CompressibleFlowSystemImplicit.cpp

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///////////////////////////////////////////////////////////////////////////////
//
// File: CompressibleFlowSystemImplicit.cpp
//
// For more information, please see: http://www.nektar.info
//
// The MIT License
//
// Copyright (c) 2006 Division of Applied Mathematics, Brown University (USA),
// Department of Aeronautics, Imperial College London (UK), and Scientific
// Computing and Imaging Institute, University of Utah (USA).
//
// 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: Compressible flow system base class with auxiliary functions
//
///////////////////////////////////////////////////////////////////////////////
 
#include <boost/core/ignore_unused.hpp>
 
#include <CompressibleFlowSolver/EquationSystems/CompressibleFlowSystemImplicit.h>
#include <SolverUtils/Advection/AdvectionWeakDG.h>
 
#include <LibUtilities/BasicUtils/Timer.h>
 
using namespace std;
 
namespace Nektar
{
CFSImplicit::CFSImplicit(const LibUtilities::SessionReaderSharedPtr &pSession,
                         const SpatialDomains::MeshGraphSharedPtr &pGraph)
    : UnsteadySystem(pSession, pGraph), CompressibleFlowSystem(pSession, pGraph)
{
}
 
/**
 * @brief Initialization object for CFSImplicit class.
 */
void CFSImplicit::v_InitObject(bool DeclareFields)
{
    CompressibleFlowSystem::v_InitObject(DeclareFields);
    m_explicitAdvection = false;
    m_explicitDiffusion = false;
 
    // initialise implicit parameters
    m_session->MatchSolverInfo("FLAGIMPLICITITSSTATISTICS", "True",
                               m_flagImplicitItsStatistics, false);
 
    m_session->LoadParameter("JacobiFreeEps", m_jacobiFreeEps, 5.0E-8);
 
    int ntmp;
    m_session->LoadParameter("AdvectionJacFlag", ntmp, 1);
    m_advectionJacFlag = (0 == ntmp) ? false : true;
 
    m_session->LoadParameter("ViscousJacFlag", ntmp, 1);
    m_viscousJacFlag = (0 == ntmp) ? false : true;
 
    // initialise implicit functors
    m_ode.DefineImplicitSolve(&CFSImplicit::DoImplicitSolve, this);
 
    InitialiseNonlinSysSolver();
}
 
void CFSImplicit::InitialiseNonlinSysSolver()
{
    std::string SolverType = "Newton";
    if (m_session->DefinesSolverInfo("NonlinSysIterSolver"))
    {
        SolverType = m_session->GetSolverInfo("NonlinSysIterSolver");
    }
    ASSERTL0(LibUtilities::GetNekNonlinSysFactory().ModuleExists(SolverType),
             "NekNonlinSys '" + SolverType + "' is not defined.\n");
    int ntotal = m_fields[0]->GetNcoeffs() * m_fields.size();
 
    LibUtilities::NekSysKey key = LibUtilities::NekSysKey();
 
    key.m_NonlinIterTolRelativeL2   = 1.0E-3;
    key.m_LinSysRelativeTolInNonlin = 5.0E-2;
    key.m_NekNonlinSysMaxIterations = 10;
    key.m_NekLinSysMaxIterations    = 30;
    key.m_LinSysMaxStorage          = 30;
 
    m_nonlinsol = LibUtilities::GetNekNonlinSysFactory().CreateInstance(
        SolverType, m_session, m_comm->GetRowComm(), ntotal, key);
 
    LibUtilities::NekSysOperators nekSysOp;
    nekSysOp.DefineNekSysResEval(&CFSImplicit::NonlinSysEvaluatorCoeff1D, this);
    nekSysOp.DefineNekSysLhsEval(&CFSImplicit::MatrixMultiplyMatrixFreeCoeff,
                                 this);
    nekSysOp.DefineNekSysPrecon(&CFSImplicit::PreconCoeff, this);
 
    int nvariables = m_fields.size();
    const MultiRegions::LocTraceToTraceMapSharedPtr locTraceToTraceMap =
        m_fields[0]->GetLocTraceToTraceMap();
 
    locTraceToTraceMap->CalcLocTracePhysToTraceIDMap(m_fields[0]->GetTrace(),
                                                     m_spacedim);
    for (int i = 1; i < nvariables; i++)
    {
        m_fields[i]->GetLocTraceToTraceMap()->SetLocTracePhysToTraceIDMap(
            locTraceToTraceMap->GetLocTracephysToTraceIDMap());
    }
 
    m_preconCfs = GetPreconCfsFactory().CreateInstance("PreconCfsBRJ", m_fields,
                                                       m_session, m_comm);
    NekPreconCfsOperators tmpPreconOp;
    tmpPreconOp.DefineCalcPreconMatBRJCoeff(&CFSImplicit::CalcPreconMatBRJCoeff,
                                            this);
    m_preconCfs->SetOperators(tmpPreconOp);
 
    m_session->LoadParameter("NewtonAbsoluteIteTol", m_newtonAbsoluteIteTol,
                             1.0E-12);
 
    m_session->LoadParameter("nPadding", m_nPadding, 4);
 
    m_nonlinsol->SetSysOperators(nekSysOp);
}
 
/**
 * @brief Destructor for CFSImplicit class.
 */
CFSImplicit::~CFSImplicit()
{
}
 
void CFSImplicit::v_DoSolve()
{
    m_TotNewtonIts = 0;
    m_TotLinIts    = 0;
    m_TotImpStages = 0;
 
    UnsteadySystem::v_DoSolve();
}
 
void CFSImplicit::v_PrintStatusInformation(const int step,
                                           const NekDouble cpuTime)
{
    UnsteadySystem::v_PrintStatusInformation(step, cpuTime);
 
    if (m_infosteps && m_session->GetComm()->GetSpaceComm()->GetRank() == 0 &&
        !((step + 1) % m_infosteps))
    {
        if (m_flagImplicitItsStatistics)
        {
            cout << "       &&"
                 << " TotImpStages= " << m_TotImpStages
                 << " TotNewtonIts= " << m_TotNewtonIts
                 << " TotLinearIts = " << m_TotLinIts << endl;
        }
    }
}
 
void CFSImplicit::v_PrintSummaryStatistics(const NekDouble intTime)
{
    UnsteadySystem::v_PrintSummaryStatistics(intTime);
 
    if (m_session->GetComm()->GetRank() == 0)
    {
        if (m_flagImplicitItsStatistics)
        {
            cout << "-------------------------------------------" << endl
                 << "Total Implicit Stages: " << m_TotImpStages << endl
                 << "Total Newton Its     : " << m_TotNewtonIts << endl
                 << "Total Linear Its     : " << m_TotLinIts << endl
                 << "-------------------------------------------" << endl;
        }
    }
}
 
void CFSImplicit::NonlinSysEvaluatorCoeff1D(
    const Array<OneD, const NekDouble> &inarray, Array<OneD, NekDouble> &out,
    const bool &flag)
{
    const Array<OneD, const NekDouble> refsource =
        m_nonlinsol->GetRefSourceVec();
    NonlinSysEvaluatorCoeff(inarray, out, flag, refsource);
}
 
void CFSImplicit::NonlinSysEvaluatorCoeff(
    const Array<OneD, const NekDouble> &inarray, Array<OneD, NekDouble> &out,
    const bool &flag, const Array<OneD, const NekDouble> &source)
{
    LibUtilities::Timer timer;
    boost::ignore_unused(flag);
    unsigned int nvariables = m_fields.size();
    unsigned int npoints    = m_fields[0]->GetNcoeffs();
    Array<OneD, Array<OneD, NekDouble>> in2D(nvariables);
    Array<OneD, Array<OneD, NekDouble>> out2D(nvariables);
    Array<OneD, Array<OneD, NekDouble>> source2D(nvariables);
    for (int i = 0; i < nvariables; ++i)
    {
        int offset  = i * npoints;
        in2D[i]     = inarray + offset;
        out2D[i]    = out + offset;
        source2D[i] = source + offset;
    }
 
    timer.Start();
    NonlinSysEvaluatorCoeff(in2D, out2D, source2D);
    timer.Stop();
    timer.AccumulateRegion("CFSImplicit::NonlinSysEvaluatorCoeff");
}
 
void CFSImplicit::NonlinSysEvaluatorCoeff(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &out,
    const Array<OneD, const Array<OneD, NekDouble>> &source)
{
    LibUtilities::Timer timer;
    unsigned int nvariable = inarray.size();
    unsigned int ncoeffs   = inarray[nvariable - 1].size();
    unsigned int npoints   = m_fields[0]->GetNpoints();
 
    Array<OneD, Array<OneD, NekDouble>> inpnts(nvariable);
 
    for (int i = 0; i < nvariable; ++i)
    {
        inpnts[i] = Array<OneD, NekDouble>(npoints, 0.0);
        m_fields[i]->BwdTrans(inarray[i], inpnts[i]);
    }
 
    timer.Start();
    DoOdeProjection(inpnts, inpnts, m_bndEvaluateTime);
    timer.Stop();
    timer.AccumulateRegion("CompressibleFlowSystem::DoOdeProjection", 1);
 
    timer.Start();
    DoOdeRhsCoeff(inpnts, out, m_bndEvaluateTime);
    timer.Stop();
    timer.AccumulateRegion("CFSImplicit::DoOdeRhsCoeff", 1);
 
    for (int i = 0; i < nvariable; ++i)
    {
        Vmath::Svtvp(ncoeffs, -m_TimeIntegLambda, out[i], 1, inarray[i], 1,
                     out[i], 1);
    }
 
    if (NullNekDoubleArrayOfArray != source)
    {
        for (int i = 0; i < nvariable; ++i)
        {
            Vmath::Vsub(ncoeffs, out[i], 1, source[i], 1, out[i], 1);
        }
    }
    return;
}
 
/**
 * @brief Compute the right-hand side.
 */
void CFSImplicit::DoOdeRhsCoeff(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray, const NekDouble time)
{
    LibUtilities::Timer timer;
 
    int nvariables = inarray.size();
    int nTracePts  = GetTraceTotPoints();
    int ncoeffs    = GetNcoeffs();
    // Store forwards/backwards space along trace space
    Array<OneD, Array<OneD, NekDouble>> Fwd(nvariables);
    Array<OneD, Array<OneD, NekDouble>> Bwd(nvariables);
 
    if (m_HomogeneousType == eHomogeneous1D)
    {
        Fwd = NullNekDoubleArrayOfArray;
        Bwd = NullNekDoubleArrayOfArray;
    }
    else
    {
        for (int i = 0; i < nvariables; ++i)
        {
            Fwd[i] = Array<OneD, NekDouble>(nTracePts, 0.0);
            Bwd[i] = Array<OneD, NekDouble>(nTracePts, 0.0);
            m_fields[i]->GetFwdBwdTracePhys(inarray[i], Fwd[i], Bwd[i]);
        }
    }
 
    // Calculate advection
    timer.Start();
    DoAdvectionCoeff(inarray, outarray, time, Fwd, Bwd);
    timer.Stop();
    timer.AccumulateRegion("CFSImplicit::DoAdvectionCoeff", 2);
 
    // Negate results
    for (int i = 0; i < nvariables; ++i)
    {
        Vmath::Neg(ncoeffs, outarray[i], 1);
    }
 
    timer.Start();
    DoDiffusionCoeff(inarray, outarray, Fwd, Bwd);
    timer.Stop();
    timer.AccumulateRegion("CFSImplicit::DoDiffusionCoeff", 2);
 
    // Add forcing terms
    for (auto &x : m_forcing)
    {
        x->ApplyCoeff(m_fields, inarray, outarray, time);
    }
 
    if (m_useLocalTimeStep)
    {
        int nElements = m_fields[0]->GetExpSize();
        int nq, offset;
        NekDouble fac;
        Array<OneD, NekDouble> tmp;
 
        Array<OneD, NekDouble> tstep(nElements, 0.0);
        GetElmtTimeStep(inarray, tstep);
 
        // Loop over elements
        for (int n = 0; n < nElements; ++n)
        {
            nq     = m_fields[0]->GetExp(n)->GetNcoeffs();
            offset = m_fields[0]->GetCoeff_Offset(n);
            fac    = tstep[n] / m_timestep;
            for (int i = 0; i < nvariables; ++i)
            {
                Vmath::Smul(nq, fac, outarray[i] + offset, 1,
                            tmp = outarray[i] + offset, 1);
            }
        }
    }
}
 
/**
 * @brief Compute the advection terms for the right-hand side
 */
void CFSImplicit::DoAdvectionCoeff(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray, const NekDouble time,
    const Array<OneD, const Array<OneD, NekDouble>> &pFwd,
    const Array<OneD, const Array<OneD, NekDouble>> &pBwd)
{
    int nvariables = inarray.size();
    Array<OneD, Array<OneD, NekDouble>> advVel(m_spacedim);
 
    auto advWeakDGObject =
        std::dynamic_pointer_cast<SolverUtils::AdvectionWeakDG>(m_advObject);
    ASSERTL0(advWeakDGObject,
             "Use WeakDG for implicit compressible flow solver!");
    advWeakDGObject->AdvectCoeffs(nvariables, m_fields, advVel, inarray,
                                  outarray, time, pFwd, pBwd);
}
 
void CFSImplicit::DoImplicitSolve(
    const Array<OneD, const Array<OneD, NekDouble>> &inpnts,
    Array<OneD, Array<OneD, NekDouble>> &outpnt, const NekDouble time,
    const NekDouble lambda)
{
    unsigned int nvariables = inpnts.size();
    unsigned int ncoeffs    = m_fields[0]->GetNcoeffs();
    unsigned int ntotal     = nvariables * ncoeffs;
 
    Array<OneD, NekDouble> inarray(ntotal);
    Array<OneD, NekDouble> out(ntotal);
    Array<OneD, NekDouble> tmpArray;
 
    // Switch flag to make sure the physical shock capturing AV is updated
    m_updateShockCaptPhys = true;
 
    for (int i = 0; i < nvariables; ++i)
    {
        int noffset = i * ncoeffs;
        tmpArray    = inarray + noffset;
        m_fields[i]->FwdTrans(inpnts[i], tmpArray);
    }
 
    DoImplicitSolveCoeff(inpnts, inarray, out, time, lambda);
 
    for (int i = 0; i < nvariables; ++i)
    {
        int noffset = i * ncoeffs;
        tmpArray    = out + noffset;
        m_fields[i]->BwdTrans(tmpArray, outpnt[i]);
    }
}
 
void CFSImplicit::DoImplicitSolveCoeff(
    const Array<OneD, const Array<OneD, NekDouble>> &inpnts,
    const Array<OneD, const NekDouble> &inarray, Array<OneD, NekDouble> &out,
    const NekDouble time, const NekDouble lambda)
{
    boost::ignore_unused(inpnts);
 
    m_TimeIntegLambda   = lambda;
    m_bndEvaluateTime   = time;
    m_solutionPhys      = inpnts;
    unsigned int ntotal = inarray.size();
 
    if (m_inArrayNorm < 0.0)
    {
        CalcRefValues(inarray);
    }
 
    NekDouble tol = std::sqrt(m_inArrayNorm) * m_newtonAbsoluteIteTol;
 
    m_nonlinsol->SetupNekNonlinSystem(ntotal, inarray, inarray, 0);
 
    m_TotNewtonIts += m_nonlinsol->SolveSystem(ntotal, inarray, out, 0, tol);
 
    m_TotLinIts += m_nonlinsol->GetNtotLinSysIts();
 
    m_TotImpStages++;
}
 
void CFSImplicit::CalcRefValues(const Array<OneD, const NekDouble> &inarray)
{
    unsigned int nvariables = m_fields.size();
    unsigned int ntotal     = inarray.size();
    unsigned int npoints    = ntotal / nvariables;
 
    unsigned int nTotalGlobal = ntotal;
    m_comm->GetSpaceComm()->AllReduce(nTotalGlobal,
                                      Nektar::LibUtilities::ReduceSum);
    unsigned int nTotalDOF = nTotalGlobal / nvariables;
    NekDouble invTotalDOF  = 1.0 / nTotalDOF;
 
    m_inArrayNorm    = 0.0;
    m_magnitdEstimat = Array<OneD, NekDouble>(nvariables, 0.0);
 
    for (int i = 0; i < nvariables; ++i)
    {
        int offset = i * npoints;
        m_magnitdEstimat[i] =
            Vmath::Dot(npoints, inarray + offset, inarray + offset);
    }
    m_comm->GetSpaceComm()->AllReduce(m_magnitdEstimat,
                                      Nektar::LibUtilities::ReduceSum);
 
    for (int i = 0; i < nvariables; ++i)
    {
        m_inArrayNorm += m_magnitdEstimat[i];
    }
 
    for (int i = 2; i < nvariables - 1; ++i)
    {
        m_magnitdEstimat[1] += m_magnitdEstimat[i];
    }
    for (int i = 2; i < nvariables - 1; ++i)
    {
        m_magnitdEstimat[i] = m_magnitdEstimat[1];
    }
 
    for (int i = 0; i < nvariables; ++i)
    {
        m_magnitdEstimat[i] = sqrt(m_magnitdEstimat[i] * invTotalDOF);
    }
    if (m_comm->GetRank() == 0 && m_verbose)
    {
        for (int i = 0; i < nvariables; ++i)
        {
            cout << "m_magnitdEstimat[" << i << "]    = " << m_magnitdEstimat[i]
                 << endl;
        }
        cout << "m_inArrayNorm    = " << m_inArrayNorm << endl;
    }
}
 
void CFSImplicit::PreconCoeff(const Array<OneD, NekDouble> &inarray,
                              Array<OneD, NekDouble> &outarray,
                              const bool &flag)
{
    LibUtilities::Timer timer, Gtimer;
 
    Gtimer.Start();
    if (m_preconCfs->UpdatePreconMatCheck(NullNekDouble1DArray,
                                          m_TimeIntegLambda))
    {
        int nvariables = m_solutionPhys.size();
 
        int nphspnt = m_solutionPhys[nvariables - 1].size();
        Array<OneD, Array<OneD, NekDouble>> intmp(nvariables);
        for (int i = 0; i < nvariables; i++)
        {
            intmp[i] = Array<OneD, NekDouble>(nphspnt, 0.0);
        }
 
        timer.Start();
        DoOdeProjection(m_solutionPhys, intmp, m_bndEvaluateTime);
        timer.Stop();
        timer.AccumulateRegion("CompressibleFlowSystem::DoOdeProjection", 1);
 
        timer.Start();
        m_preconCfs->BuildPreconCfs(m_fields, intmp, m_bndEvaluateTime,
                                    m_TimeIntegLambda);
        timer.Stop();
        timer.AccumulateRegion("PreconCfsOp::BuildPreconCfs", 1);
    }
 
    timer.Start();
    m_preconCfs->DoPreconCfs(m_fields, inarray, outarray, flag);
    timer.Stop();
    timer.AccumulateRegion("PreconCfsOp::DoPreconCfs", 1);
 
    Gtimer.Stop();
    Gtimer.AccumulateRegion("CFSImplicit::PreconCoeff");
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::AddMatNSBlkDiagVol(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    const Array<OneD, const TensorOfArray2D<NekDouble>> &qfield,
    Array<OneD, Array<OneD, TypeNekBlkMatSharedPtr>> &gmtxarray,
    TensorOfArray4D<DataType> &StdMatDataDBB,
    TensorOfArray5D<DataType> &StdMatDataDBDB)
{
    if (StdMatDataDBB.size() == 0)
    {
        CalcVolJacStdMat(StdMatDataDBB, StdMatDataDBDB);
    }
 
    int nSpaceDim = m_graph->GetSpaceDimension();
    int nvariable = inarray.size();
    int npoints   = m_fields[0]->GetTotPoints();
    int nVar2     = nvariable * nvariable;
    std::shared_ptr<LocalRegions::ExpansionVector> expvect =
        m_fields[0]->GetExp();
    int nTotElmt = (*expvect).size();
 
    Array<OneD, NekDouble> mu(npoints, 0.0);
    Array<OneD, NekDouble> DmuDT(npoints, 0.0);
    if (m_viscousJacFlag)
    {
        CalcMuDmuDT(inarray, mu, DmuDT);
    }
 
    Array<OneD, NekDouble> normals;
    Array<OneD, Array<OneD, NekDouble>> normal3D(3);
    for (int i = 0; i < 3; i++)
    {
        normal3D[i] = Array<OneD, NekDouble>(3, 0.0);
    }
    normal3D[0][0] = 1.0;
    normal3D[1][1] = 1.0;
    normal3D[2][2] = 1.0;
    Array<OneD, Array<OneD, NekDouble>> normalPnt(3);
 
    DNekMatSharedPtr wspMat =
        MemoryManager<DNekMat>::AllocateSharedPtr(nvariable, nvariable, 0.0);
    DNekMatSharedPtr wspMatDrv = MemoryManager<DNekMat>::AllocateSharedPtr(
        nvariable - 1, nvariable, 0.0);
 
    Array<OneD, DataType> GmatxData;
    Array<OneD, DataType> MatData;
 
    Array<OneD, NekDouble> tmppnts;
    TensorOfArray3D<NekDouble> PntJacCons(
        m_spacedim); // Nvar*Nvar*Ndir*Nelmt*Npnt
    TensorOfArray3D<DataType> PntJacConsStd(
        m_spacedim); // Nvar*Nvar*Ndir*Nelmt*Npnt
    Array<OneD, Array<OneD, NekDouble>> ConsStdd(m_spacedim);
    Array<OneD, Array<OneD, NekDouble>> ConsCurv(m_spacedim);
    TensorOfArray4D<NekDouble> PntJacDerv(
        m_spacedim); // Nvar*Nvar*Ndir*Nelmt*Npnt
    TensorOfArray4D<DataType> PntJacDervStd(
        m_spacedim); // Nvar*Nvar*Ndir*Nelmt*Npnt
    TensorOfArray3D<NekDouble> DervStdd(
        m_spacedim); // Nvar*Nvar*Ndir*Nelmt*Npnt
    TensorOfArray3D<NekDouble> DervCurv(
        m_spacedim); // Nvar*Nvar*Ndir*Nelmt*Npnt
    for (int ndir = 0; ndir < m_spacedim; ndir++)
    {
        PntJacDerv[ndir]    = TensorOfArray3D<NekDouble>(m_spacedim);
        PntJacDervStd[ndir] = TensorOfArray3D<DataType>(m_spacedim);
        DervStdd[ndir]      = Array<OneD, Array<OneD, NekDouble>>(m_spacedim);
        DervCurv[ndir]      = Array<OneD, Array<OneD, NekDouble>>(m_spacedim);
    }
 
    Array<OneD, NekDouble> locmu;
    Array<OneD, NekDouble> locDmuDT;
    Array<OneD, Array<OneD, NekDouble>> locVars(nvariable);
    TensorOfArray3D<NekDouble> locDerv(m_spacedim);
    for (int ndir = 0; ndir < m_spacedim; ndir++)
    {
        locDerv[ndir] = Array<OneD, Array<OneD, NekDouble>>(nvariable);
    }
 
    int nElmtCoefOld = -1;
    for (int ne = 0; ne < nTotElmt; ne++)
    {
        int nElmtCoef  = (*expvect)[ne]->GetNcoeffs();
        int nElmtCoef2 = nElmtCoef * nElmtCoef;
        int nElmtPnt   = (*expvect)[ne]->GetTotPoints();
 
        int nQuot     = nElmtCoef2 / m_nPadding;
        int nRemd     = nElmtCoef2 - nQuot * m_nPadding;
        int nQuotPlus = nQuot;
        if (nRemd > 0)
        {
            nQuotPlus++;
        }
        int nElmtCoef2Paded = nQuotPlus * m_nPadding;
 
        if (nElmtPnt > PntJacCons[0].size() || nElmtCoef > nElmtCoefOld)
        {
            nElmtCoefOld = nElmtCoef;
            for (int ndir = 0; ndir < 3; ndir++)
            {
                normalPnt[ndir] = Array<OneD, NekDouble>(npoints, 0.0);
            }
            tmppnts = Array<OneD, NekDouble>(nElmtPnt);
            MatData = Array<OneD, DataType>(nElmtCoef2Paded * nVar2);
            for (int ndir = 0; ndir < m_spacedim; ndir++)
            {
                ConsCurv[ndir] = Array<OneD, NekDouble>(nElmtPnt);
                ConsStdd[ndir] = Array<OneD, NekDouble>(nElmtPnt);
                PntJacCons[ndir] =
                    Array<OneD, Array<OneD, NekDouble>>(nElmtPnt);
                PntJacConsStd[ndir] =
                    Array<OneD, Array<OneD, DataType>>(nElmtPnt);
                for (int i = 0; i < nElmtPnt; i++)
                {
                    PntJacCons[ndir][i]    = Array<OneD, NekDouble>(nVar2);
                    PntJacConsStd[ndir][i] = Array<OneD, DataType>(nVar2);
                }
 
                for (int ndir1 = 0; ndir1 < m_spacedim; ndir1++)
                {
                    PntJacDerv[ndir][ndir1] =
                        Array<OneD, Array<OneD, NekDouble>>(nElmtPnt);
                    PntJacDervStd[ndir][ndir1] =
                        Array<OneD, Array<OneD, DataType>>(nElmtPnt);
                    DervStdd[ndir][ndir1] = Array<OneD, NekDouble>(nElmtPnt);
                    DervCurv[ndir][ndir1] = Array<OneD, NekDouble>(nElmtPnt);
                    for (int i = 0; i < nElmtPnt; i++)
                    {
                        PntJacDerv[ndir][ndir1][i] =
                            Array<OneD, NekDouble>(nVar2);
                        PntJacDervStd[ndir][ndir1][i] =
                            Array<OneD, DataType>(nVar2);
                    }
                }
            }
        }
 
        int noffset = GetPhys_Offset(ne);
        for (int j = 0; j < nvariable; j++)
        {
            locVars[j] = inarray[j] + noffset;
        }
 
        if (m_advectionJacFlag)
        {
            for (int nFluxDir = 0; nFluxDir < nSpaceDim; nFluxDir++)
            {
                normals = normal3D[nFluxDir];
                GetFluxVectorJacDirElmt(nvariable, nElmtPnt, locVars, normals,
                                        wspMat, PntJacCons[nFluxDir]);
            }
        }
 
        if (m_viscousJacFlag)
        {
            for (int j = 0; j < nSpaceDim; j++)
            {
                for (int k = 0; k < nvariable; k++)
                {
                    locDerv[j][k] = qfield[j][k] + noffset;
                }
            }
            locmu    = mu + noffset;
            locDmuDT = DmuDT + noffset;
            for (int nFluxDir = 0; nFluxDir < nSpaceDim; nFluxDir++)
            {
                normals = normal3D[nFluxDir];
                MinusDiffusionFluxJacPoint(nvariable, nElmtPnt, locVars,
                                           locDerv, locmu, locDmuDT, normals,
                                           wspMatDrv, PntJacCons[nFluxDir]);
            }
        }
 
        if (m_viscousJacFlag)
        {
            locmu = mu + noffset;
            for (int nFluxDir = 0; nFluxDir < nSpaceDim; nFluxDir++)
            {
                Vmath::Fill(npoints, 1.0, normalPnt[nFluxDir], 1);
                for (int nDervDir = 0; nDervDir < nSpaceDim; nDervDir++)
                {
                    GetFluxDerivJacDirctnElmt(
                        nvariable, nElmtPnt, nDervDir, locVars, locmu,
                        normalPnt, wspMatDrv, PntJacDerv[nFluxDir][nDervDir]);
                }
                Vmath::Fill(npoints, 0.0, normalPnt[nFluxDir], 1);
            }
        }
 
        for (int n = 0; n < nvariable; n++)
        {
            for (int m = 0; m < nvariable; m++)
            {
                int nVarOffset = m + n * nvariable;
                GmatxData      = gmtxarray[m][n]->GetBlock(ne, ne)->GetPtr();
 
                for (int ndStd0 = 0; ndStd0 < m_spacedim; ndStd0++)
                {
                    Vmath::Zero(nElmtPnt, ConsStdd[ndStd0], 1);
                }
                for (int ndir = 0; ndir < m_spacedim; ndir++)
                {
                    for (int i = 0; i < nElmtPnt; i++)
                    {
                        tmppnts[i] = PntJacCons[ndir][i][nVarOffset];
                    }
                    (*expvect)[ne]->AlignVectorToCollapsedDir(ndir, tmppnts,
                                                              ConsCurv);
                    for (int nd = 0; nd < m_spacedim; nd++)
                    {
                        Vmath::Vadd(nElmtPnt, ConsCurv[nd], 1, ConsStdd[nd], 1,
                                    ConsStdd[nd], 1);
                    }
                }
 
                for (int ndir = 0; ndir < m_spacedim; ndir++)
                {
                    (*expvect)[ne]->MultiplyByQuadratureMetric(
                        ConsStdd[ndir], ConsStdd[ndir]); // weight with metric
                    for (int i = 0; i < nElmtPnt; i++)
                    {
                        PntJacConsStd[ndir][i][nVarOffset] =
                            DataType(ConsStdd[ndir][i]);
                    }
                }
            }
        }
 
        if (m_viscousJacFlag)
        {
            for (int m = 0; m < nvariable; m++)
            {
                for (int n = 0; n < nvariable; n++)
                {
                    int nVarOffset = m + n * nvariable;
                    for (int ndStd0 = 0; ndStd0 < m_spacedim; ndStd0++)
                    {
                        for (int ndStd1 = 0; ndStd1 < m_spacedim; ndStd1++)
                        {
                            Vmath::Zero(nElmtPnt, DervStdd[ndStd0][ndStd1], 1);
                        }
                    }
                    for (int nd0 = 0; nd0 < m_spacedim; nd0++)
                    {
                        for (int nd1 = 0; nd1 < m_spacedim; nd1++)
                        {
                            for (int i = 0; i < nElmtPnt; i++)
                            {
                                tmppnts[i] =
                                    PntJacDerv[nd0][nd1][i][nVarOffset];
                            }
 
                            (*expvect)[ne]->AlignVectorToCollapsedDir(
                                nd0, tmppnts, ConsCurv);
                            for (int nd = 0; nd < m_spacedim; nd++)
                            {
                                (*expvect)[ne]->AlignVectorToCollapsedDir(
                                    nd1, ConsCurv[nd], DervCurv[nd]);
                            }
 
                            for (int ndStd0 = 0; ndStd0 < m_spacedim; ndStd0++)
                            {
                                for (int ndStd1 = 0; ndStd1 < m_spacedim;
                                     ndStd1++)
                                {
                                    Vmath::Vadd(nElmtPnt,
                                                DervCurv[ndStd0][ndStd1], 1,
                                                DervStdd[ndStd0][ndStd1], 1,
                                                DervStdd[ndStd0][ndStd1], 1);
                                }
                            }
                        }
                    }
                    for (int nd0 = 0; nd0 < m_spacedim; nd0++)
                    {
                        for (int nd1 = 0; nd1 < m_spacedim; nd1++)
                        {
                            (*expvect)[ne]->MultiplyByQuadratureMetric(
                                DervStdd[nd0][nd1],
                                DervStdd[nd0][nd1]); // weight with metric
                            for (int i = 0; i < nElmtPnt; i++)
                            {
                                PntJacDervStd[nd0][nd1][i][nVarOffset] =
                                    -DataType(DervStdd[nd0][nd1][i]);
                            }
                        }
                    }
                }
            }
        }
 
        Vmath::Zero(nElmtCoef2Paded * nVar2, MatData, 1);
        DataType one = 1.0;
        for (int ndir = 0; ndir < m_spacedim; ndir++)
        {
            for (int i = 0; i < nElmtPnt; i++)
            {
                Blas::Ger(nElmtCoef2Paded, nVar2, one,
                          &StdMatDataDBB[ne][ndir][i][0], 1,
                          &PntJacConsStd[ndir][i][0], 1, &MatData[0],
                          nElmtCoef2Paded);
            }
        }
 
        if (m_viscousJacFlag)
        {
            for (int nd0 = 0; nd0 < m_spacedim; nd0++)
            {
                for (int nd1 = 0; nd1 < m_spacedim; nd1++)
                {
                    for (int i = 0; i < nElmtPnt; i++)
                    {
                        Blas::Ger(nElmtCoef2Paded, nVar2, one,
                                  &StdMatDataDBDB[ne][nd0][nd1][i][0], 1,
                                  &PntJacDervStd[nd0][nd1][i][0], 1,
                                  &MatData[0], nElmtCoef2Paded);
                    }
                }
            }
        }
 
        Array<OneD, DataType> tmpA;
 
        for (int n = 0; n < nvariable; n++)
        {
            for (int m = 0; m < nvariable; m++)
            {
                int nVarOffset = m + n * nvariable;
                GmatxData      = gmtxarray[m][n]->GetBlock(ne, ne)->GetPtr();
                Vmath::Vcopy(nElmtCoef2,
                             tmpA = MatData + nVarOffset * nElmtCoef2Paded, 1,
                             GmatxData, 1);
            }
        }
    }
}
 
template <typename DataType>
void CFSImplicit::CalcVolJacStdMat(TensorOfArray4D<DataType> &StdMatDataDBB,
                                   TensorOfArray5D<DataType> &StdMatDataDBDB)
{
    std::shared_ptr<LocalRegions::ExpansionVector> expvect =
        m_fields[0]->GetExp();
    int nTotElmt = (*expvect).size();
 
    StdMatDataDBB  = TensorOfArray4D<DataType>(nTotElmt);
    StdMatDataDBDB = TensorOfArray5D<DataType>(nTotElmt);
 
    vector<DNekMatSharedPtr> VectStdDerivBase0;
    vector<TensorOfArray3D<DataType>> VectStdDerivBase_Base;
    vector<TensorOfArray4D<DataType>> VectStdDervBase_DervBase;
    DNekMatSharedPtr MatStdDerivBase0;
    Array<OneD, DNekMatSharedPtr> ArrayStdMat(m_spacedim);
    Array<OneD, Array<OneD, NekDouble>> ArrayStdMatData(m_spacedim);
    for (int ne = 0; ne < nTotElmt; ne++)
    {
        StdRegions::StdExpansionSharedPtr stdExp;
        stdExp = (*expvect)[ne]->GetStdExp();
        StdRegions::StdMatrixKey matkey(StdRegions::eDerivBase0,
                                        stdExp->DetShapeType(), *stdExp);
        MatStdDerivBase0 = stdExp->GetStdMatrix(matkey);
 
        int nTotStdExp   = VectStdDerivBase0.size();
        int nFoundStdExp = -1;
        for (int i = 0; i < nTotStdExp; i++)
        {
            if ((*VectStdDerivBase0[i]) == (*MatStdDerivBase0))
            {
                nFoundStdExp = i;
            }
        }
        if (nFoundStdExp >= 0)
        {
            StdMatDataDBB[ne]  = VectStdDerivBase_Base[nFoundStdExp];
            StdMatDataDBDB[ne] = VectStdDervBase_DervBase[nFoundStdExp];
        }
        else
        {
            int nElmtCoef  = (*expvect)[ne]->GetNcoeffs();
            int nElmtCoef2 = nElmtCoef * nElmtCoef;
            int nElmtPnt   = (*expvect)[ne]->GetTotPoints();
 
            int nQuot     = nElmtCoef2 / m_nPadding;
            int nRemd     = nElmtCoef2 - nQuot * m_nPadding;
            int nQuotPlus = nQuot;
            if (nRemd > 0)
            {
                nQuotPlus++;
            }
            int nPaded = nQuotPlus * m_nPadding;
 
            ArrayStdMat[0] = MatStdDerivBase0;
            if (m_spacedim > 1)
            {
                StdRegions::StdMatrixKey matkey(
                    StdRegions::eDerivBase1, stdExp->DetShapeType(), *stdExp);
                ArrayStdMat[1] = stdExp->GetStdMatrix(matkey);
 
                if (m_spacedim > 2)
                {
                    StdRegions::StdMatrixKey matkey(StdRegions::eDerivBase2,
                                                    stdExp->DetShapeType(),
                                                    *stdExp);
                    ArrayStdMat[2] = stdExp->GetStdMatrix(matkey);
                }
            }
            for (int nd0 = 0; nd0 < m_spacedim; nd0++)
            {
                ArrayStdMatData[nd0] = ArrayStdMat[nd0]->GetPtr();
            }
 
            StdRegions::StdMatrixKey matkey(StdRegions::eBwdMat,
                                            stdExp->DetShapeType(), *stdExp);
            DNekMatSharedPtr BwdMat           = stdExp->GetStdMatrix(matkey);
            Array<OneD, NekDouble> BwdMatData = BwdMat->GetPtr();
 
            TensorOfArray3D<DataType> tmpStdDBB(m_spacedim);
            TensorOfArray4D<DataType> tmpStdDBDB(m_spacedim);
 
            for (int nd0 = 0; nd0 < m_spacedim; nd0++)
            {
                tmpStdDBB[nd0] = Array<OneD, Array<OneD, DataType>>(nElmtPnt);
                for (int i = 0; i < nElmtPnt; i++)
                {
                    tmpStdDBB[nd0][i] = Array<OneD, DataType>(nPaded, 0.0);
                    for (int nc1 = 0; nc1 < nElmtCoef; nc1++)
                    {
                        int noffset = nc1 * nElmtCoef;
                        for (int nc0 = 0; nc0 < nElmtCoef; nc0++)
                        {
                            tmpStdDBB[nd0][i][nc0 + noffset] = DataType(
                                ArrayStdMatData[nd0][i * nElmtCoef + nc0] *
                                BwdMatData[i * nElmtCoef + nc1]);
                        }
                    }
                }
 
                tmpStdDBDB[nd0] = TensorOfArray3D<DataType>(m_spacedim);
                for (int nd1 = 0; nd1 < m_spacedim; nd1++)
                {
                    tmpStdDBDB[nd0][nd1] =
                        Array<OneD, Array<OneD, DataType>>(nElmtPnt);
                    for (int i = 0; i < nElmtPnt; i++)
                    {
                        tmpStdDBDB[nd0][nd1][i] =
                            Array<OneD, DataType>(nPaded, 0.0);
                        for (int nc1 = 0; nc1 < nElmtCoef; nc1++)
                        {
                            int noffset = nc1 * nElmtCoef;
                            for (int nc0 = 0; nc0 < nElmtCoef; nc0++)
                            {
                                tmpStdDBDB[nd0][nd1][i][nc0 + noffset] =
                                    DataType(
                                        ArrayStdMatData[nd0]
                                                       [i * nElmtCoef + nc0] *
                                        ArrayStdMatData[nd1]
                                                       [i * nElmtCoef + nc1]);
                            }
                        }
                    }
                }
            }
            VectStdDerivBase0.push_back(MatStdDerivBase0);
            VectStdDerivBase_Base.push_back(tmpStdDBB);
            VectStdDervBase_DervBase.push_back(tmpStdDBDB);
 
            StdMatDataDBB[ne]  = tmpStdDBB;
            StdMatDataDBDB[ne] = tmpStdDBDB;
        }
    }
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::AddMatNSBlkDiagBnd(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    TensorOfArray3D<NekDouble> &qfield,
    TensorOfArray2D<TypeNekBlkMatSharedPtr> &gmtxarray,
    Array<OneD, TypeNekBlkMatSharedPtr> &TraceJac,
    Array<OneD, TypeNekBlkMatSharedPtr> &TraceJacDeriv,
    Array<OneD, Array<OneD, DataType>> &TraceJacDerivSign,
    TensorOfArray5D<DataType> &TraceIPSymJacArray)
{
    int nvariables = inarray.size();
 
    LibUtilities::Timer timer;
    timer.Start();
    GetTraceJac(inarray, qfield, TraceJac, TraceJacDeriv, TraceJacDerivSign,
                TraceIPSymJacArray);
    timer.Stop();
    timer.AccumulateRegion("CFSImplicit::GetTraceJac", 10);
 
    Array<OneD, TypeNekBlkMatSharedPtr> tmpJac;
    Array<OneD, Array<OneD, DataType>> tmpSign;
 
    timer.Start();
    m_advObject->AddTraceJacToMat(nvariables, m_spacedim, m_fields, TraceJac,
                                  gmtxarray, tmpJac, tmpSign);
    timer.Stop();
    timer.AccumulateRegion("Advection::AddTraceJacToMap", 10);
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::ElmtVarInvMtrx(
    Array<OneD, Array<OneD, TypeNekBlkMatSharedPtr>> &gmtxarray,
    TypeNekBlkMatSharedPtr &gmtVar, const DataType &tmpDataType)
{
    boost::ignore_unused(tmpDataType);
 
    int n1d               = gmtxarray.size();
    int n2d               = gmtxarray[0].size();
    int nConvectiveFields = n1d;
 
    ASSERTL0(n1d == n2d, "ElmtVarInvMtrx requires n1d==n2d");
 
    Array<OneD, unsigned int> rowSizes;
    Array<OneD, unsigned int> colSizes;
 
    gmtxarray[0][0]->GetBlockSizes(rowSizes, colSizes);
    int nTotElmt   = rowSizes.size();
    int nElmtCoef  = rowSizes[0] - 1;
    int nElmtCoef0 = -1;
    int blocksize  = -1;
 
    Array<OneD, unsigned int> tmprow(1);
    TypeNekBlkMatSharedPtr tmpGmtx;
 
    Array<OneD, DataType> GMatData, ElmtMatData;
    Array<OneD, DataType> tmpArray1, tmpArray2;
 
    for (int nelmt = 0; nelmt < nTotElmt; nelmt++)
    {
        int nrows = gmtxarray[0][0]->GetBlock(nelmt, nelmt)->GetRows();
        int ncols = gmtxarray[0][0]->GetBlock(nelmt, nelmt)->GetColumns();
        ASSERTL0(nrows == ncols, "ElmtVarInvMtrx requires nrows==ncols");
 
        nElmtCoef = nrows;
 
        if (nElmtCoef0 != nElmtCoef)
        {
            nElmtCoef0       = nElmtCoef;
            int nElmtCoefVar = nElmtCoef0 * nConvectiveFields;
            blocksize        = nElmtCoefVar * nElmtCoefVar;
            tmprow[0]        = nElmtCoefVar;
            AllocateNekBlkMatDig(tmpGmtx, tmprow, tmprow);
            GMatData = tmpGmtx->GetBlock(0, 0)->GetPtr();
        }
 
        for (int n = 0; n < nConvectiveFields; n++)
        {
            for (int m = 0; m < nConvectiveFields; m++)
            {
                ElmtMatData = gmtxarray[m][n]->GetBlock(nelmt, nelmt)->GetPtr();
 
                for (int ncl = 0; ncl < nElmtCoef; ncl++)
                {
                    int Goffset =
                        (n * nElmtCoef + ncl) * nConvectiveFields * nElmtCoef +
                        m * nElmtCoef;
                    int Eoffset = ncl * nElmtCoef;
 
                    Vmath::Vcopy(nElmtCoef, tmpArray1 = ElmtMatData + Eoffset,
                                 1, tmpArray2         = GMatData + Goffset, 1);
                }
            }
        }
 
        tmpGmtx->GetBlock(0, 0)->Invert();
 
        for (int m = 0; m < nConvectiveFields; m++)
        {
            for (int n = 0; n < nConvectiveFields; n++)
            {
                ElmtMatData = gmtxarray[m][n]->GetBlock(nelmt, nelmt)->GetPtr();
 
                for (int ncl = 0; ncl < nElmtCoef; ncl++)
                {
                    int Goffset =
                        (n * nElmtCoef + ncl) * nConvectiveFields * nElmtCoef +
                        m * nElmtCoef;
                    int Eoffset = ncl * nElmtCoef;
 
                    Vmath::Vcopy(nElmtCoef, tmpArray1 = GMatData + Goffset, 1,
                                 tmpArray2 = ElmtMatData + Eoffset, 1);
                }
            }
        }
        ElmtMatData = gmtVar->GetBlock(nelmt, nelmt)->GetPtr();
        Vmath::Vcopy(blocksize, GMatData, 1, ElmtMatData, 1);
    }
    return;
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::GetTraceJac(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    TensorOfArray3D<NekDouble> &qfield,
    Array<OneD, TypeNekBlkMatSharedPtr> &TraceJac,
    Array<OneD, TypeNekBlkMatSharedPtr> &TraceJacDeriv,
    Array<OneD, Array<OneD, DataType>> &TraceJacDerivSign,
    TensorOfArray5D<DataType> &TraceIPSymJacArray)
{
    boost::ignore_unused(TraceJacDeriv, TraceJacDerivSign);
 
    int nvariables = inarray.size();
    int nTracePts  = GetTraceTotPoints();
 
    // Store forwards/backwards space along trace space
    Array<OneD, Array<OneD, NekDouble>> Fwd(nvariables);
    Array<OneD, Array<OneD, NekDouble>> Bwd(nvariables);
 
    TypeNekBlkMatSharedPtr FJac, BJac;
    Array<OneD, unsigned int> n_blks1(nTracePts, nvariables);
 
    if (TraceJac.size() > 0)
    {
        FJac = TraceJac[0];
        BJac = TraceJac[1];
    }
    else
    {
        TraceJac = Array<OneD, TypeNekBlkMatSharedPtr>(2);
 
        AllocateNekBlkMatDig(FJac, n_blks1, n_blks1);
        AllocateNekBlkMatDig(BJac, n_blks1, n_blks1);
    }
 
    if (m_HomogeneousType == eHomogeneous1D)
    {
        Fwd = NullNekDoubleArrayOfArray;
        Bwd = NullNekDoubleArrayOfArray;
    }
    else
    {
        for (int i = 0; i < nvariables; ++i)
        {
            Fwd[i] = Array<OneD, NekDouble>(nTracePts, 0.0);
            Bwd[i] = Array<OneD, NekDouble>(nTracePts, 0.0);
            m_fields[i]->GetFwdBwdTracePhys(inarray[i], Fwd[i], Bwd[i]);
        }
    }
 
    Array<OneD, Array<OneD, NekDouble>> AdvVel(m_spacedim);
 
    NumCalcRiemFluxJac(nvariables, m_fields, AdvVel, inarray, qfield,
                       m_bndEvaluateTime, Fwd, Bwd, FJac, BJac,
                       TraceIPSymJacArray);
 
    TraceJac[0] = FJac;
    TraceJac[1] = BJac;
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::NumCalcRiemFluxJac(
    const int nConvectiveFields,
    const Array<OneD, MultiRegions::ExpListSharedPtr> &fields,
    const Array<OneD, const Array<OneD, NekDouble>> &AdvVel,
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    TensorOfArray3D<NekDouble> &qfield, const NekDouble &time,
    const Array<OneD, const Array<OneD, NekDouble>> &Fwd,
    const Array<OneD, const Array<OneD, NekDouble>> &Bwd,
    TypeNekBlkMatSharedPtr &FJac, TypeNekBlkMatSharedPtr &BJac,
    TensorOfArray5D<DataType> &TraceIPSymJacArray)
{
    boost::ignore_unused(TraceIPSymJacArray);
 
    const NekDouble PenaltyFactor2 = 0.0;
    int nvariables                 = nConvectiveFields;
    int npoints                    = GetNpoints();
    // int nPts        = npoints;
    int nTracePts = GetTraceTotPoints();
    int nDim      = m_spacedim;
 
    Array<OneD, int> nonZeroIndex;
 
    Array<OneD, Array<OneD, NekDouble>> tmpinarry(nvariables);
    for (int i = 0; i < nvariables; i++)
    {
        tmpinarry[i] = Array<OneD, NekDouble>(npoints, 0.0);
        Vmath::Vcopy(npoints, inarray[i], 1, tmpinarry[i], 1);
    }
 
    // DmuDT of artificial diffusion is neglected
    // TODO: to consider the Jacobian of AV seperately
    Array<OneD, NekDouble> muvar      = NullNekDouble1DArray;
    Array<OneD, NekDouble> MuVarTrace = NullNekDouble1DArray;
 
    Array<OneD, Array<OneD, NekDouble>> numflux(nvariables);
    for (int i = 0; i < nvariables; ++i)
    {
        numflux[i] = Array<OneD, NekDouble>(nTracePts, 0.0);
    }
 
    const MultiRegions::AssemblyMapDGSharedPtr TraceMap =
        fields[0]->GetTraceMap();
    TensorOfArray3D<NekDouble> qBwd(nDim);
    TensorOfArray3D<NekDouble> qFwd(nDim);
    if (m_viscousJacFlag)
    {
        for (int nd = 0; nd < nDim; ++nd)
        {
            qBwd[nd] = Array<OneD, Array<OneD, NekDouble>>(nConvectiveFields);
            qFwd[nd] = Array<OneD, Array<OneD, NekDouble>>(nConvectiveFields);
            for (int i = 0; i < nConvectiveFields; ++i)
            {
                qBwd[nd][i] = Array<OneD, NekDouble>(nTracePts, 0.0);
                qFwd[nd][i] = Array<OneD, NekDouble>(nTracePts, 0.0);
 
                fields[i]->GetFwdBwdTracePhys(qfield[nd][i], qFwd[nd][i],
                                              qBwd[nd][i], true, true, false);
                TraceMap->GetAssemblyCommDG()->PerformExchange(qFwd[nd][i],
                                                               qBwd[nd][i]);
            }
        }
    }
 
    CalcTraceNumericalFlux(nConvectiveFields, nDim, npoints, nTracePts,
                           PenaltyFactor2, fields, AdvVel, inarray, time,
                           qfield, Fwd, Bwd, qFwd, qBwd, MuVarTrace,
                           nonZeroIndex, numflux);
 
    int nFields = nvariables;
    Array<OneD, Array<OneD, NekDouble>> plusFwd(nFields), plusBwd(nFields);
    Array<OneD, Array<OneD, NekDouble>> Jacvect(nFields);
    Array<OneD, Array<OneD, NekDouble>> FwdBnd(nFields);
    Array<OneD, Array<OneD, NekDouble>> plusflux(nFields);
    for (int i = 0; i < nFields; i++)
    {
        Jacvect[i]  = Array<OneD, NekDouble>(nTracePts, 0.0);
        plusFwd[i]  = Array<OneD, NekDouble>(nTracePts, 0.0);
        plusBwd[i]  = Array<OneD, NekDouble>(nTracePts, 0.0);
        plusflux[i] = Array<OneD, NekDouble>(nTracePts, 0.0);
        FwdBnd[i]   = Array<OneD, NekDouble>(nTracePts, 0.0);
    }
 
    for (int i = 0; i < nFields; i++)
    {
        Vmath::Vcopy(nTracePts, Fwd[i], 1, plusFwd[i], 1);
        Vmath::Vcopy(nTracePts, Bwd[i], 1, plusBwd[i], 1);
    }
 
    NekDouble eps = 1.0E-6;
 
    Array<OneD, DataType> tmpMatData;
    // Fwd Jacobian
    for (int i = 0; i < nFields; i++)
    {
        NekDouble epsvar  = eps * m_magnitdEstimat[i];
        NekDouble oepsvar = 1.0 / epsvar;
        Vmath::Sadd(nTracePts, epsvar, Fwd[i], 1, plusFwd[i], 1);
 
        if (m_bndConds.size())
        {
            for (int i = 0; i < nFields; i++)
            {
                Vmath::Vcopy(nTracePts, plusFwd[i], 1, FwdBnd[i], 1);
            }
            // Loop over user-defined boundary conditions
            for (auto &x : m_bndConds)
            {
                x->Apply(FwdBnd, tmpinarry, time);
            }
        }
 
        for (int j = 0; j < nFields; j++)
        {
            m_fields[j]->FillBwdWithBoundCond(plusFwd[j], plusBwd[j]);
        }
 
        CalcTraceNumericalFlux(nConvectiveFields, nDim, npoints, nTracePts,
                               PenaltyFactor2, fields, AdvVel, inarray, time,
                               qfield, plusFwd, plusBwd, qFwd, qBwd, MuVarTrace,
                               nonZeroIndex, plusflux);
 
        for (int n = 0; n < nFields; n++)
        {
            Vmath::Vsub(nTracePts, plusflux[n], 1, numflux[n], 1, Jacvect[n],
                        1);
            Vmath::Smul(nTracePts, oepsvar, Jacvect[n], 1, Jacvect[n], 1);
        }
        for (int j = 0; j < nTracePts; j++)
        {
            tmpMatData = FJac->GetBlock(j, j)->GetPtr();
            for (int n = 0; n < nFields; n++)
            {
                tmpMatData[n + i * nFields] = DataType(Jacvect[n][j]);
            }
        }
 
        Vmath::Vcopy(nTracePts, Fwd[i], 1, plusFwd[i], 1);
    }
 
    // Reset the boundary conditions
    if (m_bndConds.size())
    {
        for (int i = 0; i < nFields; i++)
        {
            Vmath::Vcopy(nTracePts, Fwd[i], 1, FwdBnd[i], 1);
        }
        // Loop over user-defined boundary conditions
        for (auto &x : m_bndConds)
        {
            x->Apply(FwdBnd, tmpinarry, time);
        }
    }
 
    for (int i = 0; i < nFields; i++)
    {
        Vmath::Vcopy(nTracePts, Bwd[i], 1, plusBwd[i], 1);
    }
 
    for (int i = 0; i < nFields; i++)
    {
        NekDouble epsvar  = eps * m_magnitdEstimat[i];
        NekDouble oepsvar = 1.0 / epsvar;
 
        Vmath::Sadd(nTracePts, epsvar, Bwd[i], 1, plusBwd[i], 1);
 
        for (int j = 0; j < nFields; j++)
        {
            m_fields[j]->FillBwdWithBoundCond(Fwd[j], plusBwd[j]);
        }
 
        CalcTraceNumericalFlux(nConvectiveFields, nDim, npoints, nTracePts,
                               PenaltyFactor2, fields, AdvVel, inarray, time,
                               qfield, Fwd, plusBwd, qFwd, qBwd, MuVarTrace,
                               nonZeroIndex, plusflux);
 
        for (int n = 0; n < nFields; n++)
        {
            Vmath::Vsub(nTracePts, plusflux[n], 1, numflux[n], 1, Jacvect[n],
                        1);
            Vmath::Smul(nTracePts, oepsvar, Jacvect[n], 1, Jacvect[n], 1);
        }
        for (int j = 0; j < nTracePts; j++)
        {
            tmpMatData = BJac->GetBlock(j, j)->GetPtr();
            for (int n = 0; n < nFields; n++)
            {
                tmpMatData[n + i * nFields] = DataType(Jacvect[n][j]);
            }
        }
 
        Vmath::Vcopy(nTracePts, Bwd[i], 1, plusBwd[i], 1);
    }
}
 
void CFSImplicit::CalcTraceNumericalFlux(
    const int nConvectiveFields, const int nDim, const int nPts,
    const int nTracePts, const NekDouble PenaltyFactor2,
    const Array<OneD, MultiRegions::ExpListSharedPtr> &fields,
    const Array<OneD, const Array<OneD, NekDouble>> &AdvVel,
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    const NekDouble time, TensorOfArray3D<NekDouble> &qfield,
    const Array<OneD, const Array<OneD, NekDouble>> &vFwd,
    const Array<OneD, const Array<OneD, NekDouble>> &vBwd,
    const Array<OneD, const TensorOfArray2D<NekDouble>> &qFwd,
    const Array<OneD, const TensorOfArray2D<NekDouble>> &qBwd,
    const Array<OneD, NekDouble> &MuVarTrace, Array<OneD, int> &nonZeroIndex,
    Array<OneD, Array<OneD, NekDouble>> &traceflux)
{
    boost::ignore_unused(nDim, nPts, PenaltyFactor2, time, qFwd, qBwd,
                         MuVarTrace);
 
    if (m_advectionJacFlag)
    {
        auto advWeakDGObject =
            std::dynamic_pointer_cast<SolverUtils::AdvectionWeakDG>(
                m_advObject);
        ASSERTL0(advWeakDGObject,
                 "Use WeakDG for implicit compressible flow solver!");
        advWeakDGObject->AdvectTraceFlux(nConvectiveFields, m_fields, AdvVel,
                                         inarray, traceflux, m_bndEvaluateTime,
                                         vFwd, vBwd);
    }
    else
    {
        for (int i = 0; i < nConvectiveFields; i++)
        {
            traceflux[i] = Array<OneD, NekDouble>(nTracePts, 0.0);
        }
    }
 
    if (m_viscousJacFlag)
    {
        Array<OneD, Array<OneD, NekDouble>> visflux(nConvectiveFields);
        for (int i = 0; i < nConvectiveFields; i++)
        {
            visflux[i] = Array<OneD, NekDouble>(nTracePts, 0.0);
        }
 
        string diffName;
        m_session->LoadSolverInfo("DiffusionType", diffName, "InteriorPenalty");
        if (diffName == "InteriorPenalty")
        {
            m_diffusion->DiffuseTraceFlux(fields, inarray, qfield,
                                          NullNekDoubleTensorOfArray3D, visflux,
                                          vFwd, vBwd, nonZeroIndex);
        }
        else
        {
            ASSERTL1(false, "LDGNS not yet validated for implicit compressible "
                            "flow solver");
            // For LDGNS, the array size should be nConvectiveFields - 1
            Array<OneD, Array<OneD, NekDouble>> inBwd(nConvectiveFields - 1);
            Array<OneD, Array<OneD, NekDouble>> inFwd(nConvectiveFields - 1);
            for (int i = 0; i < nConvectiveFields - 1; ++i)
            {
                inBwd[i] = vBwd[i];
                inFwd[i] = vFwd[i];
            }
            m_diffusion->DiffuseTraceFlux(fields, inarray, qfield,
                                          NullNekDoubleTensorOfArray3D, visflux,
                                          inFwd, inBwd, nonZeroIndex);
        }
        for (int i = 0; i < nConvectiveFields; i++)
        {
            Vmath::Vsub(nTracePts, traceflux[i], 1, visflux[i], 1, traceflux[i],
                        1);
        }
    }
}
 
void CFSImplicit::CalcPreconMatBRJCoeff(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, SNekBlkMatSharedPtr>> &gmtxarray,
    SNekBlkMatSharedPtr &gmtVar, Array<OneD, SNekBlkMatSharedPtr> &TraceJac,
    Array<OneD, SNekBlkMatSharedPtr> &TraceJacDeriv,
    Array<OneD, Array<OneD, NekSingle>> &TraceJacDerivSign,
    TensorOfArray4D<NekSingle> &TraceJacArray,
    TensorOfArray4D<NekSingle> &TraceJacDerivArray,
    TensorOfArray5D<NekSingle> &TraceIPSymJacArray)
{
    TensorOfArray3D<NekDouble> qfield;
 
    if (m_viscousJacFlag)
    {
        CalcPhysDeriv(inarray, qfield);
    }
 
    NekSingle zero = 0.0;
    Fill2DArrayOfBlkDiagonalMat(gmtxarray, zero);
 
    LibUtilities::Timer timer;
    timer.Start();
    AddMatNSBlkDiagVol(inarray, qfield, gmtxarray, m_stdSMatDataDBB,
                       m_stdSMatDataDBDB);
    timer.Stop();
    timer.AccumulateRegion("CFSImplicit::AddMatNSBlkDiagVol", 2);
 
    timer.Start();
    AddMatNSBlkDiagBnd(inarray, qfield, gmtxarray, TraceJac, TraceJacDeriv,
                       TraceJacDerivSign, TraceIPSymJacArray);
    timer.Stop();
    timer.AccumulateRegion("CFSImplicit::AddMatNSBlkDiagBnd", 2);
 
    MultiplyElmtInvMassPlusSource(gmtxarray, m_TimeIntegLambda, zero);
 
    timer.Start();
    ElmtVarInvMtrx(gmtxarray, gmtVar, zero);
    timer.Stop();
    timer.AccumulateRegion("CFSImplicit::ElmtVarInvMtrx", 2);
 
    TransTraceJacMatToArray(TraceJac, TraceJacDeriv, TraceJacArray,
                            TraceJacDerivArray);
}
 
void CFSImplicit::v_MinusDiffusionFluxJacPoint(
    const int nConvectiveFields, const int nElmtPnt,
    const Array<OneD, const Array<OneD, NekDouble>> &locVars,
    const TensorOfArray3D<NekDouble> &locDerv,
    const Array<OneD, NekDouble> &locmu, const Array<OneD, NekDouble> &locDmuDT,
    const Array<OneD, NekDouble> &normals, DNekMatSharedPtr &wspMat,
    Array<OneD, Array<OneD, NekDouble>> &PntJacArray)
{
    boost::ignore_unused(nConvectiveFields, nElmtPnt, locVars, locDerv, locmu,
                         locDmuDT, normals, wspMat, PntJacArray);
    // Do nothing by default
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::MultiplyElmtInvMassPlusSource(
    Array<OneD, Array<OneD, TypeNekBlkMatSharedPtr>> &gmtxarray,
    const NekDouble dtlamda, const DataType tmpDataType)
{
    boost::ignore_unused(tmpDataType);
 
    MultiRegions::ExpListSharedPtr explist              = m_fields[0];
    std::shared_ptr<LocalRegions::ExpansionVector> pexp = explist->GetExp();
    int nTotElmt                                        = (*pexp).size();
    int nConvectiveFields                               = m_fields.size();
 
    NekDouble Negdtlamda = -dtlamda;
 
    Array<OneD, NekDouble> pseudotimefactor(nTotElmt, 0.0);
    Vmath::Fill(nTotElmt, Negdtlamda, pseudotimefactor, 1);
 
    Array<OneD, DataType> GMatData;
    for (int m = 0; m < nConvectiveFields; m++)
    {
        for (int n = 0; n < nConvectiveFields; n++)
        {
            for (int nelmt = 0; nelmt < nTotElmt; nelmt++)
            {
                GMatData = gmtxarray[m][n]->GetBlock(nelmt, nelmt)->GetPtr();
                DataType factor = DataType(pseudotimefactor[nelmt]);
 
                Vmath::Smul(GMatData.size(), factor, GMatData, 1, GMatData, 1);
            }
        }
    }
 
    DNekMatSharedPtr MassMat;
    Array<OneD, NekDouble> BwdMatData, MassMatData, tmp;
    Array<OneD, NekDouble> tmp2;
    Array<OneD, DataType> MassMatDataDataType;
    LibUtilities::ShapeType ElmtTypePrevious = LibUtilities::eNoShapeType;
 
    for (int nelmt = 0; nelmt < nTotElmt; nelmt++)
    {
        int nelmtcoef = GetNcoeffs(nelmt);
        int nelmtpnts = GetTotPoints(nelmt);
        LibUtilities::ShapeType ElmtTypeNow =
            explist->GetExp(nelmt)->DetShapeType();
 
        if (tmp.size() != nelmtcoef || (ElmtTypeNow != ElmtTypePrevious))
        {
            StdRegions::StdExpansionSharedPtr stdExp;
            stdExp = explist->GetExp(nelmt)->GetStdExp();
            StdRegions::StdMatrixKey matkey(StdRegions::eBwdTrans,
                                            stdExp->DetShapeType(), *stdExp);
 
            DNekMatSharedPtr BwdMat = stdExp->GetStdMatrix(matkey);
            BwdMatData              = BwdMat->GetPtr();
 
            if (nelmtcoef != tmp.size())
            {
                tmp     = Array<OneD, NekDouble>(nelmtcoef, 0.0);
                MassMat = MemoryManager<DNekMat>::AllocateSharedPtr(
                    nelmtcoef, nelmtcoef, 0.0);
                MassMatData = MassMat->GetPtr();
                MassMatDataDataType =
                    Array<OneD, DataType>(nelmtcoef * nelmtcoef);
            }
 
            ElmtTypePrevious = ElmtTypeNow;
        }
 
        for (int np = 0; np < nelmtcoef; np++)
        {
            explist->GetExp(nelmt)->IProductWRTBase(BwdMatData + np * nelmtpnts,
                                                    tmp);
            Vmath::Vcopy(nelmtcoef, tmp, 1, tmp2 = MassMatData + np * nelmtcoef,
                         1);
        }
        for (int i = 0; i < MassMatData.size(); i++)
        {
            MassMatDataDataType[i] = DataType(MassMatData[i]);
        }
 
        for (int m = 0; m < nConvectiveFields; m++)
        {
            GMatData = gmtxarray[m][m]->GetBlock(nelmt, nelmt)->GetPtr();
            Vmath::Vadd(MassMatData.size(), MassMatDataDataType, 1, GMatData, 1,
                        GMatData, 1);
        }
    }
    return;
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::TransTraceJacMatToArray(
    const Array<OneD, TypeNekBlkMatSharedPtr> &TraceJac,
    const Array<OneD, TypeNekBlkMatSharedPtr> &TraceJacDeriv,
    TensorOfArray4D<DataType> &TraceJacArray,
    TensorOfArray4D<DataType> &TraceJacDerivArray)
{
    boost::ignore_unused(TraceJacArray, TraceJacDeriv, TraceJacDerivArray);
 
    int nFwdBwd, nDiagBlks, nvar0Jac, nvar1Jac;
 
    Array<OneD, unsigned int> rowSizes;
    Array<OneD, unsigned int> colSizes;
    nFwdBwd = TraceJac.size();
    TraceJac[0]->GetBlockSizes(rowSizes, colSizes);
    nDiagBlks = rowSizes.size();
    nvar0Jac  = rowSizes[1] - rowSizes[0];
    nvar1Jac  = colSizes[1] - colSizes[0];
 
    if (0 == TraceJacArray.size())
    {
        TraceJacArray = TensorOfArray4D<DataType>(nFwdBwd);
        for (int nlr = 0; nlr < nFwdBwd; nlr++)
        {
            TraceJacArray[nlr] = TensorOfArray3D<DataType>(nvar0Jac);
            for (int m = 0; m < nvar0Jac; m++)
            {
                TraceJacArray[nlr][m] =
                    Array<OneD, Array<OneD, DataType>>(nvar1Jac);
                for (int n = 0; n < nvar1Jac; n++)
                {
                    TraceJacArray[nlr][m][n] = Array<OneD, DataType>(nDiagBlks);
                }
            }
        }
    }
 
    for (int nlr = 0; nlr < nFwdBwd; nlr++)
    {
        const TypeNekBlkMatSharedPtr tmpMat = TraceJac[nlr];
        TensorOfArray3D<DataType> tmpaa     = TraceJacArray[nlr];
        TranSamesizeBlkDiagMatIntoArray(tmpMat, tmpaa);
    }
 
    return;
}
 
void CFSImplicit::v_GetFluxDerivJacDirctn(
    const MultiRegions::ExpListSharedPtr &explist,
    const Array<OneD, const Array<OneD, NekDouble>> &normals,
    const int nDervDir,
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    TensorOfArray5D<NekDouble> &ElmtJacArray, const int nFluxDir)
{
    boost::ignore_unused(explist, normals, nDervDir, inarray, ElmtJacArray,
                         nFluxDir);
    NEKERROR(ErrorUtil::efatal, "v_GetFluxDerivJacDirctn not coded");
}
 
void CFSImplicit::v_GetFluxDerivJacDirctnElmt(
    const int nConvectiveFields, const int nElmtPnt, const int nDervDir,
    const Array<OneD, const Array<OneD, NekDouble>> &locVars,
    const Array<OneD, NekDouble> &locmu,
    const Array<OneD, const Array<OneD, NekDouble>> &locnormal,
    DNekMatSharedPtr &wspMat, Array<OneD, Array<OneD, NekDouble>> &PntJacArray)
{
    boost::ignore_unused(nConvectiveFields, nElmtPnt, nDervDir, locVars, locmu,
                         locnormal, wspMat, PntJacArray);
    NEKERROR(ErrorUtil::efatal, "v_GetFluxDerivJacDirctn not coded");
}
 
void CFSImplicit::v_GetFluxDerivJacDirctn(
    const MultiRegions::ExpListSharedPtr &explist,
    const Array<OneD, const Array<OneD, NekDouble>> &normals,
    const int nDervDir,
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    TensorOfArray2D<DNekMatSharedPtr> &ElmtJac)
{
    boost::ignore_unused(explist, normals, nDervDir, inarray, ElmtJac);
}
 
void CFSImplicit::GetFluxVectorJacDirElmt(
    const int nConvectiveFields, const int nElmtPnt,
    const Array<OneD, const Array<OneD, NekDouble>> &locVars,
    const Array<OneD, NekDouble> &normals, DNekMatSharedPtr &wspMat,
    Array<OneD, Array<OneD, NekDouble>> &PntJacArray)
{
    Array<OneD, NekDouble> wspMatData = wspMat->GetPtr();
 
    int matsize = nConvectiveFields * nConvectiveFields;
 
    Array<OneD, NekDouble> pointVar(nConvectiveFields);
 
    for (int npnt = 0; npnt < nElmtPnt; npnt++)
    {
        for (int j = 0; j < nConvectiveFields; j++)
        {
            pointVar[j] = locVars[j][npnt];
        }
 
        GetFluxVectorJacPoint(nConvectiveFields, pointVar, normals, wspMat);
 
        Vmath::Vcopy(matsize, wspMatData, 1, PntJacArray[npnt], 1);
    }
    return;
}
 
void CFSImplicit::GetFluxVectorJacPoint(
    const int nConvectiveFields, const Array<OneD, NekDouble> &conservVar,
    const Array<OneD, NekDouble> &normals, DNekMatSharedPtr &fluxJac)
{
    int nvariables         = conservVar.size();
    const int nvariables3D = 5;
    int expDim             = m_spacedim;
 
    NekDouble fsw, efix_StegerWarming;
    efix_StegerWarming = 0.0;
    fsw                = 0.0; // exact flux Jacobian if fsw=0.0
    if (nvariables > expDim + 2)
    {
        NEKERROR(ErrorUtil::efatal, "nvariables > expDim+2 case not coded")
    }
 
    Array<OneD, NekDouble> fluxJacData;
    ;
    fluxJacData = fluxJac->GetPtr();
 
    if (nConvectiveFields == nvariables3D)
    {
        PointFluxJacobianPoint(conservVar, normals, fluxJac, efix_StegerWarming,
                               fsw);
    }
    else
    {
        DNekMatSharedPtr PointFJac3D =
            MemoryManager<DNekMat>::AllocateSharedPtr(nvariables3D,
                                                      nvariables3D, 0.0);
 
        Array<OneD, NekDouble> PointFJac3DData;
        PointFJac3DData = PointFJac3D->GetPtr();
 
        Array<OneD, NekDouble> PointFwd(nvariables3D, 0.0);
 
        Array<OneD, unsigned int> index(nvariables);
 
        index[nvariables - 1] = 4;
        for (int i = 0; i < nvariables - 1; i++)
        {
            index[i] = i;
        }
 
        int nj = 0;
        int nk = 0;
        for (int j = 0; j < nvariables; j++)
        {
            nj           = index[j];
            PointFwd[nj] = conservVar[j];
        }
 
        PointFluxJacobianPoint(PointFwd, normals, PointFJac3D,
                               efix_StegerWarming, fsw);
 
        for (int j = 0; j < nvariables; j++)
        {
            nj = index[j];
            for (int k = 0; k < nvariables; k++)
            {
                nk = index[k];
                fluxJacData[j + k * nConvectiveFields] =
                    PointFJac3DData[nj + nk * nvariables3D];
            }
        }
    }
}
 
/**
 * @brief Add the diffusions terms to the right-hand side
 * Similar to DoDiffusion() but with outarray in coefficient space
 */
void CFSImplicit::DoDiffusionCoeff(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray,
    const Array<OneD, const Array<OneD, NekDouble>> &pFwd,
    const Array<OneD, const Array<OneD, NekDouble>> &pBwd)
{
    v_DoDiffusionCoeff(inarray, outarray, pFwd, pBwd);
}
 
void CFSImplicit::MatrixMultiplyMatrixFreeCoeff(
    const Array<OneD, const NekDouble> &inarray, Array<OneD, NekDouble> &out,
    const bool &flag)
{
    boost::ignore_unused(flag);
    const Array<OneD, const NekDouble> refsol = m_nonlinsol->GetRefSolution();
    const Array<OneD, const NekDouble> refres = m_nonlinsol->GetRefResidual();
    const Array<OneD, const NekDouble> refsource =
        m_nonlinsol->GetRefSourceVec();
 
    NekDouble eps             = m_jacobiFreeEps;
    unsigned int nTotalGlobal = inarray.size();
    NekDouble magninarray     = Vmath::Dot(nTotalGlobal, inarray, inarray);
    m_comm->GetSpaceComm()->AllReduce(magninarray,
                                      Nektar::LibUtilities::ReduceSum);
    eps *= sqrt((sqrt(m_inArrayNorm) + 1.0) / magninarray);
 
    NekDouble oeps      = 1.0 / eps;
    unsigned int ntotal = inarray.size();
    Array<OneD, NekDouble> solplus{ntotal, 0.0};
    Array<OneD, NekDouble> resplus{ntotal, 0.0};
 
    Vmath::Svtvp(ntotal, eps, inarray, 1, refsol, 1, solplus, 1);
 
    NonlinSysEvaluatorCoeff(solplus, resplus, flag, refsource);
 
    Vmath::Vsub(ntotal, resplus, 1, refres, 1, out, 1);
    Vmath::Smul(ntotal, oeps, out, 1, out, 1);
 
    return;
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::TranSamesizeBlkDiagMatIntoArray(
    const TypeNekBlkMatSharedPtr &BlkMat, TensorOfArray3D<DataType> &MatArray)
{
    Array<OneD, unsigned int> rowSizes;
    Array<OneD, unsigned int> colSizes;
    BlkMat->GetBlockSizes(rowSizes, colSizes);
    int nDiagBlks = rowSizes.size();
    int nvar0     = rowSizes[1] - rowSizes[0];
    int nvar1     = colSizes[1] - colSizes[0];
 
    Array<OneD, DataType> ElmtMatData;
 
    for (int i = 0; i < nDiagBlks; i++)
    {
        ElmtMatData = BlkMat->GetBlock(i, i)->GetPtr();
        for (int n = 0; n < nvar1; n++)
        {
            int noffset = n * nvar0;
            for (int m = 0; m < nvar0; m++)
            {
                MatArray[m][n][i] = ElmtMatData[m + noffset];
            }
        }
    }
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::Fill2DArrayOfBlkDiagonalMat(
    Array<OneD, Array<OneD, TypeNekBlkMatSharedPtr>> &gmtxarray,
    const DataType valu)
{
 
    int n1d = gmtxarray.size();
 
    for (int n1 = 0; n1 < n1d; ++n1)
    {
        Fill1DArrayOfBlkDiagonalMat(gmtxarray[n1], valu);
    }
}
 
template <typename DataType, typename TypeNekBlkMatSharedPtr>
void CFSImplicit::Fill1DArrayOfBlkDiagonalMat(
    Array<OneD, TypeNekBlkMatSharedPtr> &gmtxarray, const DataType valu)
{
    int n1d = gmtxarray.size();
 
    Array<OneD, unsigned int> rowSizes;
    Array<OneD, unsigned int> colSizes;
 
    Array<OneD, DataType> loc_mat_arr;
 
    for (int n1 = 0; n1 < n1d; ++n1)
    {
        gmtxarray[n1]->GetBlockSizes(rowSizes, colSizes);
        int nelmts = rowSizes.size();
 
        for (int i = 0; i < nelmts; ++i)
        {
            loc_mat_arr = gmtxarray[n1]->GetBlock(i, i)->GetPtr();
 
            int nrows = gmtxarray[n1]->GetBlock(i, i)->GetRows();
            int ncols = gmtxarray[n1]->GetBlock(i, i)->GetColumns();
 
            Vmath::Fill(nrows * ncols, valu, loc_mat_arr, 1);
        }
    }
}
 
// Currently duplacate in compressibleFlowSys
// if fsw=+-1 calculate the steger-Warming flux vector splitting flux Jacobian
// if fsw=0   calculate the Jacobian of the exact flux
// efix is the numerical flux entropy fix parameter
void CFSImplicit::PointFluxJacobianPoint(const Array<OneD, NekDouble> &Fwd,
                                         const Array<OneD, NekDouble> &normals,
                                         DNekMatSharedPtr &FJac,
                                         const NekDouble efix,
                                         const NekDouble fsw)
{
    Array<OneD, NekDouble> FJacData = FJac->GetPtr();
    const int nvariables3D          = 5;
 
    NekDouble ro, vx, vy, vz, ps, gama, ae;
    NekDouble a, a2, h, h0, v2, vn, eps, eps2;
    NekDouble nx, ny, nz;
    NekDouble sn, osn, nxa, nya, nza, vna;
    NekDouble l1, l4, l5, al1, al4, al5, x1, x2, x3, y1;
    NekDouble c1, d1, c2, d2, c3, d3, c4, d4, c5, d5;
    NekDouble sml_ssf = 1.0E-12;
 
    NekDouble fExactorSplt = 2.0 - abs(fsw); // if fsw=+-1 calculate
 
    NekDouble rhoL  = Fwd[0];
    NekDouble rhouL = Fwd[1];
    NekDouble rhovL = Fwd[2];
    NekDouble rhowL = Fwd[3];
    NekDouble EL    = Fwd[4];
 
    ro = rhoL;
    vx = rhouL / rhoL;
    vy = rhovL / rhoL;
    vz = rhowL / rhoL;
 
    // Internal energy (per unit mass)
    NekDouble eL = (EL - 0.5 * (rhouL * vx + rhovL * vy + rhowL * vz)) / rhoL;
 
    ps   = m_varConv->Geteos()->GetPressure(rhoL, eL);
    gama = m_gamma;
 
    ae = gama - 1.0;
    v2 = vx * vx + vy * vy + vz * vz;
    a2 = gama * ps / ro;
    h  = a2 / ae;
 
    h0 = h + 0.5 * v2;
    a  = sqrt(a2);
 
    nx  = normals[0];
    ny  = normals[1];
    nz  = normals[2];
    vn  = nx * vx + ny * vy + nz * vz;
    sn  = std::max(sqrt(nx * nx + ny * ny + nz * nz), sml_ssf);
    osn = 1.0 / sn;
 
    nxa = nx * osn;
    nya = ny * osn;
    nza = nz * osn;
    vna = vn * osn;
    l1  = vn;
    l4  = vn + sn * a;
    l5  = vn - sn * a;
 
    eps  = efix * sn;
    eps2 = eps * eps;
 
    al1 = sqrt(l1 * l1 + eps2);
    al4 = sqrt(l4 * l4 + eps2);
    al5 = sqrt(l5 * l5 + eps2);
 
    l1 = 0.5 * (fExactorSplt * l1 + fsw * al1);
    l4 = 0.5 * (fExactorSplt * l4 + fsw * al4);
    l5 = 0.5 * (fExactorSplt * l5 + fsw * al5);
 
    x1 = 0.5 * (l4 + l5);
    x2 = 0.5 * (l4 - l5);
    x3 = x1 - l1;
    y1 = 0.5 * v2;
    c1 = ae * x3 / a2;
    d1 = x2 / a;
 
    int nVar0           = 0;
    int nVar1           = nvariables3D;
    int nVar2           = 2 * nvariables3D;
    int nVar3           = 3 * nvariables3D;
    int nVar4           = 4 * nvariables3D;
    FJacData[nVar0]     = c1 * y1 - d1 * vna + l1;
    FJacData[nVar1]     = -c1 * vx + d1 * nxa;
    FJacData[nVar2]     = -c1 * vy + d1 * nya;
    FJacData[nVar3]     = -c1 * vz + d1 * nza;
    FJacData[nVar4]     = c1;
    c2                  = c1 * vx + d1 * nxa * ae;
    d2                  = x3 * nxa + d1 * vx;
    FJacData[1 + nVar0] = c2 * y1 - d2 * vna;
    FJacData[1 + nVar1] = -c2 * vx + d2 * nxa + l1;
    FJacData[1 + nVar2] = -c2 * vy + d2 * nya;
    FJacData[1 + nVar3] = -c2 * vz + d2 * nza;
    FJacData[1 + nVar4] = c2;
    c3                  = c1 * vy + d1 * nya * ae;
    d3                  = x3 * nya + d1 * vy;
    FJacData[2 + nVar0] = c3 * y1 - d3 * vna;
    FJacData[2 + nVar1] = -c3 * vx + d3 * nxa;
    FJacData[2 + nVar2] = -c3 * vy + d3 * nya + l1;
    FJacData[2 + nVar3] = -c3 * vz + d3 * nza;
    FJacData[2 + nVar4] = c3;
    c4                  = c1 * vz + d1 * nza * ae;
    d4                  = x3 * nza + d1 * vz;
    FJacData[3 + nVar0] = c4 * y1 - d4 * vna;
    FJacData[3 + nVar1] = -c4 * vx + d4 * nxa;
    FJacData[3 + nVar2] = -c4 * vy + d4 * nya;
    FJacData[3 + nVar3] = -c4 * vz + d4 * nza + l1;
    FJacData[3 + nVar4] = c4;
    c5                  = c1 * h0 + d1 * vna * ae;
    d5                  = x3 * vna + d1 * h0;
    FJacData[4 + nVar0] = c5 * y1 - d5 * vna;
    FJacData[4 + nVar1] = -c5 * vx + d5 * nxa;
    FJacData[4 + nVar2] = -c5 * vy + d5 * nya;
    FJacData[4 + nVar3] = -c5 * vz + d5 * nza;
    FJacData[4 + nVar4] = c5 + l1;
}
 
bool CFSImplicit::v_UpdateTimeStepCheck()
{
    bool flag =
        (m_time + m_timestep > m_fintime && m_fintime > 0.0) ||
        (m_checktime && m_time + m_timestep - m_lastCheckTime >= m_checktime);
    if (m_explicitAdvection)
    {
        flag = true;
    }
    else
    {
        flag = flag || m_preconCfs->UpdatePreconMatCheck(NullNekDouble1DArray,
                                                         m_TimeIntegLambda);
    }
    return flag;
}
 
} // namespace Nektar