LinearParabolicPdeSystemWithCoupledOdeSystemSolver.hpp

/*
 
Copyright (c) 2005-2026, University of Oxford.
All rights reserved.
 
University of Oxford means the Chancellor, Masters and Scholars of the
University of Oxford, having an administrative office at Wellington
Square, Oxford OX1 2JD, UK.
 
This file is part of Chaste.
 
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
 * Redistributions of source code must retain the above copyright notice,
   this list of conditions and the following disclaimer.
 * Redistributions in binary form must reproduce the above copyright notice,
   this list of conditions and the following disclaimer in the documentation
   and/or other materials provided with the distribution.
 * Neither the name of the University of Oxford nor the names of its
   contributors may be used to endorse or promote products derived from this
   software without specific prior written permission.
 
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 
*/
#ifndef LINEARPARABOLICPDESYSTEMWITHCOUPLEDODESYSTEMSOLVER_HPP_
#define LINEARPARABOLICPDESYSTEMWITHCOUPLEDODESYSTEMSOLVER_HPP_
 
#include "AbstractAssemblerSolverHybrid.hpp"
#include "AbstractDynamicLinearPdeSolver.hpp"
#include "AbstractLinearParabolicPdeSystemForCoupledOdeSystem.hpp"
#include "TetrahedralMesh.hpp"
#include "BoundaryConditionsContainer.hpp"
#include "AbstractOdeSystemForCoupledPdeSystem.hpp"
#include "CvodeAdaptor.hpp"
#include "BackwardEulerIvpOdeSolver.hpp"
#include "Warnings.hpp"
#include "VtkMeshWriter.hpp"
 
#include <boost/shared_ptr.hpp>
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM=ELEMENT_DIM, unsigned PROBLEM_DIM=1>
class LinearParabolicPdeSystemWithCoupledOdeSystemSolver
    : public AbstractAssemblerSolverHybrid<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM, NORMAL>,
      public AbstractDynamicLinearPdeSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>
{
private:
 
    AbstractTetrahedralMesh<ELEMENT_DIM, SPACE_DIM>* mpMesh;
 
    AbstractLinearParabolicPdeSystemForCoupledOdeSystem<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>* mpPdeSystem;
 
    std::vector<AbstractOdeSystemForCoupledPdeSystem*> mOdeSystemsAtNodes;
 
    std::vector<double> mInterpolatedOdeStateVariables;
 
    boost::shared_ptr<AbstractIvpOdeSolver> mpOdeSolver;
 
    double mSamplingTimeStep;
 
    bool mOdeSystemsPresent;
 
    out_stream mpVtkMetaFile;
 
    bool mClearOutputDirectory;
 
    void WriteVtkResultsToFile();
 
    c_matrix<double, PROBLEM_DIM*(ELEMENT_DIM+1), PROBLEM_DIM*(ELEMENT_DIM+1)> ComputeMatrixTerm(
        c_vector<double, ELEMENT_DIM+1>& rPhi,
        c_matrix<double, SPACE_DIM, ELEMENT_DIM+1>& rGradPhi,
        ChastePoint<SPACE_DIM>& rX,
        c_vector<double,PROBLEM_DIM>& rU,
        c_matrix<double, PROBLEM_DIM, SPACE_DIM>& rGradU,
        Element<ELEMENT_DIM, SPACE_DIM>* pElement);
 
    c_vector<double, PROBLEM_DIM*(ELEMENT_DIM+1)> ComputeVectorTerm(
        c_vector<double, ELEMENT_DIM+1>& rPhi,
        c_matrix<double, SPACE_DIM, ELEMENT_DIM+1>& rGradPhi,
        ChastePoint<SPACE_DIM>& rX,
        c_vector<double,PROBLEM_DIM>& rU,
        c_matrix<double,PROBLEM_DIM,SPACE_DIM>& rGradU,
        Element<ELEMENT_DIM, SPACE_DIM>* pElement);
 
    void ResetInterpolatedQuantities();
 
    void IncrementInterpolatedQuantities(double phiI, const Node<SPACE_DIM>* pNode);
 
    void InitialiseForSolve(Vec initialSolution=NULL);
 
    void SetupLinearSystem(Vec currentSolution, bool computeMatrix);
 
public:
 
    LinearParabolicPdeSystemWithCoupledOdeSystemSolver(TetrahedralMesh<ELEMENT_DIM, SPACE_DIM>* pMesh,
                                                       AbstractLinearParabolicPdeSystemForCoupledOdeSystem<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>* pPdeSystem,
                                                       BoundaryConditionsContainer<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>* pBoundaryConditions,
                                                       std::vector<AbstractOdeSystemForCoupledPdeSystem*> odeSystemsAtNodes=std::vector<AbstractOdeSystemForCoupledPdeSystem*>(),
                                                       boost::shared_ptr<AbstractIvpOdeSolver> pOdeSolver=boost::shared_ptr<AbstractIvpOdeSolver>());
 
    ~LinearParabolicPdeSystemWithCoupledOdeSystemSolver();
 
    void PrepareForSetupLinearSystem(Vec currentPdeSolution);
 
    void SetOutputDirectory(std::string outputDirectory, bool clearDirectory=false);
 
    void SetSamplingTimeStep(double samplingTimeStep);
 
    void SolveAndWriteResultsToFile();
 
    void WriteVtkResultsToFile(Vec solution, unsigned numTimeStepsElapsed);
 
    AbstractOdeSystemForCoupledPdeSystem* GetOdeSystemAtNode(unsigned index);
};
 
// Implementation
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
c_matrix<double, PROBLEM_DIM*(ELEMENT_DIM+1), PROBLEM_DIM*(ELEMENT_DIM+1)> LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::ComputeMatrixTerm(
    c_vector<double, ELEMENT_DIM+1>& rPhi,
    c_matrix<double, SPACE_DIM, ELEMENT_DIM+1>& rGradPhi,
    ChastePoint<SPACE_DIM>& rX,
    c_vector<double,PROBLEM_DIM>& rU,
    c_matrix<double, PROBLEM_DIM, SPACE_DIM>& rGradU,
    Element<ELEMENT_DIM, SPACE_DIM>* pElement)
{
    double timestep_inverse = PdeSimulationTime::GetPdeTimeStepInverse();
    c_matrix<double, PROBLEM_DIM*(ELEMENT_DIM+1), PROBLEM_DIM*(ELEMENT_DIM+1)> matrix_term = zero_matrix<double>(PROBLEM_DIM*(ELEMENT_DIM+1), PROBLEM_DIM*(ELEMENT_DIM+1));
 
    // Loop over PDEs and populate matrix_term
    for (unsigned pde_index=0; pde_index<PROBLEM_DIM; pde_index++)
    {
        double this_dudt_coefficient = mpPdeSystem->ComputeDuDtCoefficientFunction(rX, pde_index);
        c_matrix<double, SPACE_DIM, SPACE_DIM> this_pde_diffusion_term = mpPdeSystem->ComputeDiffusionTerm(rX, pde_index, pElement);
        c_matrix<double, 1*(ELEMENT_DIM+1), 1*(ELEMENT_DIM+1)> this_stiffness_matrix = prod(trans(rGradPhi), c_matrix<double, SPACE_DIM, ELEMENT_DIM+1>(prod(this_pde_diffusion_term, rGradPhi)) ) + timestep_inverse * this_dudt_coefficient * outer_prod(rPhi, rPhi);
 
        for (unsigned i=0; i<ELEMENT_DIM+1; i++)
        {
            for (unsigned j=0; j<ELEMENT_DIM+1; j++)
            {
                matrix_term(i*PROBLEM_DIM + pde_index, j*PROBLEM_DIM + pde_index) = this_stiffness_matrix(i,j);
            }
        }
    }
    return matrix_term;
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
c_vector<double, PROBLEM_DIM*(ELEMENT_DIM+1)> LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::ComputeVectorTerm(
    c_vector<double, ELEMENT_DIM+1>& rPhi,
    c_matrix<double, SPACE_DIM, ELEMENT_DIM+1>& rGradPhi,
    ChastePoint<SPACE_DIM>& rX,
    c_vector<double,PROBLEM_DIM>& rU,
    c_matrix<double,PROBLEM_DIM,SPACE_DIM>& rGradU,
    Element<ELEMENT_DIM, SPACE_DIM>* pElement)
{
    double timestep_inverse = PdeSimulationTime::GetPdeTimeStepInverse();
    c_vector<double, PROBLEM_DIM*(ELEMENT_DIM+1)> vector_term;
    vector_term = zero_vector<double>(PROBLEM_DIM*(ELEMENT_DIM+1));
 
    // Loop over PDEs and populate vector_term
    for (unsigned pde_index=0; pde_index<PROBLEM_DIM; pde_index++)
    {
        double this_dudt_coefficient = mpPdeSystem->ComputeDuDtCoefficientFunction(rX, pde_index);
        double this_source_term = mpPdeSystem->ComputeSourceTerm(rX, rU, mInterpolatedOdeStateVariables, pde_index);
        c_vector<double, ELEMENT_DIM+1> this_vector_term;
        this_vector_term = (this_source_term + timestep_inverse*this_dudt_coefficient*rU(pde_index))* rPhi;
 
        for (unsigned i=0; i<ELEMENT_DIM+1; i++)
        {
            vector_term(i*PROBLEM_DIM + pde_index) = this_vector_term(i);
        }
    }
 
    return vector_term;
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::ResetInterpolatedQuantities()
{
    mInterpolatedOdeStateVariables.clear();
 
    if (mOdeSystemsPresent)
    {
        unsigned num_state_variables = mOdeSystemsAtNodes[0]->GetNumberOfStateVariables();
        mInterpolatedOdeStateVariables.resize(num_state_variables, 0.0);
    }
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::IncrementInterpolatedQuantities(double phiI, const Node<SPACE_DIM>* pNode)
{
    if (mOdeSystemsPresent)
    {
        unsigned num_state_variables = mOdeSystemsAtNodes[0]->GetNumberOfStateVariables();
 
        for (unsigned i=0; i<num_state_variables; i++)
        {
            mInterpolatedOdeStateVariables[i] += phiI * mOdeSystemsAtNodes[pNode->GetIndex()]->rGetStateVariables()[i];
        }
    }
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::InitialiseForSolve(Vec initialSolution)
{
    if (this->mpLinearSystem == NULL)
    {
        unsigned preallocation = mpMesh->CalculateMaximumContainingElementsPerProcess() + ELEMENT_DIM;
        if (ELEMENT_DIM > 1)
        {
            // Highest connectivity is closed
            preallocation--;
        }
        preallocation *= PROBLEM_DIM;
 
        /*
         * Use the current solution (ie the initial solution) as the
         * template in the alternative constructor of LinearSystem.
         * This is to avoid problems with VecScatter.
         */
        this->mpLinearSystem = new LinearSystem(initialSolution, preallocation);
    }
 
    assert(this->mpLinearSystem);
    this->mpLinearSystem->SetMatrixIsSymmetric(true);
    this->mpLinearSystem->SetKspType("cg");
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::SetupLinearSystem(Vec currentSolution, bool computeMatrix)
{
    this->SetupGivenLinearSystem(currentSolution, computeMatrix, this->mpLinearSystem);
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::LinearParabolicPdeSystemWithCoupledOdeSystemSolver(
        TetrahedralMesh<ELEMENT_DIM, SPACE_DIM>* pMesh,
        AbstractLinearParabolicPdeSystemForCoupledOdeSystem<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>* pPdeSystem,
        BoundaryConditionsContainer<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>* pBoundaryConditions,
        std::vector<AbstractOdeSystemForCoupledPdeSystem*> odeSystemsAtNodes,
        boost::shared_ptr<AbstractIvpOdeSolver> pOdeSolver)
    : AbstractAssemblerSolverHybrid<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM, NORMAL>(pMesh, pBoundaryConditions),
      AbstractDynamicLinearPdeSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>(pMesh),
      mpMesh(pMesh),
      mpPdeSystem(pPdeSystem),
      mOdeSystemsAtNodes(odeSystemsAtNodes),
      mpOdeSolver(pOdeSolver),
      mSamplingTimeStep(DOUBLE_UNSET),
      mOdeSystemsPresent(false),
      mClearOutputDirectory(false)
{
    this->mpBoundaryConditions = pBoundaryConditions;
 
    /*
     * If any ODE systems are passed in to the constructor, then we aren't just
     * solving a coupled PDE system, in which case the number of ODE system objects
     * must match the number of nodes in the finite element mesh.
     */
    if (!mOdeSystemsAtNodes.empty())
    {
        mOdeSystemsPresent = true;
        assert(mOdeSystemsAtNodes.size() == mpMesh->GetNumNodes());
 
        /*
         * In this case, if an ODE solver is not explicitly passed into the
         * constructor, then we create a default solver.
         */
        if (!mpOdeSolver)
        {
#ifdef CHASTE_CVODE
            mpOdeSolver.reset(new CvodeAdaptor);
#else
            mpOdeSolver.reset(new BackwardEulerIvpOdeSolver(mOdeSystemsAtNodes[0]->GetNumberOfStateVariables()));
#endif //CHASTE_CVODE
        }
    }
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::~LinearParabolicPdeSystemWithCoupledOdeSystemSolver()
{
    if (mOdeSystemsPresent)
    {
        for (unsigned i=0; i<mOdeSystemsAtNodes.size(); i++)
        {
            delete mOdeSystemsAtNodes[i];
        }
    }
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::PrepareForSetupLinearSystem(Vec currentPdeSolution)
{
    if (mOdeSystemsPresent)
    {
        double time = PdeSimulationTime::GetTime();
        double next_time = PdeSimulationTime::GetNextTime();
        double dt = PdeSimulationTime::GetPdeTimeStep();
 
        ReplicatableVector soln_repl(currentPdeSolution);
        std::vector<double> current_soln_this_node(PROBLEM_DIM);
 
        // Loop over nodes
        for (unsigned node_index=0; node_index<mpMesh->GetNumNodes(); node_index++)
        {
            // Store the current solution to the PDE system at this node
            for (unsigned pde_index=0; pde_index<PROBLEM_DIM; pde_index++)
            {
                double current_soln_this_pde_this_node = soln_repl[PROBLEM_DIM*node_index + pde_index];
                current_soln_this_node[pde_index] = current_soln_this_pde_this_node;
            }
 
            // Pass it into the ODE system at this node
            mOdeSystemsAtNodes[node_index]->SetPdeSolution(current_soln_this_node);
 
            // Solve ODE system at this node
            mpOdeSolver->SolveAndUpdateStateVariable(mOdeSystemsAtNodes[node_index], time, next_time, dt);
        }
    }
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::SetOutputDirectory(std::string outputDirectory, bool clearDirectory)
{
    mClearOutputDirectory = clearDirectory;
    this->mOutputDirectory = outputDirectory;
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::SetSamplingTimeStep(double samplingTimeStep)
{
    assert(samplingTimeStep >= this->mIdealTimeStep);
    mSamplingTimeStep = samplingTimeStep;
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::SolveAndWriteResultsToFile()
{
    // A number of methods must have been called prior to this method
    if (this->mOutputDirectory == "")
    {
        EXCEPTION("SetOutputDirectory() must be called prior to SolveAndWriteResultsToFile()");
    }
    if (this->mTimesSet == false)
    {
        EXCEPTION("SetTimes() must be called prior to SolveAndWriteResultsToFile()");
    }
    if (this->mIdealTimeStep <= 0.0)
    {
        EXCEPTION("SetTimeStep() must be called prior to SolveAndWriteResultsToFile()");
    }
    if (mSamplingTimeStep == DOUBLE_UNSET)
    {
        EXCEPTION("SetSamplingTimeStep() must be called prior to SolveAndWriteResultsToFile()");
    }
    if (!this->mInitialCondition)
    {
        EXCEPTION("SetInitialCondition() must be called prior to SolveAndWriteResultsToFile()");
    }
 
#ifdef CHASTE_VTK
    // Create a .pvd output file
    OutputFileHandler output_file_handler(this->mOutputDirectory, mClearOutputDirectory);
    mpVtkMetaFile = output_file_handler.OpenOutputFile("results.pvd");
    *mpVtkMetaFile << "<?xml version=\"1.0\"?>\n";
    *mpVtkMetaFile << "<VTKFile type=\"Collection\" version=\"0.1\" byte_order=\"LittleEndian\" compressor=\"vtkZLibDataCompressor\">\n";
    *mpVtkMetaFile << "    <Collection>\n";
 
    // Write initial condition to VTK
    Vec initial_condition = this->mInitialCondition;
    WriteVtkResultsToFile(initial_condition, 0);
 
    // The helper class TimeStepper deals with issues such as small final timesteps so we don't have to
    TimeStepper stepper(this->mTstart, this->mTend, mSamplingTimeStep);
 
    // Main time loop
    while (!stepper.IsTimeAtEnd())
    {
        // Reset start and end times
        this->SetTimes(stepper.GetTime(), stepper.GetNextTime());
 
        // Solve the system up to the new end time
        Vec soln = this->Solve();
 
        // Reset the initial condition for the next timestep
        if (this->mInitialCondition != initial_condition)
        {
            PetscTools::Destroy(this->mInitialCondition);
        }
        this->mInitialCondition = soln;
 
        // Move forward in time
        stepper.AdvanceOneTimeStep();
 
        // Write solution to VTK
        WriteVtkResultsToFile(soln, stepper.GetTotalTimeStepsTaken());
    }
 
    // Restore saved initial condition to avoid user confusion!
    if (this->mInitialCondition != initial_condition)
    {
        PetscTools::Destroy(this->mInitialCondition);
    }
    this->mInitialCondition = initial_condition;
 
    // Close .pvd output file
    *mpVtkMetaFile << "    </Collection>\n";
    *mpVtkMetaFile << "</VTKFile>\n";
    mpVtkMetaFile->close();
#else //CHASTE_VTK
// LCOV_EXCL_START // We only test this in weekly builds
    WARNING("VTK is not installed and is required for this functionality");
// LCOV_EXCL_STOP
#endif //CHASTE_VTK
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
void LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::WriteVtkResultsToFile(Vec solution, unsigned numTimeStepsElapsed)
{
#ifdef CHASTE_VTK
 
    // Create a new VTK file for this time step
    std::stringstream time;
    time << numTimeStepsElapsed;
    VtkMeshWriter<ELEMENT_DIM, SPACE_DIM> mesh_writer(this->mOutputDirectory, "results_"+time.str(), false);
 
    /*
     * We first loop over PDEs. For each PDE we store the solution
     * at each node in a vector, then pass this vector to the mesh
     * writer.
     */
    ReplicatableVector solution_repl(solution);
    unsigned num_nodes = mpMesh->GetNumNodes();
    for (unsigned pde_index=0; pde_index<PROBLEM_DIM; pde_index++)
    {
        // Store the solution of this PDE at each node
        std::vector<double> pde_index_data;
        pde_index_data.resize(num_nodes, 0.0);
        for (unsigned node_index=0; node_index<num_nodes; node_index++)
        {
            pde_index_data[node_index] = solution_repl[PROBLEM_DIM*node_index + pde_index];
        }
 
        // Add this data to the mesh writer
        std::stringstream data_name;
        data_name << "PDE variable " << pde_index;
        mesh_writer.AddPointData(data_name.str(), pde_index_data);
    }
 
    if (mOdeSystemsPresent)
    {
        /*
         * We cannot loop over ODEs like PDEs, since the solutions are not
         * stored in one place. Therefore we build up a large 'vector of
         * vectors', then pass each component of this vector to the mesh
         * writer.
         */
        std::vector<std::vector<double> > ode_data;
        unsigned num_odes = mOdeSystemsAtNodes[0]->rGetStateVariables().size();
        ode_data.resize(num_odes);
        for (unsigned ode_index=0; ode_index<num_odes; ode_index++)
        {
            ode_data[ode_index].resize(num_nodes, 0.0);
        }
 
        for (unsigned node_index=0; node_index<num_nodes; node_index++)
        {
            std::vector<double> all_odes_this_node = mOdeSystemsAtNodes[node_index]->rGetStateVariables();
            for (unsigned i=0; i<num_odes; i++)
            {
                ode_data[i][node_index] = all_odes_this_node[i];
            }
        }
 
        for (unsigned ode_index=0; ode_index<num_odes; ode_index++)
        {
            std::vector<double> ode_index_data = ode_data[ode_index];
 
            // Add this data to the mesh writer
            std::stringstream data_name;
            data_name << "ODE variable " << ode_index;
            mesh_writer.AddPointData(data_name.str(), ode_index_data);
        }
    }
 
    mesh_writer.WriteFilesUsingMesh(*mpMesh);
    *mpVtkMetaFile << "        <DataSet timestep=\"";
    *mpVtkMetaFile << numTimeStepsElapsed;
    *mpVtkMetaFile << "\" group=\"\" part=\"0\" file=\"results_";
    *mpVtkMetaFile << numTimeStepsElapsed;
    *mpVtkMetaFile << ".vtu\"/>\n";
#endif // CHASTE_VTK
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM, unsigned PROBLEM_DIM>
AbstractOdeSystemForCoupledPdeSystem* LinearParabolicPdeSystemWithCoupledOdeSystemSolver<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::GetOdeSystemAtNode(unsigned index)
{
    return mOdeSystemsAtNodes[index];
}
 
#endif /*LINEARPARABOLICPDESYSTEMWITHCOUPLEDODESYSTEMSOLVER_HPP_*/