RungeKuttaTimeIntegrationSchemes.h

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///////////////////////////////////////////////////////////////////////////////
//
// File: RungeKuttaTimeIntegrationSchemes.h
//
// For more information, please see: http://www.nektar.info
//
// The MIT License
//
// Copyright (c) 2018 Division of Applied Mathematics, Brown University (USA),
// Department of Aeronautics, Imperial College London (UK), and Scientific
// Computing and Imaging Institute, University of Utah (USA).
//
// License for the specific language governing rights and limitations under
// Permission is hereby granted, free of charge, to any person obtaining a
// copy of this software and associated documentation files (the "Software"),
// to deal in the Software without restriction, including without limitation
// the rights to use, copy, modify, merge, publish, distribute, sublicense,
// and/or sell copies of the Software, and to permit persons to whom the
// Software is furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included
// in all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
// THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
// DEALINGS IN THE SOFTWARE.
//
// Description: Combined header file for all Runge Kutta based time integration
// schemes.
//
///////////////////////////////////////////////////////////////////////////////
 
// Note : If adding a new integrator be sure to register the
// integrator with the Time Integration Scheme Facatory in
// SchemeInitializor.cpp.
 
#ifndef NEKTAR_LIB_UTILITIES_TIME_INTEGRATION_RK_TIME_INTEGRATION_SCHEME
#define NEKTAR_LIB_UTILITIES_TIME_INTEGRATION_RK_TIME_INTEGRATION_SCHEME
 
#define LUE LIB_UTILITIES_EXPORT
 
#include <LibUtilities/TimeIntegration/TimeIntegrationSchemeGLM.h>
 
namespace Nektar::LibUtilities
{
 
////////////////////////////////////////////////////////////////////////////////
// Runge Kutta Order N where the number of stages == order
 
class RungeKuttaTimeIntegrationScheme : public TimeIntegrationSchemeGLM
{
public:
    RungeKuttaTimeIntegrationScheme(std::string variant, size_t order,
                                    std::vector<NekDouble> freeParams)
        : TimeIntegrationSchemeGLM(variant, order, freeParams)
    {
        ASSERTL1(variant == "" || variant == "SSP",
                 "Runge Kutta Time integration scheme unknown variant: " +
                     variant + ". Must be blank or 'SSP'");
 
        // Std - Currently up to 5th order is implemented.
        // SSP - Currently 1st through 3rd order is implemented.
        ASSERTL1((variant == "" && 1 <= order && order <= 5) ||
                     (variant == "SSP" && 1 <= order && order <= 3),
                 "Runge Kutta Time integration scheme bad order, "
                 "Std (1-5) or SSP (1-3): " +
                     std::to_string(order));
 
        m_integration_phases    = TimeIntegrationAlgorithmGLMVector(1);
        m_integration_phases[0] = TimeIntegrationAlgorithmGLMSharedPtr(
            new TimeIntegrationAlgorithmGLM(this));
 
        RungeKuttaTimeIntegrationScheme::SetupSchemeData(
            m_integration_phases[0], variant, order, freeParams);
    }
 
    ~RungeKuttaTimeIntegrationScheme() override = default;
 
    static TimeIntegrationSchemeSharedPtr create(
        std::string variant, size_t order, std::vector<NekDouble> freeParams)
    {
        TimeIntegrationSchemeSharedPtr p =
            MemoryManager<RungeKuttaTimeIntegrationScheme>::AllocateSharedPtr(
                variant, order, freeParams);
 
        return p;
    }
 
    static std::string className;
 
    LUE static void SetupSchemeData(
        TimeIntegrationAlgorithmGLMSharedPtr &phase, std::string variant,
        size_t order, [[maybe_unused]] std::vector<NekDouble> freeParams)
    {
        constexpr size_t nStages[6] = {0, 1, 2, 3, 4, 6};
 
        // A Coefficients for the lower diagonal quadrant stored in a
        // contiguous fashion. For the fourth order, six coefficients
        // from the Butcher tableau would be stored as the following.
        //
        //                0 0 0 0
        //    Butcher     a 0 0 0   Stored as   a
        //    Tableau     b c 0 0               b c
        //                d e f 0               d e f 0 ... 0
 
        // clang-format off
        constexpr NekDouble Acoefficients[2][6][15] =
            { { {     0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 1st Order
                {     0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 2nd Order - midpoint
                {   1./2,      // Last entry
                      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 3rd Order - Ralston's
                {  1./2.,
                      0.,   3./4.,      // Last entry
                      0.,      0.,
                      0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 4th Order - Classic
                {  1./2.,
                      0.,   1./2.,
                      0.,      0.,      1.,      // Last entry
                      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 5th Order - 6 stages
                // Rabiei, Faranak, and Fudziah Ismail. "Fifth order improved
                // Runge-Kutta method for solving ordinary differential
                // equations." In Proceedings of the 11th WSEAS international
                // conference on Applied computer science, pp. 129-133. 2011.
                {  1./4.,
                   1./8.,   1./8.,
                      0.,  -1./2.,      1.,
                  3./16.,      0.,      0.,  9./16.,
                  -3./7.,   2./7.,  12./7., -12./7.,   8./7. } },
              // Strong Stability Preserving
              { {     0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 1st Order
                {     0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 2nd Order - strong scaling - improved
                {     1.,      // Last entry
                      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 3rd Order - strong scaling
                {     1.,
                      1./4.,   1./4.,      // Last entry
                      0,       0.,
                      0.,      0.,       0.,     0.,      0.,
                      0.,      0.,       0.,     0.,      0. },
                // 4th Order - Classic - not used
                {  1./2.,
                      0.,   1./2.,
                      0.,      0.,      1.,      // Last entry
                      0.,      0.,      0.,      0.,
                      0.,      0.,      0.,      0.,      0. },
                // 5th Order - 6 stages - not used
                // Rabiei, Faranak, and Fudziah Ismail. "Fifth order improved
                // Runge-Kutta method for solving ordinary differential
                // equations." In Proceedings of the 11th WSEAS international
                // conference on Applied computer science, pp. 129-133. 2011.
                {  1./4.,
                   1./8.,   1./8.,
                      0.,  -1./2.,      1.,
                  3./16.,      0.,      0.,  9./16.,
                  -3./7.,   2./7.,   12./7., -12./7.,   8./7. } } };
        // clang-format on
 
        // B Coefficients for the final summing.
 
        // clang-format off
        constexpr NekDouble Bcoefficients[2][6][6] =
            { { {    0.,       0.,    0.,       0.,      0.,      0. },
                // 1st Order
                {    1.,       0.,    0.,       0.,      0.,      0. },
                // 2nd Order - midpoint
                {    0.,       1.,     0.,      0.,      0.,      0. },
                // 3rd Order - Ralston's
                { 2./9.,    3./9.,  4./9.,      0.,      0.,      0. },
                // 4th Order - Classic
                { 1./6.,    2./6.,  2./6.,   1./6.,      0.,      0. },
                // 5th Order - 6 stages
                // Rabiei, Faranak, and Fudziah Ismail. "Fifth order improved
                // Runge-Kutta method for solving ordinary differential
                // equations." In Proceedings of the 11th WSEAS international
                // conference on Applied computer science, pp. 129-133. 2011.
                { 7./90.,      0., 32./90., 12./90., 32./90., 7./90.} },
              // Strong Stability Preserving
              { {    0.,       0.,     0.,      0.,      0.,      0. },
                // 1st Order
                {    1.,       0.,     0.,      0.,      0.,      0. },
                // 2nd Order - improved
                { 1./2.,    1./2.,     0.,      0.,      0.,      0. },
                // 3rd Order - strong scaling
                { 1./6.,    1./6.,  4./6.,      0.,      0.,      0. },
                // 4th Order - Classic - not used
                { 1./6.,    2./6.,  2./6.,   1./6.,      0.,      0. },
                // 5th Order - 6 stages - not used
                // Rabiei, Faranak, and Fudziah Ismail. "Fifth order improved
                // Runge-Kutta method for solving ordinary differential
                // equations." In Proceedings of the 11th WSEAS international
                // conference on Applied computer science, pp. 129-133. 2011.
                { 7./90.,      0., 32./90., 12./90., 32./90., 7./90. } } };
        // clang-format on
 
        size_t index = (variant == "SSP" || variant == "ImprovedEuler");
 
        phase->m_schemeType = eExplicit;
        phase->m_variant    = variant;
        phase->m_order      = order;
        phase->m_name       = std::string("RungeKutta") + phase->m_variant +
                        std::string("Order") + std::to_string(phase->m_order);
 
        phase->m_numsteps  = 1;
        phase->m_numstages = nStages[phase->m_order];
 
        phase->m_A = Array<OneD, Array<TwoD, NekDouble>>(1);
        phase->m_B = Array<OneD, Array<TwoD, NekDouble>>(1);
 
        phase->m_A[0] =
            Array<TwoD, NekDouble>(phase->m_numstages, phase->m_numstages, 0.0);
        phase->m_B[0] =
            Array<TwoD, NekDouble>(phase->m_numsteps, phase->m_numstages, 0.0);
        phase->m_U =
            Array<TwoD, NekDouble>(phase->m_numstages, phase->m_numsteps, 1.0);
        phase->m_V =
            Array<TwoD, NekDouble>(phase->m_numsteps, phase->m_numsteps, 1.0);
 
        // Coefficients
 
        // A Coefficients for each stages along the lower diagonal quadrant.
        size_t cc = 0;
 
        for (size_t s = 1; s < phase->m_numstages; ++s)
        {
            for (size_t i = 0; i < s; ++i)
            {
                phase->m_A[0][s][i] =
                    Acoefficients[index][phase->m_order][cc++];
            }
        }
 
        // B Coefficients for the finial summing.
        for (size_t n = 0; n < phase->m_numstages; ++n)
        {
            phase->m_B[0][0][n] = Bcoefficients[index][phase->m_order][n];
        }
 
        phase->m_numMultiStepValues         = 1;
        phase->m_numMultiStepImplicitDerivs = 0;
        phase->m_numMultiStepExplicitDerivs = 0;
        phase->m_timeLevelOffset    = Array<OneD, size_t>(phase->m_numsteps);
        phase->m_timeLevelOffset[0] = 0;
 
        phase->CheckAndVerify();
    }
 
protected:
    LUE std::string v_GetName() const override
    {
        return std::string("RungeKutta");
    }
 
    LUE NekDouble v_GetTimeStability() const override
    {
        if (GetOrder() == 1 || GetOrder() == 2)
        {
            return 2.0;
        }
        else if (GetOrder() == 3)
        {
            return 2.51274532661833;
        }
        else if (GetOrder() == 4)
        {
            // return 2.78529356340528;
            return 2.784;
        }
        else if (GetOrder() == 5)
        {
            return 3.21704786664011;
        }
        else
        {
            return 2.0;
        }
    }
 
}; // end class RungeKuttaTimeIntegrator
 
} // namespace Nektar::LibUtilities
 
#endif