @f$\mathbf{a}@f$ matrix for the Butcher Tableau. @hideinitializer @showinlinesource
  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
  !> The @f$\mathbf{b}@f$ matrix for the Butcher Tableau. @hideinitializer @showinlinesource
  real(kind=rk),    parameter :: b(s,m) = reshape([    130625.0_rk,          0.0_rk,      619520.0_rk,   604175.0_rk,    -225720.0_rk,       0.0_rk, &
       &                                               133760.0_rk,          0.0_rk,      585728.0_rk,   571220.0_rk,    -203148.0_rk,   41040.0_rk], [s, m]) /     1128600.0_rk
  !> The @f$\mathbf{c}@f$ matrix for the Butcher Tableau. @hideinitializer @showinlinesource
  real(kind=rk),    parameter :: c(s)   = [                 0.0_rk,          26.0_rk,         39.0_rk,        96.0_rk,       104.0_rk,       52.0_rk ]          /       104.0_rk
  !> The method orders
  integer,          parameter :: p(m)   = [ 4, 5 ]
  !> Number of stages for each method
  integer,          parameter :: se(m)  = [ 5, 6 ]
end module mrkiss_eerk_fehlberg_4_5