mrkiss_erk_ralston_4.f90

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! -*- Mode:F90; Coding:us-ascii-unix; fill-column:129 -*-
!.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.H.S.!!
!>
!! @file      mrkiss_erk_ralston_4.f90
!! @author    Mitch Richling http://www.mitchr.me/
!! @brief     Butcher tableau for ralston's 4 stage order (4) 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 ralston's 4 stage order (4) Runge-Kutta method
!!
!! @image html erk_ralston_4-stab.png
!!
!! @par IMO
!! This method was designed to minimize truncation error over every other consideration includeing stability, roundoff
!! error, performance, and storage.
!!
!! @par Stability Image Links
!! <a href="erk_ralston_4-stab.png">  <img src="erk_ralston_4-stab.png"  width="256px"> </a>
!! <a href="erk_ralston_4-astab.png"> <img src="erk_ralston_4-astab.png" width="256px"> </a>
!! <a href="erk_ralston_4-star1.png"> <img src="erk_ralston_4-star1.png" width="256px"> </a>
!! <a href="erk_ralston_4-star2.png"> <img src="erk_ralston_4-star2.png" width="256px"> </a>
!!
!! @par References:
!!  - Ralston (1962); Runge-Kutta Methods with Minimum Error Bounds; Math. of Comp. 16 (80); p431-437; 
!!    zotero://select/items/0_2FK7F4J3
!!
module mrkiss_erk_ralston_4
  use mrkiss_config, only: rk
  implicit none
  public
  !> The order of the overall method
  integer,          parameter :: s      = 4
  !> Number of methods
  integer,          parameter :: m      = 1
  !> 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,                             &
       &                                             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)
  !> The @f$\mathbf{b}@f$ matrix for the Butcher Tableau. @hideinitializer @showinlinesource                                         
  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)
  !> The @f$\mathbf{c}@f$ matrix for the Butcher Tableau. @hideinitializer @showinlinesource
  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)
  !> The method orders
  integer,          parameter :: p(m)   = [4]
  !> Number of stages for each method
  integer,          parameter :: se(m)  = [4]
end module mrkiss_erk_ralston_4