MMFSWE.cpp

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/////////////////////////////////////////////////////////////////////////////
//
// File MMFSWE.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: MMF solve routines
//
///////////////////////////////////////////////////////////////////////////////
 
#include <iomanip>
#include <iostream>
 
#include <boost/algorithm/string/predicate.hpp>
#include <boost/core/ignore_unused.hpp>
 
#include <LibUtilities/BasicUtils/Timer.h>
#include <LibUtilities/TimeIntegration/TimeIntegrationScheme.h>
#include <MultiRegions/AssemblyMap/AssemblyMapDG.h>
#include <ShallowWaterSolver/EquationSystems/MMFSWE.h>
 
namespace Nektar
{
std::string MMFSWE::className =
    SolverUtils::GetEquationSystemFactory().RegisterCreatorFunction(
        "MMFSWE", MMFSWE::create, "MMFSWE equation.");
 
MMFSWE::MMFSWE(const LibUtilities::SessionReaderSharedPtr &pSession,
               const SpatialDomains::MeshGraphSharedPtr &pGraph)
    : UnsteadySystem(pSession, pGraph), MMFSystem(pSession, pGraph)
{
    m_planeNumber = 0;
}
 
/**
 * @brief Initialisation object for the unsteady linear advection equation.
 */
void MMFSWE::v_InitObject(bool DeclareFields)
{
    // Call to the initialisation object
    UnsteadySystem::v_InitObject(DeclareFields);
 
    int nq       = m_fields[0]->GetNpoints();
    int shapedim = m_fields[0]->GetShapeDimension();
    Array<OneD, Array<OneD, NekDouble>> Anisotropy(shapedim);
    for (int j = 0; j < shapedim; ++j)
    {
        Anisotropy[j] = Array<OneD, NekDouble>(nq, 1.0);
    }
 
    MMFSystem::MMFInitObject(Anisotropy);
 
    // Load acceleration of gravity
    m_session->LoadParameter("Gravity", m_g, 9.81);
 
    // Add Coriolois effects
    m_session->LoadParameter("AddCoriolis", m_AddCoriolis, 1);
 
    // Add Rotation of the sphere along the pole
    m_session->LoadParameter("AddRotation", m_AddRotation, 1);
 
    m_session->LoadParameter("AddRossbyDisturbance", m_RossbyDisturbance, 0);
    m_session->LoadParameter("PurturbedJet", m_PurturbedJet, 0);
 
    // Define TestType
    ASSERTL0(m_session->DefinesSolverInfo("TESTTYPE"),
             "No TESTTYPE defined in session.");
    std::string TestTypeStr = m_session->GetSolverInfo("TESTTYPE");
    for (int i = 0; i < (int)SIZE_TestType; ++i)
    {
        if (boost::iequals(TestTypeMap[i], TestTypeStr))
        {
            m_TestType = (TestType)i;
            break;
        }
    }
 
    // Variable Setting for each test problem
    NekDouble gms = 9.80616;
 
    switch (m_TestType)
    {
        case eTestSteadyZonal:
        {
            NekDouble SecondToDay = 60.0 * 60.0 * 24.0;
            NekDouble rad_earth   = 6.37122 * 1000000;
            NekDouble Omegams;
 
            // Nondimensionalized coeffs.
            m_g = (gms * SecondToDay * SecondToDay) / rad_earth;
 
            m_session->LoadParameter("RotationAngle", m_alpha, 0.0);
            m_session->LoadParameter("u0", m_u0, 2.0 * m_pi / 12.0);
            m_session->LoadParameter("Omega", Omegams, 7.292 * 0.00001);
            m_Omega = Omegams * SecondToDay;
 
            m_session->LoadParameter("H0", m_H0, 2.94 * 10000);
            m_H0 = m_H0 / (rad_earth * gms);
 
            m_Hvar = (1.0 / m_g) * (m_Omega * m_u0 + 0.5 * m_u0 * m_u0);
        }
        break;
 
        case eTestUnsteadyZonal:
        {
            NekDouble SecondToDay = 60.0 * 60.0 * 24.0;
            NekDouble rad_earth   = 6.37122 * 1000000;
            NekDouble Omegams;
 
            // Nondimensionalized coeffs.
            m_g = (gms * SecondToDay * SecondToDay) / rad_earth;
 
            m_session->LoadParameter("RotationAngle", m_alpha, 0.0);
            m_session->LoadParameter("u0", m_u0, 2.0 * m_pi / 12.0);
            m_session->LoadParameter("Omega", Omegams, 7.292 * 0.00001);
            m_Omega = Omegams * SecondToDay;
 
            m_H0 = 133681.0 / (rad_earth * gms); // m^2 / s^2
            m_k2 = 10.0 / (rad_earth * gms);     // m^2 / s^2
        }
        break;
 
        case eTestIsolatedMountain:
        {
            NekDouble SecondToDay = 60.0 * 60.0 * 24.0;
            NekDouble rad_earth   = 6.37122 * 1000000;
            NekDouble Omegams;
 
            // Nondimensionalized coeffs.
            m_g = (gms * SecondToDay * SecondToDay) / rad_earth;
 
            m_session->LoadParameter("RotationAngle", m_alpha, 0.0);
 
            m_session->LoadParameter("u0", m_u0, 20.0);
            m_u0 = m_u0 * SecondToDay / rad_earth;
 
            m_session->LoadParameter("Omega", Omegams, 7.292 * 0.00001);
            m_Omega = Omegams * SecondToDay;
 
            m_H0  = 5960.0 / rad_earth;
            m_hs0 = 2000.0 / rad_earth;
        }
        break;
 
        case eTestUnstableJet:
        {
            NekDouble SecondToDay = 60.0 * 60.0 * 24.0;
            NekDouble rad_earth   = 6.37122 * 1000000;
            NekDouble Omegams;
 
            // Nondimensionalized coeffs.
            m_g = (gms * SecondToDay * SecondToDay) / rad_earth;
 
            m_session->LoadParameter("u0", m_u0, 80.0);
            m_u0 = m_u0 * SecondToDay / rad_earth;
 
            m_session->LoadParameter("Omega", Omegams, 7.292 * 0.00001);
            m_Omega = Omegams * SecondToDay;
 
            m_session->LoadParameter("H0", m_H0, 10000.0);
            m_H0 = m_H0 / rad_earth;
 
            m_uthetamax = 80 * SecondToDay / rad_earth;
            m_theta0    = m_pi / 7.0;
            m_theta1    = m_pi / 2.0 - m_theta0;
            m_en   = exp(-4.0 / (m_theta1 - m_theta0) / (m_theta1 - m_theta0));
            m_hbar = 120.0 / rad_earth;
 
            std::cout << "m_theta0 = " << m_theta0
                      << ", m_theta1 = " << m_theta1 << ", m_en = " << m_en
                      << ", m_hbar = " << m_hbar << std::endl;
        }
        break;
 
        case eTestRossbyWave:
        {
            NekDouble SecondToDay = 60.0 * 60.0 * 24.0;
            NekDouble rad_earth   = 6.37122 * 1000000;
            NekDouble Omegams;
 
            // Nondimensionalized coeffs.
            m_g = (gms * SecondToDay * SecondToDay) / rad_earth;
 
            m_session->LoadParameter("Omega", Omegams, 7.292 * 0.00001);
            m_Omega = Omegams * SecondToDay;
 
            m_session->LoadParameter("H0", m_H0, 8000.0);
            m_H0 = m_H0 / rad_earth;
 
            m_angfreq = 7.848 * 0.000001 * SecondToDay;
            m_K       = 7.848 * 0.000001 * SecondToDay;
        }
        break;
 
        default:
            break;
    }
 
    // TestVorticityComputation
    if (m_surfaceType == SolverUtils::eSphere)
    {
        TestVorticityComputation();
    }
 
    // If explicit it computes RHS and PROJECTION for the time integration
    if (m_explicitAdvection)
    {
        m_ode.DefineOdeRhs(&MMFSWE::DoOdeRhs, this);
        m_ode.DefineProjection(&MMFSWE::DoOdeProjection, this);
    }
    // Otherwise it gives an error (no implicit integration)
    else
    {
        ASSERTL0(false, "Implicit unsteady Advection not set up.");
    }
}
 
/**
 * @brief Unsteady linear advection equation destructor.
 */
MMFSWE::~MMFSWE()
{
}
 
void MMFSWE::v_DoSolve()
{
    ASSERTL0(m_intScheme != 0, "No time integration scheme.");
 
    int i, nchk = 1;
    int nvariables = 0;
    int nfields    = m_fields.size();
    int nq         = m_fields[0]->GetNpoints();
 
    if (m_intVariables.empty())
    {
        for (i = 0; i < nfields; ++i)
        {
            m_intVariables.push_back(i);
        }
        nvariables = nfields;
    }
    else
    {
        nvariables = m_intVariables.size();
    }
 
    // Set up wrapper to fields data storage.
    Array<OneD, Array<OneD, NekDouble>> fields(nvariables);
    Array<OneD, Array<OneD, NekDouble>> tmp(nvariables);
 
    // Order storage to list time-integrated fields first.
    for (i = 0; i < nvariables; ++i)
    {
        fields[i] = m_fields[m_intVariables[i]]->GetPhys();
        m_fields[m_intVariables[i]]->SetPhysState(false);
    }
 
    // Initialise time integration scheme
    m_intScheme->InitializeScheme(m_timestep, fields, m_time, m_ode);
 
    // Check uniqueness of checkpoint output
    ASSERTL0((m_checktime == 0.0 && m_checksteps == 0) ||
                 (m_checktime > 0.0 && m_checksteps == 0) ||
                 (m_checktime == 0.0 && m_checksteps > 0),
             "Only one of IO_CheckTime and IO_CheckSteps "
             "should be set!");
 
    LibUtilities::Timer timer;
    bool doCheckTime  = false;
    int step          = 0;
    NekDouble intTime = 0.0;
    NekDouble cpuTime = 0.0;
    NekDouble elapsed = 0.0;
 
    NekDouble Mass = 0.0, Energy = 0.0, Enstrophy = 0.0, Vorticity = 0.0;
    Array<OneD, NekDouble> zeta(nq);
    Array<OneD, Array<OneD, NekDouble>> fieldsprimitive(nvariables);
    for (int i = 0; i < nvariables; ++i)
    {
        fieldsprimitive[i] = Array<OneD, NekDouble>(nq);
    }
 
    while (step < m_steps || m_time < m_fintime - NekConstants::kNekZeroTol)
    {
        timer.Start();
        fields = m_intScheme->TimeIntegrate(step, m_timestep, m_ode);
        timer.Stop();
 
        m_time += m_timestep;
        elapsed = timer.TimePerTest(1);
        intTime += elapsed;
        cpuTime += elapsed;
 
        // Write out status information
        if (m_session->GetComm()->GetRank() == 0 && !((step + 1) % m_infosteps))
        {
            std::cout << "Steps: " << std::setw(8) << std::left << step + 1
                      << " "
                      << "Time: " << std::setw(12) << std::left << m_time;
 
            std::stringstream ss;
            ss << cpuTime << "s";
            std::cout << " CPU Time: " << std::setw(8) << std::left << ss.str()
                      << std::endl;
 
            // Printout Mass, Energy, Enstrophy
            ConservativeToPrimitive(fields, fieldsprimitive);
 
            // Vorticity zeta
            ComputeVorticity(fieldsprimitive[1], fieldsprimitive[2], zeta);
            Vorticity = std::abs(m_fields[0]->Integral(zeta) - m_Vorticity0);
 
            // Masss = h^*
            Mass = (ComputeMass(fieldsprimitive[0]) - m_Mass0) / m_Mass0;
 
            // Energy = 0.5*( h^*(u^2 + v^2) + g ( h^2 - h_s^s ) )
            Energy = (ComputeEnergy(fieldsprimitive[0], fieldsprimitive[1],
                                    fieldsprimitive[2]) -
                      m_Energy0) /
                     m_Energy0;
 
            // Enstrophy = 0.5/h^* ( \mathbf{k} \cdot (\nabla \times \mathbf{v}
            // ) + f )^2
            Enstrophy =
                (ComputeEnstrophy(fieldsprimitive[0], fieldsprimitive[1],
                                  fieldsprimitive[2]) -
                 m_Enstrophy0) /
                m_Enstrophy0;
 
            std::cout << "dMass = " << std::setw(8) << std::left << Mass << " "
                      << ", dEnergy = " << std::setw(8) << std::left << Energy
                      << " "
                      << ", dEnstrophy = " << std::setw(8) << std::left
                      << Enstrophy << " "
                      << ", dVorticity = " << std::setw(8) << std::left
                      << Vorticity << std::endl
                      << std::endl;
 
            cpuTime = 0.0;
        }
 
        // (h, hu, hv) -> (\eta, u, v)
        ConservativeToPrimitive();
 
        // Transform data into coefficient space
        for (i = 0; i < nvariables; ++i)
        {
            m_fields[m_intVariables[i]]->SetPhys(fields[i]);
            m_fields[m_intVariables[i]]->FwdTransLocalElmt(
                fields[i], m_fields[m_intVariables[i]]->UpdateCoeffs());
            m_fields[m_intVariables[i]]->SetPhysState(false);
        }
 
        // Write out checkpoint files
        if ((m_checksteps && step && !((step + 1) % m_checksteps)) ||
            doCheckTime)
        {
            Checkpoint_Output(nchk++);
            doCheckTime = false;
 
            //  (\eta, u, v) -> (h, hu, hv)
            PrimitiveToConservative();
        }
 
        // Step advance
        ++step;
    }
 
    // Print out summary statistics
    if (m_session->GetComm()->GetRank() == 0)
    {
        if (m_cflSafetyFactor > 0.0)
        {
            std::cout << "CFL safety factor : " << m_cflSafetyFactor
                      << std::endl
                      << "CFL time-step     : " << m_timestep << std::endl;
        }
 
        if (m_session->GetSolverInfo("Driver") != "SteadyState")
        {
            std::cout << "Time-integration  : " << intTime << "s" << std::endl;
        }
    }
 
    for (i = 0; i < nvariables; ++i)
    {
        m_fields[m_intVariables[i]]->SetPhys(fields[i]);
        m_fields[m_intVariables[i]]->SetPhysState(true);
    }
 
    // (h, hu, hv) -> (\eta, u, v)
    ConservativeToPrimitive();
 
    for (i = 0; i < nvariables; ++i)
    {
        m_fields[i]->FwdTrans(m_fields[i]->GetPhys(),
                              m_fields[i]->UpdateCoeffs());
    }
}
 
void MMFSWE::DoOdeRhs(const Array<OneD, const Array<OneD, NekDouble>> &inarray,
                      Array<OneD, Array<OneD, NekDouble>> &outarray,
                      const NekDouble time)
{
    boost::ignore_unused(time);
 
    int i;
    int nvariables = inarray.size();
    int ncoeffs    = GetNcoeffs();
    int nq         = GetTotPoints();
 
    // inarray in physical space
    Array<OneD, Array<OneD, NekDouble>> physarray(nvariables);
    Array<OneD, Array<OneD, NekDouble>> modarray(nvariables);
 
    for (i = 0; i < nvariables; ++i)
    {
        physarray[i] = Array<OneD, NekDouble>(nq);
        modarray[i]  = Array<OneD, NekDouble>(ncoeffs);
    }
 
    // (h, hu, hv) -> (\eta, u, v)
    ConservativeToPrimitive(inarray, physarray);
 
    // Weak Directional Derivative
    WeakDGSWEDirDeriv(physarray, modarray);
    AddDivForGradient(physarray, modarray);
 
    for (i = 0; i < nvariables; ++i)
    {
        Vmath::Neg(ncoeffs, modarray[i], 1);
    }
 
    // coriolis forcing
    if (m_AddCoriolis)
    {
        AddCoriolis(physarray, modarray);
    }
 
    // Bottom Elevation Effect
    // Add  g H \nabla m_depth
    AddElevationEffect(physarray, modarray);
 
    // Add terms concerning the rotation of the moving frame
    if (m_AddRotation)
    {
        AddRotation(physarray, modarray);
    }
 
    for (i = 0; i < nvariables; ++i)
    {
        m_fields[i]->MultiplyByElmtInvMass(modarray[i], modarray[i]);
        m_fields[i]->BwdTrans(modarray[i], outarray[i]);
    }
}
 
void MMFSWE::WeakDGSWEDirDeriv(
    const Array<OneD, Array<OneD, NekDouble>> &InField,
    Array<OneD, Array<OneD, NekDouble>> &OutField)
{
    int i;
    int nq              = GetNpoints();
    int ncoeffs         = GetNcoeffs();
    int nTracePointsTot = GetTraceNpoints();
    int nvariables      = m_fields.size();
 
    Array<OneD, Array<OneD, NekDouble>> fluxvector(m_shapedim);
    Array<OneD, Array<OneD, NekDouble>> physfield(nvariables);
 
    for (i = 0; i < m_shapedim; ++i)
    {
        fluxvector[i] = Array<OneD, NekDouble>(nq);
    }
 
    // InField is Primitive
    for (i = 0; i < nvariables; ++i)
    {
        physfield[i] = InField[i];
    }
 
    // Get the ith component of the  flux vector in (physical space)
    // fluxvector[0] = component for e^1 cdot \nabla \varphi
    // fluxvector[1] = component for e^2 cdot \nabla \varphi
    Array<OneD, NekDouble> fluxtmp(nq);
    Array<OneD, NekDouble> tmp(ncoeffs);
 
    // Compute Divergence Components
    for (i = 0; i < nvariables; ++i)
    {
        GetSWEFluxVector(i, physfield, fluxvector);
 
        OutField[i] = Array<OneD, NekDouble>(ncoeffs, 0.0);
        for (int j = 0; j < m_shapedim; ++j)
        {
            // Directional derivation with respect to the j'th moving frame
            // tmp_j = ( \nabla \phi, fluxvector[j] \mathbf{e}^j )
            m_fields[i]->IProductWRTDirectionalDerivBase(m_movingframes[j],
                                                         fluxvector[j], tmp);
            Vmath::Vadd(ncoeffs, &tmp[0], 1, &OutField[i][0], 1,
                        &OutField[i][0], 1);
        }
    }
 
    // Numerical Flux
    Array<OneD, Array<OneD, NekDouble>> numfluxFwd(nvariables);
    Array<OneD, Array<OneD, NekDouble>> numfluxBwd(nvariables);
 
    for (i = 0; i < nvariables; ++i)
    {
        numfluxFwd[i] = Array<OneD, NekDouble>(nTracePointsTot);
        numfluxBwd[i] = Array<OneD, NekDouble>(nTracePointsTot);
    }
 
    NumericalSWEFlux(physfield, numfluxFwd, numfluxBwd);
 
    // Evaulate  <\phi, \hat{F}\cdot n> - OutField[i]
    for (i = 0; i < nvariables; ++i)
    {
        Vmath::Neg(ncoeffs, OutField[i], 1);
        m_fields[i]->AddFwdBwdTraceIntegral(numfluxFwd[i], numfluxBwd[i],
                                            OutField[i]);
        m_fields[i]->SetPhysState(false);
    }
}
 
// Substract 0.5 * g * H * H  / || e^m ||^2 \nalba \cdot e^m
void MMFSWE::AddDivForGradient(Array<OneD, Array<OneD, NekDouble>> &physarray,
                               Array<OneD, Array<OneD, NekDouble>> &outarray)
{
    // routine works for both primitive and conservative formulations
    int ncoeffs = outarray[0].size();
    int nq      = physarray[0].size();
 
    Array<OneD, NekDouble> h(nq);
    Array<OneD, NekDouble> tmp(nq);
    Array<OneD, Array<OneD, NekDouble>> fluxvector(m_shapedim);
    for (int i = 0; i < m_shapedim; ++i)
    {
        fluxvector[i] = Array<OneD, NekDouble>(nq);
    }
 
    // Get 0.5 g H*H / || e^m ||^2
    GetSWEFluxVector(3, physarray, fluxvector);
 
    Array<OneD, NekDouble> tmpc(ncoeffs);
    for (int j = 0; j < m_shapedim; ++j)
    {
        Vmath::Vmul(nq, &fluxvector[0][0], 1, &m_DivMF[j][0], 1, &tmp[0], 1);
 
        Vmath::Neg(nq, &tmp[0], 1);
        m_fields[0]->IProductWRTBase(tmp, tmpc);
 
        Vmath::Vadd(ncoeffs, outarray[j + 1], 1, tmpc, 1, outarray[j + 1], 1);
    }
}
 
void MMFSWE::GetSWEFluxVector(
    const int i, const Array<OneD, const Array<OneD, NekDouble>> &physfield,
    Array<OneD, Array<OneD, NekDouble>> &flux)
{
    int nq = m_fields[0]->GetTotPoints();
 
    switch (i)
    {
        // flux function for the h equation = [(\eta + d ) u, (\eta + d) v ]
        case 0:
        {
            // h in flux 1
            Vmath::Vadd(nq, physfield[0], 1, m_depth, 1, flux[1], 1);
 
            // hu in flux 0
            Vmath::Vmul(nq, flux[1], 1, physfield[1], 1, flux[0], 1);
 
            // hv in flux 1
            Vmath::Vmul(nq, flux[1], 1, physfield[2], 1, flux[1], 1);
        }
        break;
 
        // flux function for the hu equation = [Hu^2 + 0.5 g H^2, Huv]
        case 1:
        {
            Array<OneD, NekDouble> tmp(nq);
 
            // h in tmp
            Vmath::Vadd(nq, physfield[0], 1, m_depth, 1, tmp, 1);
 
            // hu in flux 1
            Vmath::Vmul(nq, tmp, 1, physfield[1], 1, flux[1], 1);
 
            // huu in flux 0
            Vmath::Vmul(nq, flux[1], 1, physfield[1], 1, flux[0], 1);
 
            //  hh in tmp
            Vmath::Vmul(nq, tmp, 1, tmp, 1, tmp, 1);
 
            // huu + 0.5 g hh in flux 0
            // Daxpy overwrites flux[0] on exit
            Blas::Daxpy(nq, 0.5 * m_g, tmp, 1, flux[0], 1);
 
            // huv in flux 1
            Vmath::Vmul(nq, flux[1], 1, physfield[2], 1, flux[1], 1);
        }
        break;
 
        // flux function for the hv equation = [Huv, Hv^2]
        case 2:
        {
            Array<OneD, NekDouble> tmp(nq);
 
            // h in tmp
            Vmath::Vadd(nq, physfield[0], 1, m_depth, 1, tmp, 1);
 
            // hv in flux 0
            Vmath::Vmul(nq, tmp, 1, physfield[2], 1, flux[0], 1);
 
            // hvv in flux 1
            Vmath::Vmul(nq, flux[0], 1, physfield[2], 1, flux[1], 1);
 
            // huv in flux 0
            Vmath::Vmul(nq, flux[0], 1, physfield[1], 1, flux[0], 1);
 
            //  hh in tmp
            Vmath::Vmul(nq, tmp, 1, tmp, 1, tmp, 1);
 
            // hvv + 0.5 g hh in flux 1
            Blas::Daxpy(nq, 0.5 * m_g, tmp, 1, flux[1], 1);
        }
        break;
 
        // flux function 0.5 g h * h
        case 3:
        {
            Array<OneD, NekDouble> h(nq);
            flux[0] = Array<OneD, NekDouble>(nq, 0.0);
 
            // h in tmp
            Vmath::Vadd(nq, physfield[0], 1, m_depth, 1, h, 1);
 
            //  hh in tmp
            Vmath::Vmul(nq, h, 1, h, 1, h, 1);
 
            // 0.5 g hh in flux 0
            Blas::Daxpy(nq, 0.5 * m_g, h, 1, flux[0], 1);
        }
        break;
 
        default:
            break;
    }
}
 
void MMFSWE::NumericalSWEFlux(Array<OneD, Array<OneD, NekDouble>> &physfield,
                              Array<OneD, Array<OneD, NekDouble>> &numfluxFwd,
                              Array<OneD, Array<OneD, NekDouble>> &numfluxBwd)
{
 
    int i, k;
    int nTraceNumPoints = GetTraceTotPoints();
    int nvariables      = 3; // only the dependent variables
 
    // get temporary arrays
    Array<OneD, Array<OneD, NekDouble>> Fwd(nvariables);
    Array<OneD, Array<OneD, NekDouble>> Bwd(nvariables);
    Array<OneD, NekDouble> DepthFwd(nTraceNumPoints);
    Array<OneD, NekDouble> DepthBwd(nTraceNumPoints);
 
    for (i = 0; i < nvariables; ++i)
    {
        Fwd[i] = Array<OneD, NekDouble>(nTraceNumPoints);
        Bwd[i] = Array<OneD, NekDouble>(nTraceNumPoints);
    }
 
    // get the physical values at the trace
    for (i = 0; i < nvariables; ++i)
    {
        m_fields[i]->GetFwdBwdTracePhys(physfield[i], Fwd[i], Bwd[i]);
 
        // Copy Fwd to Bwd at boundaries
        CopyBoundaryTrace(Fwd[i], Bwd[i], SolverUtils::eFwdEQBwd);
    }
    m_fields[0]->GetFwdBwdTracePhys(m_depth, DepthFwd, DepthBwd);
    CopyBoundaryTrace(DepthFwd, DepthBwd, SolverUtils::eFwdEQBwd);
 
    // note that we are using the same depth - i.e. the depth is assumed
    // continuous...
    switch (m_upwindType)
    {
        case SolverUtils::eHLLC:
        {
            // rotate the values to the normal direction
            NekDouble tmpX, tmpY;
 
            // Fwd[1] = hu^+ = ( h^1^+ (e^1 \cdot n) + h^2^+ ( e^2 \cdot n) )
            // Fwd[2] = hv^+ = ( h^1^+ (e^1 \cdot n^{\perp}) + h^2^+ ( e^2 \cdot
            // n^{\perp}) )
            // Bwd[1] = hu^- = ( h^1^- (e^1 \cdot n) + h^2^- ( e^2 \cdot n) )
            // Bwd[2] = hv^- = ( h^1^- (e^1 \cdot n^{\perp}) + h^2^- ( e^2 \cdot
            // n^{\perp}) )
 
            for (k = 0; k < nTraceNumPoints; ++k)
            {
                tmpX = Fwd[1][k] * m_ncdotMFFwd[0][k] +
                       Fwd[2][k] * m_ncdotMFFwd[1][k];
                tmpY = Fwd[1][k] * m_nperpcdotMFFwd[0][k] +
                       Fwd[2][k] * m_nperpcdotMFFwd[1][k];
                Fwd[1][k] = tmpX;
                Fwd[2][k] = tmpY;
 
                tmpX = Bwd[1][k] * m_ncdotMFBwd[0][k] +
                       Bwd[2][k] * m_ncdotMFBwd[1][k];
                tmpY = Bwd[1][k] * m_nperpcdotMFBwd[0][k] +
                       Bwd[2][k] * m_nperpcdotMFBwd[1][k];
                Bwd[1][k] = tmpX;
                Bwd[2][k] = tmpY;
            }
 
            // Solve the Riemann problem
            NekDouble denomFwd, denomBwd;
            Array<OneD, NekDouble> numfluxF(nvariables);
            Array<OneD, NekDouble> numfluxB(nvariables);
 
            NekDouble eF1n, eF2n, eB1n, eB2n;
            NekDouble eF1t, eF2t, eB1t, eB2t;
            for (k = 0; k < nTraceNumPoints; ++k)
            {
                RiemannSolverHLLC(k, Fwd[0][k] + DepthFwd[k], Fwd[1][k],
                                  Fwd[2][k], Bwd[0][k] + DepthFwd[k], Bwd[1][k],
                                  Bwd[2][k], numfluxF, numfluxB);
 
                // uflux = 1/[ ( e^1 \cdot n) ( e^2 \cdot n^{\perp} ) - (e^2
                // \cdot n)(e^1 \cdot n^{\perp} ) ] ( e^2 \cdot n^{\perp} huflux
                // - e^2 \cdot n hvflux
                // vflux = -1/[ ( e^1 \cdot n) ( e^2 \cdot n^{\perp} ) - (e^2
                // \cdot n)(e^1 \cdot n^{\perp} ) ] ( -e^1 \cdot n^{\perp}
                // huflux + e^1 \cdot n hvflux
                eF1n = m_ncdotMFFwd[0][k];
                eF2n = m_ncdotMFFwd[1][k];
                eB1n = m_ncdotMFBwd[0][k];
                eB2n = m_ncdotMFBwd[1][k];
 
                eF1t = m_nperpcdotMFFwd[0][k];
                eF2t = m_nperpcdotMFFwd[1][k];
                eB1t = m_nperpcdotMFBwd[0][k];
                eB2t = m_nperpcdotMFBwd[1][k];
 
                denomFwd = eF1n * eF2t - eF2n * eF1t;
                denomBwd = eB1n * eB2t - eB2n * eB1t;
 
                numfluxFwd[0][k] = numfluxF[0];
                numfluxFwd[1][k] = (1.0 / denomFwd) *
                                   (eF2t * numfluxF[1] - eF2n * numfluxF[2]);
                numfluxFwd[2][k] =
                    (1.0 / denomFwd) *
                    (-1.0 * eF1t * numfluxF[1] + eF1n * numfluxF[2]);
 
                numfluxBwd[0][k] = 1.0 * numfluxB[0];
                numfluxBwd[1][k] = (1.0 / denomBwd) *
                                   (eB2t * numfluxB[1] - eB2n * numfluxB[2]);
                numfluxBwd[2][k] =
                    (1.0 / denomBwd) *
                    (-1.0 * eB1t * numfluxB[1] + eB1n * numfluxB[2]);
            }
        }
        break;
 
        case SolverUtils::eAverage:
        case SolverUtils::eLaxFriedrich:
        case SolverUtils::eRusanov:
        {
            Array<OneD, NekDouble> numfluxF(nvariables * (m_shapedim + 1));
            Array<OneD, NekDouble> numfluxB(nvariables * (m_shapedim + 1));
 
            for (k = 0; k < nTraceNumPoints; ++k)
            {
                if (m_upwindType == SolverUtils::eAverage)
                {
                    AverageFlux(k, Fwd[0][k] + DepthFwd[k], Fwd[1][k],
                                Fwd[2][k], Bwd[0][k] + DepthFwd[k], Bwd[1][k],
                                Bwd[2][k], numfluxF, numfluxB);
                }
 
                else if (m_upwindType == SolverUtils::eLaxFriedrich)
                {
                    LaxFriedrichFlux(k, Fwd[0][k] + DepthFwd[k], Fwd[1][k],
                                     Fwd[2][k], Bwd[0][k] + DepthFwd[k],
                                     Bwd[1][k], Bwd[2][k], numfluxF, numfluxB);
                }
 
                else if (m_upwindType == SolverUtils::eRusanov)
                {
                    RusanovFlux(k, Fwd[0][k] + DepthFwd[k], Fwd[1][k],
                                Fwd[2][k], Bwd[0][k] + DepthFwd[k], Bwd[1][k],
                                Bwd[2][k], numfluxF, numfluxB);
                }
 
                int indx;
                NekDouble tmpF0, tmpF1, tmpB0, tmpB1;
                for (i = 0; i < nvariables; ++i)
                {
                    indx = i * (m_shapedim + 1);
 
                    tmpF0 = numfluxF[indx] * m_ncdotMFFwd[0][k];
                    tmpF1 = numfluxF[indx + 1] * m_ncdotMFFwd[1][k];
 
                    tmpB0 = numfluxB[indx] * m_ncdotMFBwd[0][k];
                    tmpB1 = numfluxB[indx + 1] * m_ncdotMFBwd[1][k];
 
                    numfluxFwd[i][k] = tmpF0 + tmpF1 + numfluxF[indx + 2];
                    numfluxBwd[i][k] = tmpB0 + tmpB1 + numfluxB[indx + 2];
                }
            }
        }
        break;
 
        default:
        {
            ASSERTL0(false, "populate switch statement for upwind flux");
        }
        break;
    }
}
 
void MMFSWE::RiemannSolverHLLC(const int index, NekDouble hL, NekDouble uL,
                               NekDouble vL, NekDouble hR, NekDouble uR,
                               NekDouble vR, Array<OneD, NekDouble> &numfluxF,
                               Array<OneD, NekDouble> &numfluxB)
{
    boost::ignore_unused(index);
 
    NekDouble g = m_g;
 
    NekDouble cL = sqrt(g * hL);
    NekDouble cR = sqrt(g * hR);
 
    NekDouble uRF, vRF, uLB, vLB;
    NekDouble hstarF, hstarB;
 
    // Temporary assignments
    uRF = uR;
    vRF = vR;
    uLB = uL;
    vLB = vL;
 
    // the two-rarefaction wave assumption
    hstarF = 0.5 * (cL + cR) + 0.25 * (uL - uRF);
    hstarF *= hstarF;
    hstarF *= (1.0 / g);
 
    hstarB = 0.5 * (cL + cR) + 0.25 * (uLB - uR);
    hstarB *= hstarB;
    hstarB *= (1.0 / g);
 
    NekDouble hfluxF, hufluxF, hvfluxF;
    NekDouble hfluxB, hufluxB, hvfluxB;
    Computehhuhvflux(hL, uL, vL, hR, uRF, vRF, hstarF, hfluxF, hufluxF,
                     hvfluxF);
    Computehhuhvflux(hL, uLB, vLB, hR, uR, vR, hstarB, hfluxB, hufluxB,
                     hvfluxB);
 
    numfluxF[0] = hfluxF;
    numfluxF[1] = hufluxF;
    numfluxF[2] = hvfluxF;
 
    numfluxB[0] = hfluxB;
    numfluxB[1] = hufluxB;
    numfluxB[2] = hvfluxB;
}
 
void MMFSWE::Computehhuhvflux(NekDouble hL, NekDouble uL, NekDouble vL,
                              NekDouble hR, NekDouble uR, NekDouble vR,
                              NekDouble hstar, NekDouble &hflux,
                              NekDouble &huflux, NekDouble &hvflux)
{
    NekDouble g = m_g;
 
    NekDouble hC, huC, hvC, SL, SR, Sstar;
    NekDouble cL = sqrt(g * hL);
    NekDouble cR = sqrt(g * hR);
 
    // Compute SL
    if (hstar > hL)
        SL = uL - cL * sqrt(0.5 * ((hstar * hstar + hstar * hL) / (hL * hL)));
    else
        SL = uL - cL;
 
    // Compute SR
    if (hstar > hR)
        SR = uR + cR * sqrt(0.5 * ((hstar * hstar + hstar * hR) / (hR * hR)));
    else
        SR = uR + cR;
 
    if (fabs(hR * (uR - SR) - hL * (uL - SL)) <= 1.0e-15)
        Sstar = 0.0;
    else
        Sstar = (SL * hR * (uR - SR) - SR * hL * (uL - SL)) /
                (hR * (uR - SR) - hL * (uL - SL));
 
    if (SL >= 0)
    {
        hflux  = hL * uL;
        huflux = uL * uL * hL + 0.5 * g * hL * hL;
        hvflux = hL * uL * vL;
    }
    else if (SR <= 0)
    {
        hflux  = hR * uR;
        huflux = uR * uR * hR + 0.5 * g * hR * hR;
        hvflux = hR * uR * vR;
    }
    else
    {
        if ((SL < 0) && (Sstar >= 0))
        {
            hC  = hL * ((SL - uL) / (SL - Sstar));
            huC = hC * Sstar;
            hvC = hC * vL;
 
            hflux  = hL * uL + SL * (hC - hL);
            huflux = (uL * uL * hL + 0.5 * g * hL * hL) + SL * (huC - hL * uL);
            hvflux = (uL * vL * hL) + SL * (hvC - hL * vL);
        }
        else
        {
            hC  = hR * ((SR - uR) / (SR - Sstar));
            huC = hC * Sstar;
            hvC = hC * vR;
 
            hflux  = hR * uR + SR * (hC - hR);
            huflux = (uR * uR * hR + 0.5 * g * hR * hR) + SR * (huC - hR * uR);
            hvflux = (uR * vR * hR) + SR * (hvC - hR * vR);
        }
    }
}
 
void MMFSWE::AverageFlux(const int index, NekDouble hL, NekDouble uL,
                         NekDouble vL, NekDouble hR, NekDouble uR, NekDouble vR,
                         Array<OneD, NekDouble> &numfluxF,
                         Array<OneD, NekDouble> &numfluxB)
{
    NekDouble MageF1, MageF2, MageB1, MageB2;
    NekDouble eF1_cdot_eB1, eF1_cdot_eB2;
    NekDouble eF2_cdot_eB1, eF2_cdot_eB2;
 
    NekDouble g = m_g;
    NekDouble uRF, vRF, uLB, vLB;
 
    ComputeMagAndDot(index, MageF1, MageF2, MageB1, MageB2, eF1_cdot_eB1,
                     eF1_cdot_eB2, eF2_cdot_eB1, eF2_cdot_eB2);
 
    // uRF = uR component in moving frames e^{Fwd}
    // vRF = vR component in moving frames e^{Fwd}
    uRF = (uR * eF1_cdot_eB1 + vR * eF1_cdot_eB2) / MageF1;
    vRF = (uR * eF2_cdot_eB1 + vR * eF2_cdot_eB2) / MageF2;
 
    numfluxF[0] = 0.5 * (hL * uL + hR * uRF);
    numfluxF[1] = 0.5 * (hL * vL + hR * vRF);
    numfluxF[2] = 0.0;
 
    numfluxF[3] =
        0.5 * (hL * uL * uL + hR * uRF * uRF + 0.5 * g * (hL * hL + hR * hR));
    numfluxF[4] = 0.5 * (hL * uL * vL + hR * uRF * vRF);
    numfluxF[5] = 0.0;
 
    numfluxF[6] = 0.5 * (hL * uL * vL + hR * uRF * vRF);
    numfluxF[7] =
        0.5 * (hL * vL * vL + hR * vRF * vRF + 0.5 * g * (hL * hL + hR * hR));
    numfluxF[8] = 0.0;
 
    // uLB = uL component in moving frames e^{Bwd}
    // vLB = vL component in moving frames e^{Bwd}
    uLB = (uL * eF1_cdot_eB1 + vL * eF2_cdot_eB1) / MageB1;
    vLB = (uL * eF1_cdot_eB2 + vL * eF2_cdot_eB2) / MageB2;
 
    numfluxB[0] = 0.5 * (hR * uR + hR * uLB);
    numfluxB[1] = 0.5 * (hR * vR + hR * vLB);
    numfluxB[2] = 0.0;
 
    numfluxB[3] =
        0.5 * (hR * uR * uR + hR * uLB * uLB + 0.5 * g * (hR * hR + hL * hL));
    numfluxB[4] = 0.5 * (hR * uR * vR + hR * uLB * vLB);
    numfluxB[5] = 0.0;
 
    numfluxB[6] = 0.5 * (hR * uR * vR + hR * uLB * vLB);
    numfluxB[7] =
        0.5 * (hR * vR * vR + hR * vLB * vLB + 0.5 * g * (hR * hR + hL * hL));
    numfluxB[8] = 0.0;
}
 
void MMFSWE::LaxFriedrichFlux(const int index, NekDouble hL, NekDouble uL,
                              NekDouble vL, NekDouble hR, NekDouble uR,
                              NekDouble vR, Array<OneD, NekDouble> &numfluxF,
                              Array<OneD, NekDouble> &numfluxB)
{
    int nvariables = 3;
    NekDouble MageF1, MageF2, MageB1, MageB2;
    NekDouble eF1_cdot_eB1, eF1_cdot_eB2;
    NekDouble eF2_cdot_eB1, eF2_cdot_eB2;
 
    NekDouble g = m_g;
    NekDouble uRF, vRF, uLB, vLB;
    NekDouble velL, velR, lambdaF, lambdaB;
 
    Array<OneD, NekDouble> EigF(nvariables);
    Array<OneD, NekDouble> EigB(nvariables);
 
    // Compute Magnitude and Dot product of moving frames for the index
    ComputeMagAndDot(index, MageF1, MageF2, MageB1, MageB2, eF1_cdot_eB1,
                     eF1_cdot_eB2, eF2_cdot_eB1, eF2_cdot_eB2);
 
    // Get the velocity in the normal to the edge
    velL = uL * m_ncdotMFFwd[0][index] + vL * m_ncdotMFFwd[1][index];
    velR = -1.0 * (uR * m_ncdotMFBwd[0][index] + vR * m_ncdotMFBwd[1][index]);
 
    EigF[0] = velL - sqrt(g * hL);
    EigF[1] = velL;
    EigF[2] = velL + sqrt(g * hL);
 
    EigB[0] = velR - sqrt(g * hR);
    EigB[1] = velR;
    EigB[2] = velR + sqrt(g * hR);
 
    lambdaF = Vmath::Vamax(nvariables, EigF, 1);
    lambdaB = Vmath::Vamax(nvariables, EigB, 1);
 
    // uRF = uR component in moving frames e^{Fwd}
    // vRF = vR component in moving frames e^{Fwd}
    uRF = (uR * eF1_cdot_eB1 + vR * eF1_cdot_eB2) / MageF1;
    vRF = (uR * eF2_cdot_eB1 + vR * eF2_cdot_eB2) / MageF2;
 
    numfluxF[0] = 0.5 * (hL * uL + hR * uRF);
    numfluxF[1] = 0.5 * (hL * vL + hR * vRF);
    numfluxF[2] = 0.5 * lambdaF * (hL - hR);
 
    numfluxF[3] = 0.5 * (hL * uL * uL * MageF1 + hR * uRF * uRF * MageB1 +
                         0.5 * g * (hL * hL + hR * hR));
    numfluxF[4] = 0.5 * (hL * uL * vL * MageF1 + hR * uRF * vRF * MageB1);
    numfluxF[5] = 0.5 * lambdaF * (uL * hL - uRF * hR);
 
    numfluxF[6] = 0.5 * (hL * uL * vL * MageF2 + hR * uRF * vRF * MageB2);
    numfluxF[7] = 0.5 * (hL * vL * vL * MageF2 + hR * vRF * vRF * MageB2 +
                         0.5 * g * (hL * hL + hR * hR));
    numfluxF[8] = 0.5 * lambdaF * (vL * hL - vRF * hR);
 
    // uLB = uL component in moving frames e^{Bwd}
    // vLB = vL component in moving frames e^{Bwd}
    uLB = (uL * eF1_cdot_eB1 + vL * eF2_cdot_eB1) / MageB1;
    vLB = (uL * eF1_cdot_eB2 + vL * eF2_cdot_eB2) / MageB2;
 
    numfluxB[0] = 0.5 * (hR * uR + hR * uLB);
    numfluxB[1] = 0.5 * (hR * vR + hR * vLB);
    numfluxB[2] = 0.5 * lambdaB * (hL - hR);
 
    numfluxB[3] = 0.5 * (hR * uR * uR * MageB1 + hR * uLB * uLB * MageF1 +
                         0.5 * g * (hR * hR + hL * hL));
    numfluxB[4] = 0.5 * (hR * uR * vR * MageB1 + hR * uLB * vLB * MageF1);
    numfluxB[5] = 0.5 * lambdaB * (uLB * hL - uR * hR);
 
    numfluxB[6] = 0.5 * (hR * uR * vR * MageB2 + hR * uLB * vLB * MageF2);
    numfluxB[7] = 0.5 * (hR * vR * vR * MageB2 + hR * vLB * vLB * MageF2 +
                         0.5 * g * (hR * hR + hL * hL));
    numfluxB[8] = 0.5 * lambdaB * (vLB * hL - vR * hR);
}
 
void MMFSWE::RusanovFlux(const int index, NekDouble hL, NekDouble uL,
                         NekDouble vL, NekDouble hR, NekDouble uR, NekDouble vR,
                         Array<OneD, NekDouble> &numfluxF,
                         Array<OneD, NekDouble> &numfluxB)
{
    int nvariables = 3;
    NekDouble MageF1, MageF2, MageB1, MageB2;
    NekDouble eF1_cdot_eB1, eF1_cdot_eB2;
    NekDouble eF2_cdot_eB1, eF2_cdot_eB2;
 
    NekDouble g = m_g;
    NekDouble uRF, vRF, uLB, vLB;
    NekDouble velL, velR;
 
    Array<OneD, NekDouble> EigF(nvariables);
    Array<OneD, NekDouble> EigB(nvariables);
 
    // Compute Magnitude and Dot product of moving frames for the index
    ComputeMagAndDot(index, MageF1, MageF2, MageB1, MageB2, eF1_cdot_eB1,
                     eF1_cdot_eB2, eF2_cdot_eB1, eF2_cdot_eB2);
 
    // Get the velocity in the normal to the edge
    velL = uL * m_ncdotMFFwd[0][index] + vL * m_ncdotMFFwd[1][index];
    velR = -1.0 * (uR * m_ncdotMFBwd[0][index] + vR * m_ncdotMFBwd[1][index]);
 
    NekDouble SL, SR;
    SL = fabs(velL) + sqrt(g * hL);
    SR = fabs(velR) + sqrt(g * hR);
 
    NekDouble S;
    if (SL > SR)
        S = SL;
    else
        S = SR;
 
    // uRF = uR component in moving frames e^{Fwd}
    // vRF = vR component in moving frames e^{Fwd}
    uRF = (uR * eF1_cdot_eB1 + vR * eF1_cdot_eB2) / MageF1;
    vRF = (uR * eF2_cdot_eB1 + vR * eF2_cdot_eB2) / MageF2;
 
    numfluxF[0] = 0.5 * (hL * uL + hR * uRF);
    numfluxF[1] = 0.5 * (hL * vL + hR * vRF);
    numfluxF[2] = 0.5 * S * (hL - hR);
 
    numfluxF[3] =
        0.5 * (hL * uL * uL + hR * uRF * uRF + 0.5 * g * (hL * hL + hR * hR));
    numfluxF[4] = 0.5 * (hL * uL * vL + hR * uRF * vRF);
    numfluxF[5] = 0.5 * S * (uL * hL - uRF * hR);
 
    numfluxF[6] = 0.5 * (hL * uL * vL + hR * uRF * vRF);
    numfluxF[7] =
        0.5 * (hL * vL * vL + hR * vRF * vRF + 0.5 * g * (hL * hL + hR * hR));
    numfluxF[8] = 0.5 * S * (vL * hL - vRF * hR);
 
    // uLB = uL component in moving frames e^{Bwd}
    // vLB = vL component in moving frames e^{Bwd}
    uLB = (uL * eF1_cdot_eB1 + vL * eF2_cdot_eB1) / MageB1;
    vLB = (uL * eF1_cdot_eB2 + vL * eF2_cdot_eB2) / MageB2;
 
    numfluxB[0] = 0.5 * (hR * uR + hR * uLB);
    numfluxB[1] = 0.5 * (hR * vR + hR * vLB);
    numfluxB[2] = 0.5 * S * (hL - hR);
 
    numfluxB[3] =
        0.5 * (hR * uR * uR + hR * uLB * uLB + 0.5 * g * (hR * hR + hL * hL));
    numfluxB[4] = 0.5 * (hR * uR * vR + hR * uLB * vLB);
    numfluxB[5] = 0.5 * S * (uLB * hL - uR * hR);
 
    numfluxB[6] = 0.5 * (hR * uR * vR + hR * uLB * vLB);
    numfluxB[7] =
        0.5 * (hR * vR * vR + hR * vLB * vLB + 0.5 * g * (hR * hR + hL * hL));
    numfluxB[8] = 0.5 * S * (vLB * hL - vR * hR);
}
 
void MMFSWE::ComputeMagAndDot(const int index, NekDouble &MageF1,
                              NekDouble &MageF2, NekDouble &MageB1,
                              NekDouble &MageB2, NekDouble &eF1_cdot_eB1,
                              NekDouble &eF1_cdot_eB2, NekDouble &eF2_cdot_eB1,
                              NekDouble &eF2_cdot_eB2)
{
    NekDouble MF1x, MF1y, MF1z, MF2x, MF2y, MF2z;
    NekDouble MB1x, MB1y, MB1z, MB2x, MB2y, MB2z;
 
    MF1x = m_MFtraceFwd[0][0][index];
    MF1y = m_MFtraceFwd[0][1][index];
    MF1z = m_MFtraceFwd[0][2][index];
 
    MF2x = m_MFtraceFwd[1][0][index];
    MF2y = m_MFtraceFwd[1][1][index];
    MF2z = m_MFtraceFwd[1][2][index];
 
    MB1x = m_MFtraceBwd[0][0][index];
    MB1y = m_MFtraceBwd[0][1][index];
    MB1z = m_MFtraceBwd[0][2][index];
 
    MB2x = m_MFtraceBwd[1][0][index];
    MB2y = m_MFtraceBwd[1][1][index];
    MB2z = m_MFtraceBwd[1][2][index];
 
    // MFtrace = MFtrace [ j*spacedim + k ], j = shape, k = sapce
    MageF1 = MF1x * MF1x + MF1y * MF1y + MF1z * MF1z;
    MageF2 = MF2x * MF2x + MF2y * MF2y + MF2z * MF2z;
    MageB1 = MB1x * MB1x + MB1y * MB1y + MB1z * MB1z;
    MageB2 = MB2x * MB2x + MB2y * MB2y + MB2z * MB2z;
 
    eF1_cdot_eB1 = MF1x * MB1x + MF1y * MB1y + MF1z * MB1z;
    eF1_cdot_eB2 = MF1x * MB2x + MF1y * MB2y + MF1z * MB2z;
    eF2_cdot_eB1 = MF2x * MB1x + MF2y * MB1y + MF2z * MB1z;
    eF2_cdot_eB2 = MF2x * MB2x + MF2y * MB2y + MF2z * MB2z;
}
 
// Add Coriolis factors
void MMFSWE::AddCoriolis(Array<OneD, Array<OneD, NekDouble>> &physarray,
                         Array<OneD, Array<OneD, NekDouble>> &outarray)
{
    int ncoeffs = outarray[0].size();
    int nq      = physarray[0].size();
 
    Array<OneD, NekDouble> h(nq);
    Array<OneD, NekDouble> tmp(nq);
    Array<OneD, NekDouble> tmpc(ncoeffs);
 
    // physarray is primitive
    // conservative formulation compute h
    // h = \eta + d
    Vmath::Vadd(nq, physarray[0], 1, m_depth, 1, h, 1);
 
    int indx = 0;
    for (int j = 0; j < m_shapedim; ++j)
    {
        if (j == 0)
        {
            indx = 2;
        }
 
        else if (j == 1)
        {
            indx = 1;
        }
 
        // add to hu equation
        Vmath::Vmul(nq, m_coriolis, 1, physarray[indx], 1, tmp, 1);
        Vmath::Vmul(nq, h, 1, tmp, 1, tmp, 1);
 
        if (j == 1)
        {
            Vmath::Neg(nq, tmp, 1);
        }
 
        // N \cdot (e^1 \times e^2 )
        // Vmath::Vmul(nq, &m_MF1crossMF2dotSN[0], 1, &tmp[0], 1, &tmp[0], 1);
        m_fields[0]->IProductWRTBase(tmp, tmpc);
        Vmath::Vadd(ncoeffs, tmpc, 1, outarray[j + 1], 1, outarray[j + 1], 1);
    }
}
 
// Compuate g H \nabla m_depth
void MMFSWE::AddElevationEffect(Array<OneD, Array<OneD, NekDouble>> &physarray,
                                Array<OneD, Array<OneD, NekDouble>> &outarray)
{
    int ncoeffs = outarray[0].size();
    int nq      = physarray[0].size();
 
    Array<OneD, NekDouble> h(nq);
    Array<OneD, NekDouble> tmp(nq);
    Array<OneD, NekDouble> tmpc(ncoeffs);
 
    // physarray is primitive
    // conservative formulation compute h
    // h = \eta + d
    Vmath::Vadd(nq, physarray[0], 1, m_depth, 1, h, 1);
 
    for (int j = 0; j < m_shapedim; ++j)
    {
        Vmath::Vmul(nq, &h[0], 1, &m_Derivdepth[j][0], 1, &tmp[0], 1);
        Vmath::Smul(nq, m_g, tmp, 1, tmp, 1);
 
        m_fields[0]->IProductWRTBase(tmp, tmpc);
 
        Vmath::Vadd(ncoeffs, tmpc, 1, outarray[j + 1], 1, outarray[j + 1], 1);
    }
}
 
// =================================================
// Add rotational factors
// =================================================
void MMFSWE::AddRotation(Array<OneD, Array<OneD, NekDouble>> &physarray,
                         Array<OneD, Array<OneD, NekDouble>> &outarray)
{
    // routine works for both primitive and conservative formulations
    int ncoeffs = outarray[0].size();
    int nq      = physarray[0].size();
 
    // Compute h
    Array<OneD, NekDouble> h(nq);
    Vmath::Vadd(nq, &physarray[0][0], 1, &m_depth[0], 1, &h[0], 1);
 
    Array<OneD, NekDouble> de0dt_cdot_e0;
    Array<OneD, NekDouble> de0dt_cdot_e1;
    Array<OneD, NekDouble> de1dt_cdot_e0;
    Array<OneD, NekDouble> de1dt_cdot_e1;
    Compute_demdt_cdot_ek(0, 0, physarray, de0dt_cdot_e0);
    Compute_demdt_cdot_ek(1, 0, physarray, de1dt_cdot_e0);
    Compute_demdt_cdot_ek(0, 1, physarray, de0dt_cdot_e1);
    Compute_demdt_cdot_ek(1, 1, physarray, de1dt_cdot_e1);
 
    Array<OneD, NekDouble> Rott1(nq);
    Array<OneD, NekDouble> Rott2(nq);
    Vmath::Vmul(nq, physarray[1], 1, de0dt_cdot_e0, 1, Rott1, 1);
    Vmath::Vmul(nq, physarray[1], 1, de0dt_cdot_e1, 1, Rott2, 1);
    Vmath::Vvtvp(nq, physarray[2], 1, de1dt_cdot_e0, 1, Rott1, 1, Rott1, 1);
    Vmath::Vvtvp(nq, physarray[2], 1, de1dt_cdot_e1, 1, Rott2, 1, Rott2, 1);
 
    // Multiply H and \partial \phi / \partial t which is assumed to be u_{\phi}
    Vmath::Vmul(nq, &h[0], 1, &Rott1[0], 1, &Rott1[0], 1);
    Vmath::Vmul(nq, &h[0], 1, &Rott2[0], 1, &Rott2[0], 1);
 
    Vmath::Neg(nq, Rott1, 1);
    Vmath::Neg(nq, Rott2, 1);
 
    Array<OneD, NekDouble> tmpc1(ncoeffs);
    Array<OneD, NekDouble> tmpc2(ncoeffs);
    m_fields[0]->IProductWRTBase(Rott1, tmpc1);
    m_fields[0]->IProductWRTBase(Rott2, tmpc2);
 
    Vmath::Vadd(ncoeffs, tmpc1, 1, outarray[1], 1, outarray[1], 1);
    Vmath::Vadd(ncoeffs, tmpc2, 1, outarray[2], 1, outarray[2], 1);
}
 
// =====================================================================
// Compute \frac{d e^{indm}}{dt} \cdot e^{indk}  / \| e^{indk} \|
// Equivalently, [ \frac{d e^{indm}}{d \xi_1} \frac{ d \xi_1}{d \phi} + \frac{d
// e^{indm}}{d \xi_2} \frac{ d \xi_2}{d \phi} ] \frac{d \phi}{dt}
void MMFSWE::Compute_demdt_cdot_ek(
    const int indm, const int indk,
    const Array<OneD, const Array<OneD, NekDouble>> &physarray,
    Array<OneD, NekDouble> &outarray)
{
    int j, k;
    int nq = m_fields[0]->GetNpoints();
 
    Array<OneD, NekDouble> tmp(nq, 0.0);
    Array<OneD, NekDouble> tmpderiv(nq);
 
    outarray = Array<OneD, NekDouble>(nq, 0.0);
    for (j = 0; j < m_shapedim; ++j)
    {
        for (k = 0; k < m_spacedim; ++k)
        {
            // Compute d e^m / d \xi_1 and d e^m / d \xi_2
            Vmath::Vcopy(nq, &m_movingframes[indm][k * nq], 1, &tmp[0], 1);
            m_fields[0]->PhysDirectionalDeriv(m_movingframes[j], tmp, tmpderiv);
 
            Vmath::Vmul(nq, &physarray[j + 1][0], 1, &tmpderiv[0], 1,
                        &tmpderiv[0], 1);
 
            Vmath::Vvtvp(nq, &tmpderiv[0], 1, &m_movingframes[indk][k * nq], 1,
                         &outarray[0], 1, &outarray[0], 1);
        }
    }
}
 
/**
 * @brief Compute the projection for the linear advection equation.
 *
 * @param inarray    Given fields.
 * @param outarray   Calculated solution.
 * @param time       Time.
 */
void MMFSWE::DoOdeProjection(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray, const NekDouble time)
{
    switch (m_projectionType)
    {
        case MultiRegions::eDiscontinuous:
        {
            ConservativeToPrimitive(inarray, outarray);
            SetBoundaryConditions(outarray, time);
            PrimitiveToConservative(outarray, outarray);
        }
        break;
        default:
            ASSERTL0(false, "Unknown projection scheme");
            break;
    }
}
 
void MMFSWE::SetBoundaryConditions(Array<OneD, Array<OneD, NekDouble>> &inarray,
                                   NekDouble time)
{
 
    int nvariables = m_fields.size();
    int cnt        = 0;
 
    // loop over Boundary Regions
    for (int n = 0; n < m_fields[0]->GetBndConditions().size(); ++n)
    {
 
        // Zonal Boundary Condition
        if (m_fields[0]->GetBndConditions()[n]->GetUserDefined() == "eMG")
        {
            if (m_expdim == 1)
            {
                ASSERTL0(false, "Illegal dimension");
            }
            else if (m_expdim == 2)
            {
                // ZonalBoundary2D(n,cnt,inarray);
            }
        }
 
        // Wall Boundary Condition
        if (m_fields[0]->GetBndConditions()[n]->GetUserDefined() == "eWall")
        {
            if (m_expdim == 1)
            {
                ASSERTL0(false, "Illegal dimension");
            }
            else if (m_expdim == 2)
            {
                WallBoundary2D(n, cnt, inarray);
            }
        }
 
        // Time Dependent Boundary Condition (specified in meshfile)
        if (m_fields[0]->GetBndConditions()[n]->GetUserDefined() ==
            "eTimeDependent")
        {
            for (int i = 0; i < nvariables; ++i)
            {
                m_fields[i]->EvaluateBoundaryConditions(time);
            }
        }
        cnt += m_fields[0]->GetBndCondExpansions()[n]->GetExpSize();
    }
}
 
void MMFSWE::WallBoundary2D(int bcRegion, int cnt,
                            Array<OneD, Array<OneD, NekDouble>> &physarray)
{
 
    int i;
    int nTraceNumPoints = GetTraceTotPoints();
    int nvariables      = physarray.size();
 
    // get physical values of the forward trace
    Array<OneD, Array<OneD, NekDouble>> Fwd0(nvariables);
    for (i = 0; i < nvariables; ++i)
    {
        Fwd0[i] = Array<OneD, NekDouble>(nTraceNumPoints);
        m_fields[i]->ExtractTracePhys(physarray[i], Fwd0[i]);
    }
 
    Array<OneD, Array<OneD, NekDouble>> Fwd(nvariables);
    for (i = 0; i < nvariables; ++i)
    {
        Fwd[i] = Array<OneD, NekDouble>(nTraceNumPoints);
        m_fields[i]->ExtractTracePhys(physarray[i], Fwd[i]);
    }
 
    // Adjust the physical values of the trace to take
    // user defined boundaries into account
    int e, id1, id2, npts;
 
    for (e = 0; e < m_fields[0]->GetBndCondExpansions()[bcRegion]->GetExpSize();
         ++e)
    {
        npts = m_fields[0]
                   ->GetBndCondExpansions()[bcRegion]
                   ->GetExp(e)
                   ->GetTotPoints();
        id1 = m_fields[0]->GetBndCondExpansions()[bcRegion]->GetPhys_Offset(e);
        id2 = m_fields[0]->GetTrace()->GetPhys_Offset(
            m_fields[0]->GetTraceMap()->GetBndCondIDToGlobalTraceID(cnt + e));
 
        switch (m_expdim)
        {
            case 1:
            {
                // negate the forward flux
                Vmath::Neg(npts, &Fwd[1][id2], 1);
            }
            break;
            case 2:
            {
                Array<OneD, NekDouble> tmp_n(npts);
                Array<OneD, NekDouble> tmp_t(npts);
 
                Vmath::Vmul(npts, &Fwd[1][id2], 1, &m_ncdotMFFwd[0][id2], 1,
                            &tmp_n[0], 1);
                Vmath::Vvtvp(npts, &Fwd[2][id2], 1, &m_ncdotMFFwd[1][id2], 1,
                             &tmp_n[0], 1, &tmp_n[0], 1);
 
                Vmath::Vmul(npts, &Fwd[1][id2], 1, &m_nperpcdotMFFwd[0][id2], 1,
                            &tmp_t[0], 1);
                Vmath::Vvtvp(npts, &Fwd[2][id2], 1, &m_nperpcdotMFFwd[1][id2],
                             1, &tmp_t[0], 1, &tmp_t[0], 1);
 
                // negate the normal flux
                Vmath::Neg(npts, tmp_n, 1);
 
                Array<OneD, NekDouble> denom(npts);
                Array<OneD, NekDouble> tmp_u(npts);
                Array<OneD, NekDouble> tmp_v(npts);
 
                // denom = (e^1 \cdot n ) (e^2 \cdot t) - (e^2 \cdot n ) (e^1
                // \cdot t)
                Vmath::Vmul(npts, &m_ncdotMFFwd[1][id2], 1,
                            &m_nperpcdotMFFwd[0][id2], 1, &denom[0], 1);
                Vmath::Vvtvm(npts, &m_ncdotMFFwd[0][id2], 1,
                             &m_nperpcdotMFFwd[1][id2], 1, &denom[0], 1,
                             &denom[0], 1);
 
                Vmath::Vmul(npts, &m_ncdotMFFwd[1][id2], 1, &tmp_t[0], 1,
                            &tmp_u[0], 1);
                Vmath::Vvtvm(npts, &m_nperpcdotMFFwd[1][id2], 1, &tmp_n[0], 1,
                             &tmp_u[0], 1, &tmp_u[0], 1);
                Vmath::Vdiv(npts, &tmp_u[0], 1, &denom[0], 1, &tmp_u[0], 1);
 
                Vmath::Vcopy(npts, &tmp_u[0], 1, &Fwd[1][id2], 1);
 
                Vmath::Vmul(npts, &m_nperpcdotMFFwd[0][id2], 1, &tmp_n[0], 1,
                            &tmp_v[0], 1);
                Vmath::Vvtvm(npts, &m_ncdotMFFwd[0][id2], 1, &tmp_t[0], 1,
                             &tmp_v[0], 1, &tmp_v[0], 1);
                Vmath::Vdiv(npts, &tmp_v[0], 1, &denom[0], 1, &tmp_v[0], 1);
 
                Vmath::Vcopy(npts, &tmp_v[0], 1, &Fwd[2][id2], 1);
            }
            break;
 
            default:
                ASSERTL0(false, "Illegal expansion dimension");
        }
 
        // copy boundary adjusted values into the boundary expansion
        for (i = 0; i < nvariables; ++i)
        {
            Vmath::Vcopy(npts, &Fwd[i][id2], 1,
                         &(m_fields[i]
                               ->GetBndCondExpansions()[bcRegion]
                               ->UpdatePhys())[id1],
                         1);
        }
    }
}
 
/**
 *
 */
void MMFSWE::v_DoInitialise()
{
    // Compute m_depth and m_Derivdepth
    EvaluateWaterDepth();
    EvaluateCoriolis();
    SetInitialConditions();
    PrimitiveToConservative();
 
    // transfer the initial conditions to modal values
    for (int i = 0; i < m_fields.size(); ++i)
    {
        m_fields[i]->SetPhysState(true);
        m_fields[i]->FwdTrans(m_fields[i]->GetPhys(),
                              m_fields[i]->UpdateCoeffs());
    }
}
 
void MMFSWE::EvaluateWaterDepth(void)
{
    int nq = GetTotPoints();
 
    m_depth = Array<OneD, NekDouble>(nq, 0.0);
 
    switch (m_TestType)
    {
        case eTestPlane:
        {
            Vmath::Fill(nq, 1.0, m_depth, 1);
        }
        break;
 
        case eTestUnsteadyZonal:
        {
            // H_0 = k_1 - k_2 - (1/2/g) (Omega sin \phi)
            Array<OneD, NekDouble> x(nq);
            Array<OneD, NekDouble> y(nq);
            Array<OneD, NekDouble> z(nq);
 
            m_fields[0]->GetCoords(x, y, z);
 
            NekDouble x0j, x1j, x2j;
            NekDouble sin_varphi, cos_varphi, sin_theta, cos_theta;
 
            for (int j = 0; j < nq; ++j)
            {
                x0j = x[j];
                x1j = y[j];
                x2j = z[j];
 
                CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi,
                                     sin_theta, cos_theta);
                m_depth[j] =
                    m_H0 - m_k2 -
                    (0.5 / m_g) * (m_Omega * sin_theta) * (m_Omega * sin_theta);
            }
        }
        break;
 
        case eTestIsolatedMountain:
        {
            // H_0 = k_1 - k_2 - (1/2/g) (Omega sin \phi)
            Array<OneD, NekDouble> x(nq);
            Array<OneD, NekDouble> y(nq);
            Array<OneD, NekDouble> z(nq);
 
            m_fields[0]->GetCoords(x, y, z);
 
            int indx = 0;
            NekDouble x0j, x1j, x2j, dist2;
            NekDouble phi, theta, sin_varphi, cos_varphi, sin_theta, cos_theta;
            NekDouble hRad, phic, thetac;
            NekDouble Tol = 0.000001;
 
            hRad   = m_pi / 9.0;
            phic   = -m_pi / 2.0;
            thetac = m_pi / 6.0;
 
            for (int j = 0; j < nq; ++j)
            {
                x0j = x[j];
                x1j = y[j];
                x2j = z[j];
 
                CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi,
                                     sin_theta, cos_theta);
 
                if ((std::abs(sin(phic) - sin_varphi) +
                     std::abs(sin(thetac) - sin_theta)) < Tol)
                {
                    std::cout << "A point " << j
                              << " is coincient with the singularity "
                              << std::endl;
                    indx = 1;
                }
 
                phi   = atan2(sin_varphi, cos_varphi);
                theta = atan2(sin_theta, cos_theta);
 
                // Compute r
                dist2 = (phi - phic) * (phi - phic) +
                        (theta - thetac) * (theta - thetac);
 
                if (dist2 > hRad * hRad)
                {
                    dist2 = hRad * hRad;
                }
 
                m_depth[j] = m_H0 - m_hs0 * (1.0 - sqrt(dist2) / hRad);
            }
 
            if (!indx)
            {
                std::cout << "No point is coincident with the singularity point"
                          << std::endl;
            }
        }
        break;
 
        case eTestUnstableJet:
        {
            for (int j = 0; j < nq; ++j)
            {
                m_depth[j] = m_H0;
            }
        }
        break;
 
        case eTestSteadyZonal:
        case eTestRossbyWave:
        {
            Vmath::Zero(nq, m_depth, 1);
        }
        break;
 
        default:
        {
            Vmath::Zero(nq, m_depth, 1);
        }
        break;
    }
 
    // Comptue \nabla m_depth \cdot e^i
    m_Derivdepth = Array<OneD, Array<OneD, NekDouble>>(m_shapedim);
 
    for (int j = 0; j < m_shapedim; j++)
    {
        m_Derivdepth[j] = Array<OneD, NekDouble>(nq);
        m_fields[0]->PhysDirectionalDeriv(m_movingframes[j], m_depth,
                                          m_Derivdepth[j]);
    }
 
    std::cout << "Water Depth (m_depth) was generated with mag = "
              << AvgAbsInt(m_depth) << " with max. deriv = ( "
              << Vmath::Vamax(nq, m_Derivdepth[0], 1) << " , "
              << Vmath::Vamax(nq, m_Derivdepth[1], 1) << " ) " << std::endl;
}
 
void MMFSWE::EvaluateCoriolis(void)
{
    switch (m_TestType)
    {
        case eTestPlane:
        {
            GetFunction("Coriolis")->Evaluate("f", m_coriolis);
        }
        break;
 
        case eTestSteadyZonal:
        {
            EvaluateCoriolisForZonalFlow(m_coriolis);
        }
        break;
 
        case eTestUnsteadyZonal:
        case eTestIsolatedMountain:
        case eTestUnstableJet:
        case eTestRossbyWave:
        {
            EvaluateStandardCoriolis(m_coriolis);
        }
        break;
 
        default:
            break;
    }
}
 
void MMFSWE::EvaluateCoriolisForZonalFlow(Array<OneD, NekDouble> &outarray)
{
    int nq = GetTotPoints();
    NekDouble sin_theta, cos_theta, sin_varphi, cos_varphi;
 
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    NekDouble x0j, x1j, x2j;
    NekDouble tmp;
 
    outarray = Array<OneD, NekDouble>(nq, 0.0);
    for (int j = 0; j < nq; ++j)
    {
        x0j = x[j];
        x1j = y[j];
        x2j = z[j];
 
        CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi, sin_theta,
                             cos_theta);
 
        // H = 2 \Omega *(- \cos \phi \cos \theta \sin \alpha + \sin \theta \cos
        // \alpha )
        tmp = -1.0 * cos_varphi * cos_theta * sin(m_alpha) +
              sin_theta * cos(m_alpha);
        outarray[j] = 2.0 * m_Omega * tmp;
    }
}
 
void MMFSWE::EvaluateStandardCoriolis(Array<OneD, NekDouble> &outarray)
{
    int nq = GetTotPoints();
 
    NekDouble x0j, x1j, x2j;
    NekDouble sin_theta, cos_theta, sin_varphi, cos_varphi;
 
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    outarray = Array<OneD, NekDouble>(nq, 0.0);
    for (int j = 0; j < nq; ++j)
    {
        x0j = x[j];
        x1j = y[j];
        x2j = z[j];
 
        CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi, sin_theta,
                             cos_theta);
 
        outarray[j] = 2.0 * m_Omega * sin_theta;
    }
}
 
void MMFSWE::v_SetInitialConditions(const NekDouble initialtime,
                                    bool dumpInitialConditions,
                                    const int domain)
{
    boost::ignore_unused(domain);
 
    int nq = GetTotPoints();
 
    switch (m_TestType)
    {
        case eTestPlane:
        {
            Array<OneD, NekDouble> eta0(nq);
            Array<OneD, NekDouble> u0(nq);
            Array<OneD, NekDouble> v0(nq);
 
            TestSWE2Dproblem(initialtime, 0, eta0);
            m_fields[0]->SetPhys(eta0);
 
            TestSWE2Dproblem(initialtime, 1, u0);
            m_fields[1]->SetPhys(u0);
 
            TestSWE2Dproblem(initialtime, 2, v0);
            m_fields[2]->SetPhys(v0);
 
            // forward transform to fill the modal coeffs
            for (int i = 0; i < m_fields.size(); ++i)
            {
                m_fields[i]->SetPhysState(true);
                m_fields[i]->FwdTrans(m_fields[i]->GetPhys(),
                                      m_fields[i]->UpdateCoeffs());
            }
        }
        break;
 
        case eTestSteadyZonal:
        {
            Array<OneD, NekDouble> eta0(nq);
            Array<OneD, NekDouble> u0(nq);
            Array<OneD, NekDouble> v0(nq);
            Array<OneD, NekDouble> zeta0(nq);
 
            SteadyZonalFlow(0, eta0);
            m_fields[0]->SetPhys(eta0);
 
            SteadyZonalFlow(1, u0);
            m_fields[1]->SetPhys(u0);
 
            SteadyZonalFlow(2, v0);
            m_fields[2]->SetPhys(v0);
 
            // ComputeVorticity(u0, v0, zeta0);
            m_Vorticity0 = m_fields[0]->Integral(zeta0);
 
            m_Mass0      = ComputeMass(eta0);
            m_Energy0    = ComputeEnergy(eta0, u0, v0);
            m_Enstrophy0 = ComputeEnstrophy(eta0, u0, v0);
 
            // forward transform to fill the modal coeffs
            for (int i = 0; i < m_fields.size(); ++i)
            {
                m_fields[i]->SetPhysState(true);
                m_fields[i]->FwdTrans(m_fields[i]->GetPhys(),
                                      m_fields[i]->UpdateCoeffs());
            }
        }
        break;
 
        case eTestUnsteadyZonal:
        {
            Array<OneD, NekDouble> eta0(nq);
            Array<OneD, NekDouble> u0(nq);
            Array<OneD, NekDouble> v0(nq);
 
            UnsteadyZonalFlow(0, initialtime, eta0);
            m_fields[0]->SetPhys(eta0);
 
            UnsteadyZonalFlow(1, initialtime, u0);
            m_fields[1]->SetPhys(u0);
 
            UnsteadyZonalFlow(2, initialtime, v0);
            m_fields[2]->SetPhys(v0);
 
            m_Mass0      = ComputeMass(eta0);
            m_Energy0    = ComputeEnergy(eta0, u0, v0);
            m_Enstrophy0 = ComputeEnstrophy(eta0, u0, v0);
 
            // forward transform to fill the modal coeffs
            for (int i = 0; i < m_fields.size(); ++i)
            {
                m_fields[i]->SetPhysState(true);
                m_fields[i]->FwdTrans(m_fields[i]->GetPhys(),
                                      m_fields[i]->UpdateCoeffs());
            }
        }
        break;
 
        case eTestIsolatedMountain:
        {
            Array<OneD, NekDouble> eta0(nq);
            Array<OneD, NekDouble> u0(nq);
            Array<OneD, NekDouble> v0(nq);
 
            IsolatedMountainFlow(0, initialtime, eta0);
            m_fields[0]->SetPhys(eta0);
 
            IsolatedMountainFlow(1, initialtime, u0);
            m_fields[1]->SetPhys(u0);
 
            IsolatedMountainFlow(2, initialtime, v0);
            m_fields[2]->SetPhys(v0);
 
            m_Mass0      = ComputeMass(eta0);
            m_Energy0    = ComputeEnergy(eta0, u0, v0);
            m_Enstrophy0 = ComputeEnstrophy(eta0, u0, v0);
 
            // forward transform to fill the modal coeffs
            for (int i = 0; i < m_fields.size(); ++i)
            {
                m_fields[i]->SetPhysState(true);
                m_fields[i]->FwdTrans(m_fields[i]->GetPhys(),
                                      m_fields[i]->UpdateCoeffs());
            }
        }
        break;
 
        case eTestUnstableJet:
        {
            Array<OneD, NekDouble> eta0(nq);
            Array<OneD, NekDouble> u0(nq);
            Array<OneD, NekDouble> v0(nq);
 
            UnstableJetFlow(0, initialtime, eta0);
            m_fields[0]->SetPhys(eta0);
 
            UnstableJetFlow(1, initialtime, u0);
            m_fields[1]->SetPhys(u0);
 
            UnstableJetFlow(2, initialtime, v0);
            m_fields[2]->SetPhys(v0);
 
            m_Mass0      = ComputeMass(eta0);
            m_Energy0    = ComputeEnergy(eta0, u0, v0);
            m_Enstrophy0 = ComputeEnstrophy(eta0, u0, v0);
 
            // forward transform to fill the modal coeffs
            for (int i = 0; i < m_fields.size(); ++i)
            {
                m_fields[i]->SetPhysState(true);
                m_fields[i]->FwdTrans(m_fields[i]->GetPhys(),
                                      m_fields[i]->UpdateCoeffs());
            }
        }
        break;
 
        case eTestRossbyWave:
        {
            Array<OneD, NekDouble> eta0(nq);
            Array<OneD, NekDouble> u0(nq);
            Array<OneD, NekDouble> v0(nq);
 
            RossbyWave(0, eta0);
            m_fields[0]->SetPhys(eta0);
 
            RossbyWave(1, u0);
            m_fields[1]->SetPhys(u0);
 
            RossbyWave(2, v0);
            m_fields[2]->SetPhys(v0);
 
            m_Mass0      = ComputeMass(eta0);
            m_Energy0    = ComputeEnergy(eta0, u0, v0);
            m_Enstrophy0 = ComputeEnstrophy(eta0, u0, v0);
 
            // forward transform to fill the modal coeffs
            for (int i = 0; i < m_fields.size(); ++i)
            {
                m_fields[i]->SetPhysState(true);
                m_fields[i]->FwdTrans(m_fields[i]->GetPhys(),
                                      m_fields[i]->UpdateCoeffs());
            }
        }
        break;
 
        default:
            break;
    }
 
    if (dumpInitialConditions)
    {
        // dump initial conditions to file
        std::string outname = m_sessionName + "_initial.chk";
        WriteFld(outname);
 
        outname = m_sessionName + "_initialCART.chk";
        Checkpoint_Output_Cartesian(outname);
    }
}
 
void MMFSWE::TestSWE2Dproblem(const NekDouble time, unsigned int field,
                              Array<OneD, NekDouble> &outfield)
{
    boost::ignore_unused(time);
 
    int nq = m_fields[0]->GetNpoints();
 
    Array<OneD, NekDouble> x0(nq);
    Array<OneD, NekDouble> x1(nq);
    Array<OneD, NekDouble> x2(nq);
 
    m_fields[0]->GetCoords(x0, x1, x2);
 
    Array<OneD, NekDouble> eta0(nq);
    Array<OneD, NekDouble> u0(nq);
    Array<OneD, NekDouble> v0(nq);
 
    Array<OneD, Array<OneD, NekDouble>> uvec(m_spacedim);
 
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvec[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    for (int i = 0; i < nq; ++i)
    {
        eta0[i] = (0.771 * 0.395 * 0.395 * (1.0 / cosh(0.395 * x0[i])) *
                   (1.0 / cosh(0.395 * x0[i]))) *
                  (3.0 + 6.0 * x1[i] * x1[i]) / (4.0) *
                  exp(-0.5 * x1[i] * x1[i]);
        uvec[0][i] = (0.771 * 0.395 * 0.395 * (1.0 / cosh(0.395 * x0[i])) *
                      (1.0 / cosh(0.395 * x0[i]))) *
                     (-9.0 + 6.0 * x1[i] * x1[i]) / (4.0) *
                     exp(-0.5 * x1[i] * x1[i]);
        uvec[1][i] = (-2.0 * 0.395 * tanh(0.395 * x0[i])) *
                     (0.771 * 0.395 * 0.395 * (1.0 / cosh(0.395 * x0[i])) *
                      (1.0 / cosh(0.395 * x0[i]))) *
                     (2.0 * x1[i]) * exp(-0.5 * x1[i] * x1[i]);
    }
 
    u0 = CartesianToMovingframes(uvec, 0);
    v0 = CartesianToMovingframes(uvec, 1);
 
    switch (field)
    {
        case (0):
        {
            outfield = eta0;
        }
        break;
 
        case (1):
        {
            outfield = u0;
        }
        break;
 
        case (2):
        {
            outfield = v0;
        }
        break;
    }
}
 
void MMFSWE::SteadyZonalFlow(unsigned int field,
                             Array<OneD, NekDouble> &outfield)
{
    int nq = GetTotPoints();
    NekDouble uhat, vhat;
    NekDouble sin_theta, cos_theta, sin_varphi, cos_varphi;
    NekDouble x0j, x1j, x2j, tmp;
 
    Array<OneD, NekDouble> eta(nq, 0.0);
    Array<OneD, NekDouble> u(nq, 0.0);
    Array<OneD, NekDouble> v(nq, 0.0);
 
    Array<OneD, Array<OneD, NekDouble>> uvec(m_spacedim);
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvec[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    for (int j = 0; j < nq; ++j)
    {
        x0j = x[j];
        x1j = y[j];
        x2j = z[j];
 
        CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi, sin_theta,
                             cos_theta);
 
        // H = H_0 - (1/g)*(a \Omega u_0 + 0.5*u_0^2 )*(- \cos \phi \cos \theta
        // \sin \alpha + \sin \theta \cos \alpha )^2
        tmp = -1.0 * cos_varphi * cos_theta * sin(m_alpha) +
              sin_theta * cos(m_alpha);
        eta[j] = m_H0 - m_Hvar * tmp * tmp;
 
        // u = (\vec{u} \cdot e^1 )/ || e^1 ||^2 ,   v = (\vec{u} \cdot e^2 )/
        // || e^2 ||^2
        uhat = m_u0 * (cos_theta * cos(m_alpha) +
                       sin_theta * cos_varphi * sin(m_alpha));
        vhat = -1.0 * m_u0 * sin_varphi * sin(m_alpha);
 
        uvec[0][j] = -1.0 * uhat * sin_varphi - vhat * sin_theta * cos_varphi;
        uvec[1][j] = uhat * cos_varphi - vhat * sin_theta * sin_varphi;
        uvec[2][j] = vhat * cos_theta;
    }
 
    // Projection of u onto the tangent plane with conserving the mag. of the
    // velocity.
    Array<OneD, Array<OneD, NekDouble>> uvecproj(m_spacedim);
 
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvecproj[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    // u is projected on the tangent plane with preserving its length
    // GramSchumitz(m_surfaceNormal, uvec, uvecproj, true);
 
    // Change it to the coordinate of moving frames
    // CartesianToMovingframes(0,uvecproj,u);
    // CartesianToMovingframes(1,uvecproj,v);
 
    u = CartesianToMovingframes(uvec, 0);
    v = CartesianToMovingframes(uvec, 1);
 
    switch (field)
    {
        case (0):
        {
            outfield = eta;
        }
        break;
 
        case (1):
        {
            outfield = u;
        }
        break;
 
        case (2):
        {
            outfield = v;
        }
        break;
    }
}
 
NekDouble MMFSWE::ComputeMass(const Array<OneD, const NekDouble> &eta)
{
    int nq = m_fields[0]->GetTotPoints();
 
    Array<OneD, NekDouble> tmp(nq);
    Vmath::Vadd(nq, eta, 1, m_depth, 1, tmp, 1);
 
    return m_fields[0]->Integral(tmp);
}
 
NekDouble MMFSWE::ComputeEnergy(const Array<OneD, const NekDouble> &eta,
                                const Array<OneD, const NekDouble> &u,
                                const Array<OneD, const NekDouble> &v)
{
    int nq = m_fields[0]->GetTotPoints();
 
    Array<OneD, NekDouble> tmp(nq);
    Array<OneD, NekDouble> htmp(nq);
    Array<OneD, NekDouble> hstmp(nq);
 
    Vmath::Vmul(nq, u, 1, u, 1, tmp, 1);
    Vmath::Vvtvp(nq, v, 1, v, 1, tmp, 1, tmp, 1);
    Vmath::Vmul(nq, eta, 1, tmp, 1, tmp, 1);
 
    Vmath::Sadd(nq, m_H0, eta, 1, htmp, 1);
    Vmath::Vmul(nq, htmp, 1, htmp, 1, htmp, 1);
 
    Vmath::Sadd(nq, -1.0 * m_H0, m_depth, 1, hstmp, 1);
    Vmath::Vmul(nq, hstmp, 1, hstmp, 1, hstmp, 1);
 
    Vmath::Vsub(nq, htmp, 1, hstmp, 1, htmp, 1);
    Vmath::Smul(nq, m_g, htmp, 1, htmp, 1);
 
    Vmath::Vadd(nq, htmp, 1, tmp, 1, tmp, 1);
    Vmath::Smul(nq, 0.5, tmp, 1, tmp, 1);
 
    return m_fields[0]->Integral(tmp);
}
 
NekDouble MMFSWE::ComputeEnstrophy(const Array<OneD, const NekDouble> &eta,
                                   const Array<OneD, const NekDouble> &u,
                                   const Array<OneD, const NekDouble> &v)
{
    int nq = m_fields[0]->GetTotPoints();
 
    Array<OneD, NekDouble> hstartmp(nq);
    Array<OneD, NekDouble> tmp(nq);
 
    Vmath::Vadd(nq, eta, 1, m_depth, 1, hstartmp, 1);
 
    Array<OneD, Array<OneD, NekDouble>> uvec(m_spacedim);
    Array<OneD, Array<OneD, NekDouble>> Curlu(m_spacedim);
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvec[i]  = Array<OneD, NekDouble>(nq);
        Curlu[i] = Array<OneD, NekDouble>(nq);
    }
 
    ComputeVorticity(u, v, tmp);
 
    Vmath::Vadd(nq, m_coriolis, 1, tmp, 1, tmp, 1);
    Vmath::Vmul(nq, tmp, 1, tmp, 1, tmp, 1);
    Vmath::Vdiv(nq, tmp, 1, hstartmp, 1, tmp, 1);
    Vmath::Smul(nq, 0.5, tmp, 1, tmp, 1);
 
    return m_fields[0]->Integral(tmp);
}
 
// Vorticity = \nabla v \cdot e^1 + v \nabla \cdot e^1 - ( \nabla u \cdot e^2 +
// u \nabla \cdot e^2 )
void MMFSWE::ComputeVorticity(const Array<OneD, const NekDouble> &u,
                              const Array<OneD, const NekDouble> &v,
                              Array<OneD, NekDouble> &Vorticity)
{
    int nq = m_fields[0]->GetTotPoints();
 
    Array<OneD, NekDouble> tmp(nq);
 
    Vorticity = Array<OneD, NekDouble>(nq, 0.0);
 
    m_fields[0]->PhysDirectionalDeriv(m_movingframes[0], v, Vorticity);
    Vmath::Vvtvp(nq, &v[0], 1, &m_CurlMF[1][2][0], 1, &Vorticity[0], 1,
                 &Vorticity[0], 1);
 
    m_fields[0]->PhysDirectionalDeriv(m_movingframes[1], u, tmp);
    Vmath::Neg(nq, tmp, 1);
    Vmath::Vvtvp(nq, &u[0], 1, &m_CurlMF[0][2][0], 1, &tmp[0], 1, &tmp[0], 1);
 
    Vmath::Vadd(nq, tmp, 1, Vorticity, 1, Vorticity, 1);
}
 
void MMFSWE::ComputeNablaCdotVelocity(Array<OneD, NekDouble> &vellc)
{
    int nq = m_fields[0]->GetNpoints();
 
    Array<OneD, NekDouble> velcoeff(nq, 0.0);
 
    Array<OneD, NekDouble> Dtmp0(nq);
    Array<OneD, NekDouble> Dtmp1(nq);
    Array<OneD, NekDouble> Dtmp2(nq);
    Array<OneD, NekDouble> Drv(nq);
 
    vellc = Array<OneD, NekDouble>(nq, 0.0);
 
    // m_vellc = \nabla m_vel \cdot tan_i
    Array<OneD, NekDouble> tmp(nq);
    Array<OneD, NekDouble> vessel(nq);
 
    for (int j = 0; j < m_shapedim; ++j)
    {
        Vmath::Zero(nq, velcoeff, 1);
        for (int k = 0; k < m_spacedim; ++k)
        {
            // a_j = tan_j cdot m_vel
            Vmath::Vvtvp(nq, &m_movingframes[j][k * nq], 1, &m_velocity[k][0],
                         1, &velcoeff[0], 1, &velcoeff[0], 1);
        }
 
        // d a_j / d x^k
        m_fields[0]->PhysDeriv(velcoeff, Dtmp0, Dtmp1, Dtmp2);
 
        for (int k = 0; k < m_spacedim; ++k)
        {
            // tan_j^k ( d a_j / d x^k )
            switch (k)
            {
                case (0):
                {
                    Vmath::Vvtvp(nq, &Dtmp0[0], 1, &m_movingframes[j][k * nq],
                                 1, &vellc[0], 1, &vellc[0], 1);
                }
                break;
 
                case (1):
                {
                    Vmath::Vvtvp(nq, &Dtmp1[0], 1, &m_movingframes[j][k * nq],
                                 1, &vellc[0], 1, &vellc[0], 1);
                }
                break;
 
                case (2):
                {
                    Vmath::Vvtvp(nq, &Dtmp2[0], 1, &m_movingframes[j][k * nq],
                                 1, &vellc[0], 1, &vellc[0], 1);
                }
                break;
            }
        }
    }
}
 
void MMFSWE::UnsteadyZonalFlow(unsigned int field, const NekDouble time,
                               Array<OneD, NekDouble> &outfield)
{
    int nq = GetTotPoints();
    NekDouble uhat, vhat;
    NekDouble sin_theta, cos_theta, sin_varphi, cos_varphi;
    NekDouble x0j, x1j, x2j, tmp;
 
    NekDouble TR, Ttheta;
 
    Array<OneD, NekDouble> eta(nq, 0.0);
    Array<OneD, NekDouble> u(nq, 0.0);
    Array<OneD, NekDouble> v(nq, 0.0);
 
    Array<OneD, Array<OneD, NekDouble>> uvec(m_spacedim);
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvec[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    for (int j = 0; j < nq; ++j)
    {
        x0j = x[j];
        x1j = y[j];
        x2j = z[j];
 
        CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi, sin_theta,
                             cos_theta);
 
        // \eta = ( - ( u_0 ( - T_R sin \alpha \cos \theta + \cos \alpha \sin
        // \theta ) + \Omega \sin \theta )^2 + \Omega \sin \theta )^2 + (\Omega
        // \sin \theta )^2
        TR =
            cos_varphi * cos(m_Omega * time) - sin_varphi * sin(m_Omega * time);
        Ttheta =
            sin_varphi * cos(m_Omega * time) + cos_varphi * sin(m_Omega * time);
        tmp = -1.0 * TR * sin(m_alpha) * cos_theta + cos(m_alpha) * sin_theta;
 
        eta[j] = -1.0 * (m_u0 * tmp + m_Omega * sin_theta) *
                     (m_u0 * tmp + m_Omega * sin_theta) +
                 m_Omega * m_Omega * sin_theta * sin_theta;
        eta[j] = 0.5 * eta[j] / m_g;
 
        // u = u_0*(TR*\sin \alpha * \sin \theta + \cos \alpha * \cos \theta
        // v = - u_0 Ttheta * \sin \alpha
        uhat =
            m_u0 * (TR * sin(m_alpha) * sin_theta + cos(m_alpha) * cos_theta);
        vhat = -1.0 * m_u0 * Ttheta * sin(m_alpha);
 
        uvec[0][j] = -1.0 * uhat * sin_varphi - vhat * sin_theta * cos_varphi;
        uvec[1][j] = uhat * cos_varphi - vhat * sin_theta * sin_varphi;
        uvec[2][j] = vhat * cos_theta;
    }
 
    // Projection of u onto the tangent plane with conserving the mag. of the
    // velocity.
    Array<OneD, Array<OneD, NekDouble>> uvecproj(m_spacedim);
 
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvecproj[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    // u is projected on the tangent plane with preserving its length
    // GramSchumitz(m_surfaceNormal, uvec, uvecproj, true);
 
    // Change it to the coordinate of moving frames
    // CartesianToMovingframes(0,uvecproj,u);
    // CartesianToMovingframes(1,uvecproj,v);
 
    u = CartesianToMovingframes(uvec, 0);
    v = CartesianToMovingframes(uvec, 1);
 
    switch (field)
    {
        case (0):
        {
            outfield = eta;
        }
        break;
 
        case (1):
        {
            outfield = u;
        }
        break;
 
        case (2):
        {
            outfield = v;
        }
        break;
    }
}
 
void MMFSWE::IsolatedMountainFlow(unsigned int field, const NekDouble time,
                                  Array<OneD, NekDouble> &outfield)
{
    boost::ignore_unused(time);
 
    int nq = GetTotPoints();
 
    NekDouble uhat, vhat;
    NekDouble sin_theta, cos_theta, sin_varphi, cos_varphi;
    NekDouble x0j, x1j, x2j;
 
    Array<OneD, NekDouble> eta(nq, 0.0);
    Array<OneD, NekDouble> u(nq, 0.0);
    Array<OneD, NekDouble> v(nq, 0.0);
 
    Array<OneD, Array<OneD, NekDouble>> uvec(m_spacedim);
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvec[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    for (int j = 0; j < nq; ++j)
    {
        x0j = x[j];
        x1j = y[j];
        x2j = z[j];
 
        CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi, sin_theta,
                             cos_theta);
 
        // eta = - (1/g) (\Omega u_0 + 0.4 u^2 ) \sin^2 \theta
        eta[j] = (-1.0 / m_g) * (m_Omega * m_u0 + 0.5 * m_u0 * m_u0) *
                 sin_theta * sin_theta;
 
        uhat = m_u0 * cos_theta;
        vhat = 0.0;
 
        uvec[0][j] = -1.0 * uhat * sin_varphi - vhat * sin_theta * cos_varphi;
        uvec[1][j] = uhat * cos_varphi - vhat * sin_theta * sin_varphi;
        uvec[2][j] = vhat * cos_theta;
    }
 
    // Projection of u onto the tangent plane with conserving the mag. of the
    // velocity.
    Array<OneD, Array<OneD, NekDouble>> uvecproj(m_spacedim);
 
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvecproj[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    // u is projected on the tangent plane with preserving its length
    // GramSchumitz(m_surfaceNormal, uvec, uvecproj, true);
 
    // Change it to the coordinate of moving frames
    // CartesianToMovingframes(0,uvecproj,u);
    // CartesianToMovingframes(1,uvecproj,v);
 
    u = CartesianToMovingframes(uvec, 0);
    v = CartesianToMovingframes(uvec, 1);
 
    switch (field)
    {
        case (0):
        {
            outfield = eta;
        }
        break;
 
        case (1):
        {
            outfield = u;
        }
        break;
 
        case (2):
        {
            outfield = v;
        }
        break;
    }
}
 
void MMFSWE::UnstableJetFlow(unsigned int field, const NekDouble time,
                             Array<OneD, NekDouble> &outfield)
{
    boost::ignore_unused(time);
 
    int nq = GetTotPoints();
 
    NekDouble uhat, vhat;
    NekDouble sin_theta, cos_theta, sin_varphi, cos_varphi;
    NekDouble x0j, x1j, x2j;
    NekDouble Ttheta, Tphi;
 
    Array<OneD, NekDouble> eta(nq, 0.0);
    Array<OneD, NekDouble> u(nq, 0.0);
    Array<OneD, NekDouble> v(nq, 0.0);
 
    Array<OneD, Array<OneD, NekDouble>> uvec(m_spacedim);
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvec[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    int Nint = 1000;
    NekDouble dth, intj;
    for (int j = 0; j < nq; ++j)
    {
        x0j = x[j];
        x1j = y[j];
        x2j = z[j];
 
        CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi, sin_theta,
                             cos_theta);
 
        Ttheta = atan2(sin_theta, cos_theta);
        Tphi   = atan2(sin_varphi, cos_varphi);
 
        uhat = ComputeUnstableJetuphi(Ttheta);
        vhat = 0.0;
 
        // eta = - (1/g) (\Omega u_0 + 0.4 u^2 ) \sin^2 \theta
        dth    = Ttheta / Nint;
        eta[j] = dth * 0.5 *
                 (ComputeUnstableJetEta(0.0) + ComputeUnstableJetEta(Ttheta));
        for (int i = 1; i < Nint - 1; i++)
        {
            intj   = i * dth;
            eta[j] = eta[j] + dth * ComputeUnstableJetEta(intj);
        }
        eta[j] = (-1.0 / m_g) * eta[j];
 
        // Add perturbation
        if (m_PurturbedJet)
        {
            eta[j] = eta[j] + m_hbar * cos_theta * exp(-9.0 * Tphi * Tphi) *
                                  exp(-225.0 * (m_pi / 4.0 - Ttheta) *
                                      (m_pi / 4.0 - Ttheta));
        }
 
        uvec[0][j] = -1.0 * uhat * sin_varphi - vhat * sin_theta * cos_varphi;
        uvec[1][j] = uhat * cos_varphi - vhat * sin_theta * sin_varphi;
        uvec[2][j] = vhat * cos_theta;
    }
 
    // Projection of u onto the tangent plane with conserving the mag. of the
    // velocity.
    Array<OneD, Array<OneD, NekDouble>> uvecproj(m_spacedim);
 
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvecproj[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    // u is projected on the tangent plane with preserving its length
    // GramSchumitz(m_surfaceNormal, uvec, uvecproj, true);
 
    // Change it to the coordinate of moving frames
    // CartesianToMovingframes(0,uvecproj,u);
    // CartesianToMovingframes(1,uvecproj,v);
 
    u = CartesianToMovingframes(uvec, 0);
    v = CartesianToMovingframes(uvec, 1);
 
    switch (field)
    {
        case (0):
        {
            outfield = eta;
        }
        break;
 
        case (1):
        {
            outfield = u;
        }
        break;
 
        case (2):
        {
            outfield = v;
        }
        break;
    }
}
 
void MMFSWE::RossbyWave(unsigned int field, Array<OneD, NekDouble> &outfield)
{
    int nq = GetTotPoints();
    NekDouble uhat, vhat;
    NekDouble sin_theta, cos_theta, sin_varphi, cos_varphi;
    NekDouble x0j, x1j, x2j;
    NekDouble Ath, Bth, Cth, tmp;
 
    Array<OneD, NekDouble> eta(nq, 0.0);
    Array<OneD, NekDouble> u(nq, 0.0);
    Array<OneD, NekDouble> v(nq, 0.0);
 
    Array<OneD, Array<OneD, NekDouble>> uvec(m_spacedim);
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvec[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    NekDouble R = 4.0;
    NekDouble cos2theta, cosRtheta, cos6theta, cos2Rtheta, cosRm1theta;
    NekDouble cos2phi, cos4phi, sin4phi, cos8phi;
 
    // disturbancees of Rossby-Haurwitz Wave
    NekDouble x0d, y0d, z0d, phi0, theta0;
    // NekDouble rad_earth = 6.37122 * 1000000;
 
    phi0   = 40.0 * m_pi / 180.0;
    theta0 = 50.0 * m_pi / 180.0;
 
    x0d = cos(phi0) * cos(theta0);
    y0d = sin(phi0) * cos(theta0);
    z0d = sin(theta0);
 
    for (int j = 0; j < nq; ++j)
    {
        x0j = x[j];
        x1j = y[j];
        x2j = z[j];
 
        CartesianToSpherical(x0j, x1j, x2j, sin_varphi, cos_varphi, sin_theta,
                             cos_theta);
 
        // H = H_0 - (1/g)*(a \Omega u_0 + 0.5*u_0^2 )*(- \cos \phi \cos \theta
        // \sin \alpha + \sin \theta \cos \alpha )^2
 
        // tmp = cos^{2R} \theta, R = 4;
        cos2theta   = cos_theta * cos_theta;
        cosRm1theta = cos_theta * cos2theta;
        cosRtheta   = cos2theta * cos2theta;
        cos6theta   = cos2theta * cosRtheta;
        cos2Rtheta  = cosRtheta * cosRtheta;
 
        tmp = (0.5 * m_angfreq) * (2.0 * m_Omega + m_angfreq);
        Ath = tmp * cos2theta +
              0.25 * m_K * m_K * cos6theta *
                  ((R + 1.0) * cosRtheta + (2 * R * R - R - 2.0) * cos2theta -
                   2 * R * R);
 
        tmp = (2.0 * m_K) * (m_Omega + m_angfreq) / ((R + 1.0) * (R + 2.0));
        Bth = tmp * cosRtheta *
              ((R * R + 2 * R + 2) - (R + 1.0) * (R + 1.0) * cos2theta);
 
        Cth =
            0.25 * m_K * m_K * cos2Rtheta * ((R + 1.0) * cos2theta - (R + 2.0));
 
        // cos (2 \phi) = 2 * cos \phi * cos \phi - 1.0
        cos2phi = 2.0 * cos_varphi * cos_varphi - 1.0;
        cos4phi = 2.0 * cos2phi * cos2phi - 1.0;
        cos8phi = 2.0 * cos4phi * cos4phi - 1.0;
 
        // sin (2 \phi) = 2 * cos \phi * sin \phi
        sin4phi = 4.0 * sin_varphi * cos_varphi * cos2phi;
 
        eta[j] = m_H0 + (1.0 / m_g) * (Ath + Bth * cos4phi + Cth * cos8phi);
 
        // disturbances is added
        if (m_RossbyDisturbance)
        {
            eta[j] = eta[j] *
                     (1.0 + (1.0 / 40.0) * (x0j * x0d + x1j * y0d + x2j * z0d));
        }
 
        // u = (\vec{u} \cdot e^1 )/ || e^1 ||^2 ,   v = (\vec{u} \cdot e^2 )/
        // || e^2 ||^2
        uhat = m_angfreq * cos_theta +
               m_K * cosRm1theta * (R * sin_theta * sin_theta - cos2theta) *
                   cos4phi;
        vhat = -1.0 * m_K * R * cosRm1theta * sin_theta * sin4phi;
 
        uvec[0][j] = -1.0 * uhat * sin_varphi - vhat * sin_theta * cos_varphi;
        uvec[1][j] = uhat * cos_varphi - vhat * sin_theta * sin_varphi;
        uvec[2][j] = vhat * cos_theta;
    }
 
    // NekDouble etamin, etaaver;
    // etamin = Vmath::Vmin(nq, eta, 1)*rad_earth;
    // etaaver = Vmath::Vsum(nq, eta, 1)/nq*rad_earth;
 
    // Projection of u onto the tangent plane with conserving the mag. of the
    // velocity.
    Array<OneD, Array<OneD, NekDouble>> uvecproj(m_spacedim);
 
    for (int i = 0; i < m_spacedim; ++i)
    {
        uvecproj[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    // u is projected on the tangent plane with preserving its length
    // GramSchumitz(m_surfaceNormal, uvec, uvecproj, true);
 
    // Change it to the coordinate of moving frames
    // CartesianToMovingframes(0,uvecproj,u);
    // CartesianToMovingframes(1,uvecproj,v);
 
    u = CartesianToMovingframes(uvec, 0);
    v = CartesianToMovingframes(uvec, 1);
 
    switch (field)
    {
        case (0):
        {
            outfield = eta;
        }
        break;
 
        case (1):
        {
            outfield = u;
        }
        break;
 
        case (2):
        {
            outfield = v;
        }
        break;
    }
}
 
NekDouble MMFSWE::ComputeUnstableJetEta(const NekDouble theta)
{
    NekDouble uphi, f, dh;
 
    uphi = ComputeUnstableJetuphi(theta);
    f    = 2.0 * m_Omega * sin(theta);
 
    dh = f * uphi + tan(theta) * uphi * uphi;
 
    return dh;
}
 
NekDouble MMFSWE::ComputeUnstableJetuphi(const NekDouble theta)
{
    NekDouble uphi;
 
    if ((theta > m_theta0) && (theta < m_theta1))
    {
        uphi = (m_uthetamax / m_en) *
               exp(1.0 / (theta - m_theta0) / (theta - m_theta1));
    }
 
    else
    {
        uphi = 0.0;
    }
 
    return uphi;
}
 
void MMFSWE::Checkpoint_Output_Cartesian(std::string outname)
{
    int nq      = m_fields[0]->GetTotPoints();
    int ncoeffs = m_fields[0]->GetNcoeffs();
 
    NekDouble rad_earth = 6.37122 * 1000000;
 
    int nvariables = 7;
 
    // vector u in Cartesian coordinates
    std::vector<std::string> variables(nvariables);
 
    variables[0] = "eta";
    variables[1] = "hstar";
    variables[2] = "vorticity";
    variables[3] = "ux";
    variables[4] = "uy";
    variables[5] = "uz";
    variables[6] = "null";
 
    // Obtain \vec{u} in cartesian coordinate
    Array<OneD, Array<OneD, NekDouble>> fieldphys(nvariables);
    std::vector<Array<OneD, NekDouble>> fieldcoeffs(nvariables);
    for (int i = 0; i < nvariables; ++i)
    {
        fieldphys[i]   = Array<OneD, NekDouble>(nq, 0.0);
        fieldcoeffs[i] = Array<OneD, NekDouble>(ncoeffs);
    }
 
    Vmath::Smul(nq, rad_earth, &(m_fields[0]->GetPhys())[0], 1,
                &fieldphys[0][0], 1);
    // Vmath::Vadd(nq, &(m_fields[0]->GetPhys())[0], 1, &m_depth[0], 1,
    // &fieldphys[1][0], 1);
 
    Vmath::Vcopy(nq, &m_depth[0], 1, &fieldphys[1][0], 1);
    Vmath::Neg(nq, &fieldphys[1][0], 1);
    Vmath::Sadd(nq, m_H0, &fieldphys[1][0], 1, &fieldphys[1][0], 1);
    Vmath::Smul(nq, rad_earth, &fieldphys[1][0], 1, &fieldphys[1][0], 1);
 
    Array<OneD, NekDouble> utmp(nq);
    Array<OneD, NekDouble> vtmp(nq);
 
    Vmath::Vcopy(nq, &(m_fields[1]->GetPhys())[0], 1, &utmp[0], 1);
    Vmath::Vcopy(nq, &(m_fields[2]->GetPhys())[0], 1, &vtmp[0], 1);
 
    ComputeVorticity(utmp, vtmp, fieldphys[2]);
    // u_x = u e^1_x + v e^2_x
    int indx = 3;
    for (int k = 0; k < m_spacedim; ++k)
    {
        Vmath::Vmul(nq, &utmp[0], 1, &m_movingframes[0][k * nq], 1,
                    &fieldphys[k + indx][0], 1);
        Vmath::Vvtvp(nq, &vtmp[0], 1, &m_movingframes[1][k * nq], 1,
                     &fieldphys[k + indx][0], 1, &fieldphys[k + indx][0], 1);
    }
 
    for (int i = 0; i < nvariables; ++i)
    {
        m_fields[0]->FwdTrans(fieldphys[i], fieldcoeffs[i]);
    }
 
    WriteFld(outname, m_fields[0], fieldcoeffs, variables);
}
 
// (h, hu, hv) -> (\eta, u, v)
void MMFSWE::ConservativeToPrimitive(
    const Array<OneD, const Array<OneD, NekDouble>> &physin,
    Array<OneD, Array<OneD, NekDouble>> &physout)
{
    int nq = GetTotPoints();
 
    if (physin[0].get() == physout[0].get())
    {
        // copy indata and work with tmp array
        Array<OneD, Array<OneD, NekDouble>> tmp(3);
        for (int i = 0; i < 3; ++i)
        {
            // deep copy
            tmp[i] = Array<OneD, NekDouble>(nq);
            Vmath::Vcopy(nq, physin[i], 1, tmp[i], 1);
        }
 
        // \eta = h - d
        Vmath::Vsub(nq, tmp[0], 1, m_depth, 1, physout[0], 1);
 
        // u = hu/h
        Vmath::Vdiv(nq, tmp[1], 1, tmp[0], 1, physout[1], 1);
 
        // v = hv/h
        Vmath::Vdiv(nq, tmp[2], 1, tmp[0], 1, physout[2], 1);
    }
    else
    {
        // \eta = h - d
        Vmath::Vsub(nq, physin[0], 1, m_depth, 1, physout[0], 1);
 
        // u = hu/h
        Vmath::Vdiv(nq, physin[1], 1, physin[0], 1, physout[1], 1);
 
        // v = hv/h
        Vmath::Vdiv(nq, physin[2], 1, physin[0], 1, physout[2], 1);
    }
}
 
void MMFSWE::PrimitiveToConservative(
    const Array<OneD, const Array<OneD, NekDouble>> &physin,
    Array<OneD, Array<OneD, NekDouble>> &physout)
{
 
    int nq = GetTotPoints();
 
    if (physin[0].get() == physout[0].get())
    {
        // copy indata and work with tmp array
        Array<OneD, Array<OneD, NekDouble>> tmp(3);
        for (int i = 0; i < 3; ++i)
        {
            // deep copy
            tmp[i] = Array<OneD, NekDouble>(nq);
            Vmath::Vcopy(nq, physin[i], 1, tmp[i], 1);
        }
 
        // h = \eta + d
        Vmath::Vadd(nq, tmp[0], 1, m_depth, 1, physout[0], 1);
 
        // hu = h * u
        Vmath::Vmul(nq, physout[0], 1, tmp[1], 1, physout[1], 1);
 
        // hv = h * v
        Vmath::Vmul(nq, physout[0], 1, tmp[2], 1, physout[2], 1);
    }
    else
    {
        // h = \eta + d
        Vmath::Vadd(nq, physin[0], 1, m_depth, 1, physout[0], 1);
 
        // hu = h * u
        Vmath::Vmul(nq, physout[0], 1, physin[1], 1, physout[1], 1);
 
        // hv = h * v
        Vmath::Vmul(nq, physout[0], 1, physin[2], 1, physout[2], 1);
    }
}
 
// To obtain [eta, u, v ] from [H, Hu, Hv]
void MMFSWE::ConservativeToPrimitive(void)
{
    int nq = GetTotPoints();
 
    // u = hu/h
    Vmath::Vdiv(nq, m_fields[1]->GetPhys(), 1, m_fields[0]->GetPhys(), 1,
                m_fields[1]->UpdatePhys(), 1);
 
    // v = hv/ v
    Vmath::Vdiv(nq, m_fields[2]->GetPhys(), 1, m_fields[0]->GetPhys(), 1,
                m_fields[2]->UpdatePhys(), 1);
 
    // \eta = h - d
    Vmath::Vsub(nq, m_fields[0]->GetPhys(), 1, m_depth, 1,
                m_fields[0]->UpdatePhys(), 1);
}
 
void MMFSWE::PrimitiveToConservative(void)
{
    int nq = GetTotPoints();
 
    // h = \eta + d
    Vmath::Vadd(nq, m_fields[0]->GetPhys(), 1, m_depth, 1,
                m_fields[0]->UpdatePhys(), 1);
 
    // hu = h * u
    Vmath::Vmul(nq, m_fields[0]->GetPhys(), 1, m_fields[1]->GetPhys(), 1,
                m_fields[1]->UpdatePhys(), 1);
 
    // hv = h * v
    Vmath::Vmul(nq, m_fields[0]->GetPhys(), 1, m_fields[2]->GetPhys(), 1,
                m_fields[2]->UpdatePhys(), 1);
}
 
void MMFSWE::TestVorticityComputation(void)
{
    // Construct beta
    int i, k;
    int n = 1, m = 1;
    int nq = m_fields[0]->GetTotPoints();
 
    NekDouble alpha, beta_theta, beta_phi;
 
    NekDouble xp, yp, zp, Re;
    NekDouble theta, phi, sin_theta, cos_theta, sin_varphi, cos_varphi;
    NekDouble cosntheta3;
 
    NekDouble thetax, thetay, thetaz;
    NekDouble phix, phiy, phiz;
 
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    Array<OneD, NekDouble> u(nq);
    Array<OneD, NekDouble> v(nq);
 
    Array<OneD, NekDouble> vorticitycompt(nq);
    Array<OneD, NekDouble> vorticityexact(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    Array<OneD, Array<OneD, NekDouble>> uvec(m_spacedim);
    for (i = 0; i < m_spacedim; ++i)
    {
        uvec[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    // Generate SphericalCoords
    for (k = 0; k < nq; ++k)
    {
        xp = x[k];
        yp = y[k];
        zp = z[k];
 
        Re = sqrt(xp * xp + yp * yp + zp * zp);
 
        CartesianToSpherical(xp, yp, zp, sin_varphi, cos_varphi, sin_theta,
                             cos_theta);
 
        alpha = sin_varphi;
 
        theta = atan2(sin_theta, cos_theta);
        phi   = atan2(sin_varphi, cos_varphi);
 
        cosntheta3 = cos(n * theta) * cos(n * theta) * cos(n * theta);
 
        beta_theta = -4.0 * n * cosntheta3 * cos(m * phi) * sin(n * theta) / Re;
        beta_phi   = -m * cosntheta3 * sin(m * phi) / Re;
 
        thetax = -1.0 * cos_varphi * sin_theta;
        thetay = -1.0 * sin_varphi * sin_theta;
        thetaz = cos_theta;
 
        phix = -1.0 * sin_varphi;
        phiy = cos_varphi;
        phiz = 0.0;
 
        uvec[0][k] = alpha * (beta_theta * thetax + beta_phi * phix);
        uvec[1][k] = alpha * (beta_theta * thetay + beta_phi * phiy);
        uvec[2][k] = alpha * (beta_theta * thetaz + beta_phi * phiz);
 
        vorticityexact[k] = -4.0 * n / Re / Re * cos_theta * cos_theta *
                            cos_varphi * cos(m * phi) * sin(n * theta);
    }
 
    u = CartesianToMovingframes(uvec, 0);
    v = CartesianToMovingframes(uvec, 1);
 
    std::cout << "chi migi1" << std::endl;
 
    ComputeVorticity(u, v, vorticitycompt);
    /*for (int k=0; k < nq; k++)
    {
 
        std::cout << "vorticitycompt[ " << k << "]"<< "\t"<<vorticitycompt[k]<<
    std::endl;
 
    }*/
 
    Vmath::Vsub(nq, vorticityexact, 1, vorticitycompt, 1, vorticitycompt, 1);
 
    std::cout << "Vorticity: L2 error = " << AvgAbsInt(vorticitycompt)
              << ", Linf error =  " << Vmath::Vamax(nq, vorticitycompt, 1)
              << std::endl;
}
 
NekDouble MMFSWE::v_L2Error(unsigned int field,
                            const Array<OneD, NekDouble> &exactsoln,
                            bool Normalised)
{
    boost::ignore_unused(exactsoln);
 
    int nq            = m_fields[field]->GetNpoints();
    NekDouble L2error = -1.0;
 
    if (m_NumQuadPointsError == 0)
    {
        if (m_fields[field]->GetPhysState() == false)
        {
            m_fields[field]->BwdTrans(m_fields[field]->GetCoeffs(),
                                      m_fields[field]->UpdatePhys());
        }
 
        switch (field)
        {
            case (0):
            {
                // I(h) = I (h - h_exact ) / I (h_exact)
                Array<OneD, NekDouble> exactsolution(nq);
                v_EvaluateExactSolution(0, exactsolution, m_time);
 
                // exactsoln = u - u_T so that L2 compute u_T
                NekDouble L2exact = m_fields[0]->Integral(exactsolution);
 
                Vmath::Vsub(nq, &(m_fields[0]->GetPhys())[0], 1,
                            &exactsolution[0], 1, &exactsolution[0], 1);
                Vmath::Vabs(nq, exactsolution, 1, exactsolution, 1);
 
                L2error = (m_fields[0]->Integral(exactsolution)) / L2exact;
            }
            break;
 
            case (1):
            {
                // I2 (u) = I( (u - u_ext)^2 + (v - v_ext)^2 )^{1/2} / I(
                // u_ext^2 + v_ext^2 )^{1/2}
                Array<OneD, NekDouble> exactu(nq);
                Array<OneD, NekDouble> exactv(nq);
                Array<OneD, NekDouble> tmp(nq);
 
                // L2exact = \int (\sqrt{exactu*exactu+exactv*exactv})
                NekDouble L2exact;
                v_EvaluateExactSolution(1, exactu, m_time);
                v_EvaluateExactSolution(2, exactv, m_time);
                Vmath::Vmul(nq, exactu, 1, exactu, 1, tmp, 1);
                Vmath::Vvtvp(nq, exactv, 1, exactv, 1, tmp, 1, tmp, 1);
                Vmath::Vsqrt(nq, tmp, 1, tmp, 1);
 
                L2exact = m_fields[1]->Integral(tmp);
 
                // L2exact = \int
                // (\sqrt{(u-exactu)*(u-exactu)+(v-exactv)*(v-exactv)})
                Vmath::Vsub(nq, &(m_fields[1]->GetPhys())[0], 1, &exactu[0], 1,
                            &exactu[0], 1);
                Vmath::Vsub(nq, &(m_fields[2]->GetPhys())[0], 1, &exactv[0], 1,
                            &exactv[0], 1);
                Vmath::Vmul(nq, exactu, 1, exactu, 1, tmp, 1);
                Vmath::Vvtvp(nq, exactv, 1, exactv, 1, tmp, 1, tmp, 1);
                Vmath::Vsqrt(nq, tmp, 1, tmp, 1);
 
                L2error = (m_fields[1]->Integral(tmp)) / L2exact;
            }
            break;
 
            case (2):
            {
                L2error = 0.0;
            }
            break;
 
            default:
                break;
        }
 
        if (Normalised == true)
        {
            Array<OneD, NekDouble> one(m_fields[field]->GetNpoints(), 1.0);
 
            NekDouble Vol = m_fields[field]->Integral(one);
            m_comm->AllReduce(Vol, LibUtilities::ReduceSum);
 
            L2error = sqrt(L2error * L2error / Vol);
        }
    }
    else
    {
        Array<OneD, NekDouble> L2INF(2);
        L2INF   = ErrorExtraPoints(field);
        L2error = L2INF[0];
    }
 
    return L2error;
}
 
NekDouble MMFSWE::v_LinfError(unsigned int field,
                              const Array<OneD, NekDouble> &exactsoln)
{
    boost::ignore_unused(exactsoln);
 
    NekDouble LinfError = -1.0;
 
    if (m_fields[field]->GetPhysState() == false)
    {
        m_fields[field]->BwdTrans(m_fields[field]->GetCoeffs(),
                                  m_fields[field]->UpdatePhys());
    }
 
    int nq = m_fields[field]->GetNpoints();
 
    // Obtain \vec{u} in cartesian coordinate
    Array<OneD, NekDouble> x(nq);
    Array<OneD, NekDouble> y(nq);
    Array<OneD, NekDouble> z(nq);
 
    m_fields[0]->GetCoords(x, y, z);
 
    switch (field)
    {
        case (0):
        {
            NekDouble Letaint;
 
            Array<OneD, NekDouble> exactsolution(nq);
 
            EvaluateExactSolution(field, exactsolution, m_time);
            LinfError = m_fields[field]->Linf(m_fields[field]->GetPhys(),
                                              exactsolution);
 
            Letaint = Vmath::Vamax(nq, exactsolution, 1);
 
            Vmath::Vsub(nq, &(m_fields[0]->GetPhys())[0], 1, &exactsolution[0],
                        1, &exactsolution[0], 1);
 
            LinfError = fabs(LinfError / Letaint);
        }
        break;
 
        case (1):
        {
            Array<OneD, NekDouble> exactu(nq);
            Array<OneD, NekDouble> exactv(nq);
            Array<OneD, NekDouble> tmpu(nq);
            Array<OneD, NekDouble> tmpv(nq);
            Array<OneD, NekDouble> Lerr(nq);
            Array<OneD, NekDouble> uT(nq);
 
            EvaluateExactSolution(1, exactu, m_time);
            EvaluateExactSolution(2, exactv, m_time);
 
            // Compute max[sqrt{(u-uex)^2 + (v-vex)^2}]
            Vmath::Vcopy(nq, &(m_fields[1]->UpdatePhys())[0], 1, &tmpu[0], 1);
            Vmath::Vcopy(nq, &(m_fields[2]->UpdatePhys())[0], 1, &tmpv[0], 1);
 
            Vmath::Vsub(nq, &exactu[0], 1, &tmpu[0], 1, &tmpu[0], 1);
            Vmath::Vsub(nq, &exactv[0], 1, &tmpv[0], 1, &tmpv[0], 1);
 
            Vmath::Vmul(nq, &tmpu[0], 1, &tmpu[0], 1, &tmpu[0], 1);
            Vmath::Vmul(nq, &tmpv[0], 1, &tmpv[0], 1, &tmpv[0], 1);
 
            Vmath::Vadd(nq, &tmpu[0], 1, &tmpv[0], 1, &Lerr[0], 1);
            Vmath::Vsqrt(nq, &Lerr[0], 1, &Lerr[0], 1);
 
            // uT = max[sqrt( u_T^2 + v_T^2 ) ]
            Vmath::Vmul(nq, &exactu[0], 1, &exactu[0], 1, &tmpu[0], 1);
            Vmath::Vmul(nq, &exactv[0], 1, &exactv[0], 1, &tmpv[0], 1);
            Vmath::Vadd(nq, &tmpu[0], 1, &tmpv[0], 1, &uT[0], 1);
            Vmath::Vsqrt(nq, &uT[0], 1, &uT[0], 1);
 
            LinfError = Vmath::Vamax(nq, Lerr, 1) / Vmath::Vamax(nq, uT, 1);
        }
        break;
 
        case (2):
        {
            LinfError = 0.0;
        }
        break;
 
        default:
            break;
    }
 
    return LinfError;
}
 
void MMFSWE::v_EvaluateExactSolution(unsigned int field,
                                     Array<OneD, NekDouble> &outfield,
                                     const NekDouble time)
{
    switch (m_TestType)
    {
        case eTestSteadyZonal:
        {
            SteadyZonalFlow(field, outfield);
        }
        break;
 
        case eTestUnsteadyZonal:
        {
            UnsteadyZonalFlow(field, time, outfield);
        }
        break;
 
        case eTestIsolatedMountain:
        {
            IsolatedMountainFlow(field, time, outfield);
        }
        break;
 
        case eTestUnstableJet:
        {
            UnstableJetFlow(field, time, outfield);
        }
        break;
 
        case eTestRossbyWave:
        {
            RossbyWave(field, outfield);
        }
        break;
 
        case eTestPlane:
        {
            TestSWE2Dproblem(time, field, outfield);
        }
        break;
 
        default:
            break;
    }
}
 
void MMFSWE::v_GenerateSummary(SolverUtils::SummaryList &s)
{
    MMFSystem::v_GenerateSummary(s);
    SolverUtils::AddSummaryItem(s, "TestType", TestTypeMap[m_TestType]);
 
    switch (m_TestType)
    {
        case eTestSteadyZonal:
        {
            SolverUtils::AddSummaryItem(s, "Rotation Angle", m_alpha);
        }
        break;
 
        case eTestRossbyWave:
        {
            SolverUtils::AddSummaryItem(s, "RossbyDistrubance",
                                        m_RossbyDisturbance);
        }
        break;
 
        case eTestUnstableJet:
        {
            SolverUtils::AddSummaryItem(s, "PurturbedJet", m_PurturbedJet);
        }
        break;
 
        default:
            break;
    }
}
 
} // end namespace Nektar