///
/// This file is part of Rheolef.
///
/// Copyright (C) 2000-2009 Pierre Saramito <Pierre.Saramito@imag.fr>
///
/// Rheolef is free software; you can redistribute it and/or modify
/// it under the terms of the GNU General Public License as published by
/// the Free Software Foundation; either version 2 of the License, or
/// (at your option) any later version.
///
/// Rheolef is distributed in the hope that it will be useful,
/// but WITHOUT ANY WARRANTY; without even the implied warranty of
/// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
/// GNU General Public License for more details.
///
/// You should have received a copy of the GNU General Public License
/// along with Rheolef; if not, write to the Free Software
/// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
/// 
/// =========================================================================
//! @examplefile runge_kutta_semiimplicit.icc The semi-implicit Runge-Kutta scheme -- coefficients
namespace rk {
  // see WanShuZha-2015 page 212
  constexpr size_t pmax = 3;
  const Float _g2 = 1 - 1/sqrt(Float(2));
  const Float _d2 = 1 - 1/(2*_g2);
  const Float _g3 = 0.435866521508459;
  const Float _b31 = -3*(_g3*_g3)/2 + 4*_g3 - 1./4;
  const Float _b32 =  3*(_g3*_g3)/2 - 5*_g3 + 5./4;
  const Float _a31 = -0.356;
  const Float _a32 = (1/Float(3) - 2*(_g3*_g3) - 2*_b32*_a31*_g3)/(_g3*(1-_g3));
  Float alpha[][pmax+1][pmax+1]
                  = { {},
                      { {0},
                        {0, 1} },
                      { {0},
                        {0, _g2},
                        {0, 1-_g2, _g2} },
                      { {0},
                        {0, _g3},
                        {0, (1-_g3)/2, _g3},
                        {0, _b31,      _b32, _g3} } };

  Float tilde_alpha[][pmax+1][pmax+1]
                  = { {},
                      { {},
                        {1} },
                      { {},
                        {_g2},
                        {_d2, 1-_d2} },
                      { {},
                        {_g3},
                        {(1+_g3)/2 - _a31, _a31},
                        {0,                1-_a32, _a32} } };

  Float beta [][pmax+1]
                  = { {},
                      {0, 1},
                      {0, 1-_g2, _g2},
                      {0, _b31,  _b32, _g3} };
  Float tilde_beta [][pmax+1]
                  = { {},
                      {1},
                      {_d2, 1-_d2, 0},
                      {0,   _b31,  _b32, _g3} };
  Float gamma [][pmax+1]
                  = { {},
                      {0, 1},
                      {0, _g2, 1},
                      {0, _g3, (1+_g3)/2, 1} };
} // namespace rk
