EulerExponentialTimeIntegrationSchemes.h

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///////////////////////////////////////////////////////////////////////////////
//
// File: EulerExponentialTimeIntegrationSchemes.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
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//
// 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
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// DEALINGS IN THE SOFTWARE.
//
// Description: Combined header file for all basic Euler exponential
// 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_EULER_EXP_TIME_INTEGRATION_SCHEME
#define NEKTAR_LIB_UTILITIES_TIME_INTEGRATION_EULER_EXP_TIME_INTEGRATION_SCHEME
 
#define LUE LIB_UTILITIES_EXPORT
 
#include <LibUtilities/TimeIntegration/TimeIntegrationSchemeGLM.h>
 
#include <NekMesh/Module/ProcessModules/ProcessVarOpti/Evaluator.hxx>
 
namespace Nektar
{
 
using namespace NekMesh;
 
namespace LibUtilities
{
 
///////////////////////////////////////////////////////////////////////////////
//  EulerExponential
 
class EulerExponentialTimeIntegrationScheme : public TimeIntegrationSchemeGLM
{
public:
    EulerExponentialTimeIntegrationScheme(std::string variant, size_t order,
                                          std::vector<NekDouble> freeParams)
        : TimeIntegrationSchemeGLM(variant, order, freeParams)
    {
        // Currently up to 4th order is implemented because the number
        // of steps is the same as the order.
        ASSERTL1(variant == "Lawson" || variant == "Norsett",
                 "EulerExponential Time integration scheme unknown variant: " +
                     variant + ". Must be 'Lawson' or 'Norsett'.");
 
        ASSERTL1(((variant == "Lawson" && 1 <= order && order <= 1) ||
                  (variant == "Norsett" && 1 <= order && order <= 4)),
                 "EulerExponential Time integration scheme bad order, "
                 "Lawson (1) or Norsett (1-4): " +
                     std::to_string(order));
 
        m_integration_phases = TimeIntegrationAlgorithmGLMVector(order);
 
        // Currently the next lowest order is used to seed the current
        // order. This is not correct but is an okay approximation.
        for (size_t n = 0; n < order; ++n)
        {
            m_integration_phases[n] = TimeIntegrationAlgorithmGLMSharedPtr(
                new TimeIntegrationAlgorithmGLM(this));
 
            EulerExponentialTimeIntegrationScheme::SetupSchemeData(
                m_integration_phases[n], variant, n + 1);
        }
    }
 
    ~EulerExponentialTimeIntegrationScheme() override
    {
    }
 
    static TimeIntegrationSchemeSharedPtr create(
        std::string variant, size_t order, std::vector<NekDouble> freeParams)
    {
        TimeIntegrationSchemeSharedPtr p =
            MemoryManager<EulerExponentialTimeIntegrationScheme>::
                AllocateSharedPtr(variant, order, freeParams);
 
        return p;
    }
 
    static std::string className;
 
    LUE static void SetupSchemeData(TimeIntegrationAlgorithmGLMSharedPtr &phase,
                                    std::string variant, size_t order)
    {
        phase->m_schemeType = eExponential;
        phase->m_variant    = variant;
        phase->m_order      = order;
        phase->m_name       = variant + std::string("EulerExponentialOrder") +
                        std::to_string(phase->m_order);
 
        // Parameters for the compact 1 step implementation.
        phase->m_numstages = 1;
        phase->m_numsteps  = phase->m_order; // Okay up to 4th 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, 0.0);
        phase->m_V =
            Array<TwoD, NekDouble>(phase->m_numsteps, phase->m_numsteps, 0.0);
 
        // Coefficients
 
        // B Phi function for first row first column
        phase->m_B[0][0][0] = 1.0 / phase->m_order; // phi_func(0 or 1)
 
        // B evaluation value shuffling second row first column
        if (phase->m_order > 1)
        {
            phase->m_B[0][1][0] = 1.0; // constant 1
        }
 
        // U Curent time step evaluation first row first column
        phase->m_U[0][0] = 1.0; // constant 1
 
        // V Phi function for first row first column
        phase->m_V[0][0] = 1.0; // phi_func(0)
 
        // V Phi function for first row additional columns
        for (size_t n = 1; n < phase->m_order; ++n)
        {
            phase->m_V[0][n] = 1.0 / phase->m_order; // phi_func(n+1)
        }
 
        // V evaluation value shuffling row n column n-1
        for (size_t n = 2; n < phase->m_order; ++n)
        {
            phase->m_V[n][n - 1] = 1.0; // constant 1
        }
 
        phase->m_numMultiStepValues         = 1;
        phase->m_numMultiStepImplicitDerivs = 0;
        phase->m_numMultiStepExplicitDerivs = phase->m_order - 1;
        phase->m_timeLevelOffset    = Array<OneD, size_t>(phase->m_numsteps);
        phase->m_timeLevelOffset[0] = 0;
 
        // For order > 1 derivatives are needed.
        for (size_t n = 1; n < phase->m_order; ++n)
        {
            phase->m_timeLevelOffset[n] = n;
        }
 
        phase->CheckAndVerify();
    }
 
    LUE void SetExponentialCoefficients(
        Array<OneD, std::complex<NekDouble>> &Lambda)
    {
        ASSERTL0(!m_integration_phases.empty(), "No scheme")
 
        /**
         * \brief Lambda Matrix Assumption, parameter Lambda.
         *
         * The one-dimensional Lambda matrix is a diagonal
         * matrix thus values are non zero if and only i=j. As such,
         * the diagonal Lambda values are stored in an array of
         * complex numbers.
         */
 
        // Assume that each phase is an exponential integrator.
        for (size_t i = 0; i < m_integration_phases.size(); i++)
        {
            m_integration_phases[i]->m_L = Lambda;
 
            // Anytime the coefficents are updated reset the nVars to
            // be assured that the exponential matrices are
            // recalculated (e.g. the number of variables may remain
            // the same but the coefficients have changed).
            m_integration_phases[i]->m_lastNVars = 0;
        }
    }
 
protected:
    LUE std::string v_GetName() const override
    {
        return std::string("EulerExponential");
    }
 
    LUE std::string v_GetFullName() const override
    {
        return GetVariant() + GetName() + "Order" + std::to_string(GetOrder());
    }
 
    LUE NekDouble v_GetTimeStability() const override
    {
        return 1.0;
    }
 
    void v_InitializeSecondaryData(TimeIntegrationAlgorithmGLM *phase,
                                   NekDouble deltaT) const override
    {
        /**
         * \brief Lambda Matrix Assumption, member variable phase->m_L
         *
         * The one-dimensional Lambda matrix is a diagonal
         * matrix thus values are non zero if and only i=j. As such,
         * the diagonal Lambda values are stored in an array of
         * complex numbers.
         */
 
        ASSERTL1(phase->m_nvars == phase->m_L.size(),
                 "The number of variables does not match "
                 "the number of exponential coefficents.");
 
        phase->m_A_phi = Array<OneD, Array<TwoD, NekDouble>>(phase->m_nvars);
        phase->m_B_phi = Array<OneD, Array<TwoD, NekDouble>>(phase->m_nvars);
        phase->m_U_phi = Array<OneD, Array<TwoD, NekDouble>>(phase->m_nvars);
        phase->m_V_phi = Array<OneD, Array<TwoD, NekDouble>>(phase->m_nvars);
 
        Array<OneD, NekDouble> phi = Array<OneD, NekDouble>(phase->m_order);
 
        for (size_t k = 0; k < phase->m_nvars; ++k)
        {
            // B Phi function for first row first column
            if (phase->m_variant == "Lawson")
            {
                phi[0] = phi_function(0, deltaT * phase->m_L[k]).real();
            }
            else if (phase->m_variant == "Norsett")
            {
                phi[0] = phi_function(1, deltaT * phase->m_L[k]).real();
            }
            else
            {
                ASSERTL1(false, "Cannot call EulerExponential directly "
                                "use the variant 'Lawson' or 'Norsett'.");
            }
 
            // Set up for multiple steps. For multiple steps one needs
            // to weight the phi functions in much the same way there
            // are weights for multi-step Adams-Bashfort methods.
 
            // For order N the weights are an N x N matrix with
            // values: W[j][i] = std::pow(i, j) and phi_func = W phi.
            // There are other possible wieghting schemes.
            if (phase->m_order == 1)
            {
                // Nothing to do as the value is set above and the
                // weight is just 1.
            }
            else if (phase->m_order == 2)
            {
                // For Order 2 the weights are simply : 1 1
                //                                      0 1
 
                // If one were to solve the system of equations it
                // simply results in subtracting the second order
                // value from the first order value.
                phi[1] = phi_function(2, deltaT * phase->m_L[k]).real();
 
                phi[0] -= phi[1];
            }
            else if (phase->m_order == 3)
            {
                Array<OneD, NekDouble> phi_func =
                    Array<OneD, NekDouble>(phase->m_order);
 
                phi_func[0] = phi[0];
 
                for (size_t m = 1; m < phase->m_order; ++m)
                {
                    phi_func[m] =
                        phi_function(m + 1, deltaT * phase->m_L[k]).real();
                }
 
                NekDouble W[3][3];
 
                // Set up the wieghts and calculate the determinant.
                for (size_t j = 0; j < phase->m_order; ++j)
                {
                    for (size_t i = 0; i < phase->m_order; ++i)
                    {
                        W[j][i] = std::pow(i, j);
                    }
                }
 
                NekDouble W_det = Determinant<3>(W);
 
                // Solve the series of equations using Cramer's rule.
                for (size_t m = 0; m < phase->m_order; ++m)
                {
                    // Assemble the working matrix for this solution.
                    for (size_t j = 0; j < phase->m_order; ++j)
                    {
                        for (size_t i = 0; i < phase->m_order; ++i)
                        {
                            // Fill in the mth column for the mth
                            // solution using the phi function value
                            // otherwise utilize the weights.
                            W[i][j] = (j == m) ? phi_func[i] : std::pow(j, i);
                        }
                    }
 
                    // Get the mth solutiion.
                    phi[m] = Determinant<3>(W) / W_det;
                }
            }
            else if (phase->m_order == 4)
            {
                Array<OneD, NekDouble> phi_func =
                    Array<OneD, NekDouble>(phase->m_order);
 
                phi_func[0] = phi[0];
 
                for (size_t m = 1; m < phase->m_order; ++m)
                {
                    phi_func[m] =
                        phi_function(m + 1, deltaT * phase->m_L[k]).real();
                }
 
                NekDouble W[4][4];
 
                // Set up the weights and calculate the determinant.
                for (size_t j = 0; j < phase->m_order; ++j)
                {
                    for (size_t i = 0; i < phase->m_order; ++i)
                    {
                        W[j][i] = std::pow(i, j);
                    }
                }
 
                NekDouble W_det = Determinant<4>(W);
 
                // Solve the series of equations using Cramer's rule.
                for (size_t m = 0; m < phase->m_order; ++m)
                {
                    // Assemble the working matrix for this solution.
                    for (size_t j = 0; j < phase->m_order; ++j)
                    {
                        for (size_t i = 0; i < phase->m_order; ++i)
                        {
                            // Fill in the mth column for the mth
                            // solution using the phi function value
                            // otherwise utilize the weights.
                            W[i][j] = (j == m) ? phi_func[i] : std::pow(j, i);
                        }
                    }
 
                    // Get the mth solutiion.
                    phi[m] = Determinant<4>(W) / W_det;
                }
            }
            else
            {
                ASSERTL1(false, "Not set up for more than 4th Order.");
            }
 
            // Create the phi based Butcher tableau matrices. Note
            // these matrices are set up using a general formational
            // based on the number of steps (i.e. the order).
            phase->m_A_phi[k] = Array<TwoD, NekDouble>(phase->m_numstages,
                                                       phase->m_numstages, 0.0);
            phase->m_B_phi[k] = Array<TwoD, NekDouble>(phase->m_numsteps,
                                                       phase->m_numstages, 0.0);
            phase->m_U_phi[k] = Array<TwoD, NekDouble>(phase->m_numstages,
                                                       phase->m_numsteps, 0.0);
            phase->m_V_phi[k] = Array<TwoD, NekDouble>(phase->m_numsteps,
                                                       phase->m_numsteps, 0.0);
 
            // B Phi function for first row first column.
            phase->m_B_phi[k][0][0] = phi[0];
 
            // B evaluation value shuffling second row first column.
            if (phase->m_order > 1)
            {
                phase->m_B_phi[k][1][0] = 1.0; // constant 1
            }
 
            // U Curent time step evaluation first row first column.
            phase->m_U_phi[k][0][0] = 1.0; // constant 1
 
            // V Phi function for first row first column.
            phase->m_V_phi[k][0][0] =
                phi_function(0, deltaT * phase->m_L[k]).real();
 
            // V Phi function for first row additional columns.
            for (size_t n = 1; n < phase->m_order; ++n)
            {
                phase->m_V_phi[k][0][n] = phi[n];
            }
 
            // V evaluation value shuffling row n column n-1.
            for (size_t n = 2; n < phase->m_order; ++n)
            {
                phase->m_V_phi[k][n][n - 1] = 1.0; // constant 1
            }
        }
    }
 
private:
    inline NekDouble factorial(size_t n) const
    {
        return (n == 1 || n == 0) ? 1 : n * factorial(n - 1);
    }
 
    std::complex<NekDouble> phi_function(const size_t order,
                                         const std::complex<NekDouble> z) const
    {
        /**
         * \brief Phi function
         *
         * Central to the implementation of exponential integrators is
         * the evaluation of exponential-like functions, commonly
         * denoted by phi (φ) functions. It is convenient to define
         * then as φ0(z) = e^z, in which case the functions obey the
         * recurrence relation.
 
         * 0:  exp(z);
         * 1: (exp(z)     - 1.0) / (z);
         * 2: (exp(z) - z - 1.0) / (z * z);
         */
 
        if (z == 0.0)
        {
            return 1.0 / factorial(order);
        }
 
        if (order == 0)
        {
            return exp(z);
        }
        else
        {
            return (phi_function(order - 1, z) - 1.0 / factorial(order - 1)) /
                   z;
        }
    }
 
}; // end class EulerExponentialTimeIntegrator
 
} // end namespace LibUtilities
} // end namespace Nektar
 
#endif