mrkiss_erk_kutta_three_eight_4.f90

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!>
!! @file      mrkiss_erk_kutta_three_eight_4.f90
!! @author    Mitch Richling http://www.mitchr.me/
!! @brief     Butcher tableau for the three eights rule.  A 4 step, 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
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!!
!!  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES,
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!----------------------------------------------------------------------------------------------------------------------------------
!> Butcher tableau for the three eights rule.  A 4 step, order (4) Runge-Kutta method
!!
!! @image html erk_kutta_three_eight_4-stab.png
!!
!! @par IMO
!! Included for historical reasons.  More precise than RK4.
!!
!! @par Known Aliases
!! '3/8 rule', 'RK42' (Butcher), 'GIRK438' (OrdinaryDiffEq.jl).
!!
!! @par Stability Image Links
!! <a href="erk_kutta_three_eight_4-stab.png">  <img src="erk_kutta_three_eight_4-stab.png"  width="256px"> </a>
!! <a href="erk_kutta_three_eight_4-astab.png"> <img src="erk_kutta_three_eight_4-astab.png" width="256px"> </a>
!! <a href="erk_kutta_three_eight_4-star1.png"> <img src="erk_kutta_three_eight_4-star1.png" width="256px"> </a>
!! <a href="erk_kutta_three_eight_4-star2.png"> <img src="erk_kutta_three_eight_4-star2.png" width="256px"> </a>
!!
!! @par References:
!!  - Kutta (1901), Beitrag zur n\"herungsweisen Integration totaler Differentialgleichungen; Z. Math. Phys. 46; p435-453.
!!  - Hairer, Norsett & Wanner (2009). Solving Ordinary Differential Equations. I: Nonstiff Problems. p138; 
!!    zotero://select/items/0_VLZWN2CT
!!
module mrkiss_erk_kutta_three_eight_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, &
                                                    1.0_rk,  0.0_rk, 0.0_rk, 0.0_rk, &
                                                   -1.0_rk,  3.0_rk, 0.0_rk, 0.0_rk, &
                                                    3.0_rk, -3.0_rk, 3.0_rk, 0.0_rk], [s, s]) / 3.0_rk
  !> The @f$\mathbf{b}@f$ matrix for the Butcher Tableau. @hideinitializer @showinlinesource
  real(kind=rk),    parameter :: b(s,m) = reshape([ 1.0_rk,  3.0_rk, 3.0_rk, 1.0_rk], [s, m]) / 8.0_rk
  !> The @f$\mathbf{c}@f$ matrix for the Butcher Tableau. @hideinitializer @showinlinesource
  real(kind=rk),    parameter :: c(s)   = [         0.0_rk,  1.0_rk, 2.0_rk, 3.0_rk]          / 3.0_rk
  !> The method orders
  integer,          parameter :: p(m)   = [4]
  !> Number of stages for each method
  integer,          parameter :: se(m)  = [4]
end module mrkiss_erk_kutta_three_eight_4