Tempus_DIRKTest.cpp

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// @HEADER
// ****************************************************************************
//                Tempus: Copyright (2017) Sandia Corporation
//
// Distributed under BSD 3-clause license (See accompanying file Copyright.txt)
// ****************************************************************************
// @HEADER

#include "Teuchos_UnitTestHarness.hpp"
#include "Teuchos_XMLParameterListHelpers.hpp"
#include "Teuchos_TimeMonitor.hpp"

#include "Thyra_VectorStdOps.hpp"

#include "Tempus_IntegratorBasic.hpp"
#include "Tempus_StepperDIRK.hpp"

#include "../TestModels/SinCosModel.hpp"
#include "../TestModels/VanDerPolModel.hpp"
#include "../TestUtils/Tempus_ConvergenceTestUtils.hpp"

#include <fstream>
#include <vector>

namespace Tempus_Test {

using Teuchos::RCP;
using Teuchos::rcp;
using Teuchos::rcp_const_cast;
using Teuchos::ParameterList;
using Teuchos::parameterList;
using Teuchos::sublist;
using Teuchos::getParametersFromXmlFile;

using Tempus::IntegratorBasic;
using Tempus::SolutionHistory;
using Tempus::SolutionState;

// Comment out any of the following tests to exclude from build/run.
#define TEST_PARAMETERLIST
#define TEST_CONSTRUCTING_FROM_DEFAULTS
#define TEST_SINCOS
#define TEST_VANDERPOL
#define TEST_EMBEDDED_DIRK


#ifdef TEST_PARAMETERLIST
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(DIRK, ParameterList)
{
  std::vector<std::string> RKMethods;
  RKMethods.push_back("RK Backward Euler");
  RKMethods.push_back("IRK 1 Stage Theta Method");
  RKMethods.push_back("Implicit Midpoint");
  RKMethods.push_back("SDIRK 1 Stage 1st order");
  RKMethods.push_back("SDIRK 2 Stage 2nd order");
  RKMethods.push_back("SDIRK 2 Stage 3rd order");
  RKMethods.push_back("EDIRK 2 Stage 3rd order");
  RKMethods.push_back("EDIRK 2 Stage Theta Method");
  RKMethods.push_back("SDIRK 3 Stage 4th order");
  RKMethods.push_back("SDIRK 5 Stage 4th order");
  RKMethods.push_back("SDIRK 5 Stage 5th order");
  RKMethods.push_back("SDIRK 2(1) Pair");

  for(std::vector<std::string>::size_type m = 0; m != RKMethods.size(); m++) {

    std::string RKMethod = RKMethods[m];

    // Read params from .xml file
    RCP<ParameterList> pList =
      getParametersFromXmlFile("Tempus_DIRK_SinCos.xml");

    // Setup the SinCosModel
    RCP<ParameterList> scm_pl = sublist(pList, "SinCosModel", true);
    auto model = rcp(new SinCosModel<double>(scm_pl));

    RCP<ParameterList> tempusPL  = sublist(pList, "Tempus", true);
    tempusPL->sublist("Default Stepper").set("Stepper Type", RKMethods[m]);

    if (RKMethods[m] == "IRK 1 Stage Theta Method" ||
        RKMethods[m] == "Implicit Midpoint" ||
        RKMethods[m] == "EDIRK 2 Stage Theta Method") {
      // Construct in the same order as default.
      RCP<ParameterList> stepperPL = sublist(tempusPL, "Default Stepper", true);
      RCP<ParameterList> solverPL = parameterList();
      *solverPL  = *(sublist(stepperPL, "Default Solver", true));
      tempusPL->sublist("Default Stepper").remove("Default Solver");
      tempusPL->sublist("Default Stepper")
           .set<double>("theta", 0.5);
      tempusPL->sublist("Default Stepper").remove("Zero Initial Guess");
      tempusPL->sublist("Default Stepper").set<bool>("Zero Initial Guess", 0);
      tempusPL->sublist("Default Stepper").set("Default Solver", *solverPL);
    } else if (RKMethods[m] == "SDIRK 2 Stage 2nd order") {
      // Construct in the same order as default.
      RCP<ParameterList> stepperPL = sublist(tempusPL, "Default Stepper", true);
      RCP<ParameterList> solverPL = parameterList();
      *solverPL  = *(sublist(stepperPL, "Default Solver", true));
      tempusPL->sublist("Default Stepper").remove("Default Solver");
      tempusPL->sublist("Default Stepper")
           .set<double>("gamma", 0.2928932188134524);
      tempusPL->sublist("Default Stepper").remove("Zero Initial Guess");
      tempusPL->sublist("Default Stepper").set<bool>("Zero Initial Guess", 0);
      tempusPL->sublist("Default Stepper").set("Default Solver", *solverPL);
    } else if (RKMethods[m] == "SDIRK 2 Stage 3rd order") {
      // Construct in the same order as default.
      RCP<ParameterList> stepperPL = sublist(tempusPL, "Default Stepper", true);
      RCP<ParameterList> solverPL = parameterList();
      *solverPL  = *(sublist(stepperPL, "Default Solver", true));
      tempusPL->sublist("Default Stepper").remove("Default Solver");
      tempusPL->sublist("Default Stepper").set("3rd Order A-stable", true);
      tempusPL->sublist("Default Stepper").set("2nd Order L-stable", false);
      tempusPL->sublist("Default Stepper")
           .set<double>("gamma", 0.7886751345948128);
      tempusPL->sublist("Default Stepper").remove("Zero Initial Guess");
      tempusPL->sublist("Default Stepper").set<bool>("Zero Initial Guess", 0);
      tempusPL->sublist("Default Stepper").set("Default Solver", *solverPL);
    }

    // Test constructor IntegratorBasic(tempusPL, model)
    {
      RCP<Tempus::IntegratorBasic<double> > integrator =
        Tempus::integratorBasic<double>(tempusPL, model);

      RCP<ParameterList> stepperPL = sublist(tempusPL, "Default Stepper", true);
      RCP<ParameterList> defaultPL =
        integrator->getStepper()->getDefaultParameters();
      defaultPL->remove("Description");

      // Adjust default parameters for pseudonyms.
      if ( RKMethods[m] == "Implicit Midpoint" ) {
        defaultPL->set("Stepper Type", "Implicit Midpoint");
      }

      bool pass = haveSameValues(*stepperPL, *defaultPL, true);
      if (!pass) {
        std::cout << std::endl;
        std::cout << "stepperPL -------------- \n" << *stepperPL << std::endl;
        std::cout << "defaultPL -------------- \n" << *defaultPL << std::endl;
      }
      TEST_ASSERT(pass)
    }

    // Test constructor IntegratorBasic(model, stepperType)
    {
      RCP<Tempus::IntegratorBasic<double> > integrator =
        Tempus::integratorBasic<double>(model, RKMethods[m]);

      RCP<ParameterList> stepperPL = sublist(tempusPL, "Default Stepper", true);
      RCP<ParameterList> defaultPL =
        integrator->getStepper()->getDefaultParameters();
      defaultPL->remove("Description");

      // Adjust default parameters for pseudonyms.
      if ( RKMethods[m] == "Implicit Midpoint" ) {
        defaultPL->set("Stepper Type", "Implicit Midpoint");
      }

      bool pass = haveSameValues(*stepperPL, *defaultPL, true);
      if (!pass) {
        std::cout << std::endl;
        std::cout << "stepperPL -------------- \n" << *stepperPL << std::endl;
        std::cout << "defaultPL -------------- \n" << *defaultPL << std::endl;
      }
      TEST_ASSERT(pass)
    }
  }
}
#endif // TEST_PARAMETERLIST


#ifdef TEST_CONSTRUCTING_FROM_DEFAULTS
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(DIRK, ConstructingFromDefaults)
{
  double dt = 0.025;

  // Read params from .xml file
  RCP<ParameterList> pList =
    getParametersFromXmlFile("Tempus_DIRK_SinCos.xml");
  RCP<ParameterList> pl = sublist(pList, "Tempus", true);

  // Setup the SinCosModel
  RCP<ParameterList> scm_pl = sublist(pList, "SinCosModel", true);
  //RCP<SinCosModel<double> > model = sineCosineModel(scm_pl);
  auto model = rcp(new SinCosModel<double>(scm_pl));

  // Setup Stepper for field solve ----------------------------
  auto stepper = rcp(new Tempus::StepperDIRK<double>());
  stepper->setModel(model);
  stepper->initialize();

  // Setup TimeStepControl ------------------------------------
  auto timeStepControl = rcp(new Tempus::TimeStepControl<double>());
  ParameterList tscPL = pl->sublist("Default Integrator")
                           .sublist("Time Step Control");
  timeStepControl->setStepType (tscPL.get<std::string>("Integrator Step Type"));
  timeStepControl->setInitIndex(tscPL.get<int>   ("Initial Time Index"));
  timeStepControl->setInitTime (tscPL.get<double>("Initial Time"));
  timeStepControl->setFinalTime(tscPL.get<double>("Final Time"));
  timeStepControl->setInitTimeStep(dt);
  timeStepControl->initialize();

  // Setup initial condition SolutionState --------------------
  Thyra::ModelEvaluatorBase::InArgs<double> inArgsIC =
    stepper->getModel()->getNominalValues();
  auto icSolution = rcp_const_cast<Thyra::VectorBase<double> > (inArgsIC.get_x());
  auto icState = rcp(new Tempus::SolutionState<double>(icSolution));
  icState->setTime    (timeStepControl->getInitTime());
  icState->setIndex   (timeStepControl->getInitIndex());
  icState->setTimeStep(0.0);
  icState->setOrder   (stepper->getOrder());
  icState->setSolutionStatus(Tempus::Status::PASSED);  // ICs are passing.

  // Setup SolutionHistory ------------------------------------
  auto solutionHistory = rcp(new Tempus::SolutionHistory<double>());
  solutionHistory->setName("Forward States");
  solutionHistory->setStorageType(Tempus::STORAGE_TYPE_STATIC);
  solutionHistory->setStorageLimit(2);
  solutionHistory->addState(icState);

  // Setup Integrator -----------------------------------------
  RCP<Tempus::IntegratorBasic<double> > integrator =
    Tempus::integratorBasic<double>();
  integrator->setStepperWStepper(stepper);
  integrator->setTimeStepControl(timeStepControl);
  integrator->setSolutionHistory(solutionHistory);
  //integrator->setObserver(...);
  integrator->initialize();


  // Integrate to timeMax
  bool integratorStatus = integrator->advanceTime();
  TEST_ASSERT(integratorStatus)


  // Test if at 'Final Time'
  double time = integrator->getTime();
  double timeFinal =pl->sublist("Default Integrator")
     .sublist("Time Step Control").get<double>("Final Time");
  TEST_FLOATING_EQUALITY(time, timeFinal, 1.0e-14);

  // Time-integrated solution and the exact solution
  RCP<Thyra::VectorBase<double> > x = integrator->getX();
  RCP<const Thyra::VectorBase<double> > x_exact =
    model->getExactSolution(time).get_x();

  // Calculate the error
  RCP<Thyra::VectorBase<double> > xdiff = x->clone_v();
  Thyra::V_StVpStV(xdiff.ptr(), 1.0, *x_exact, -1.0, *(x));

  // Check the order and intercept
  std::cout << "  Stepper = SDIRK 2 Stage 2nd order" << std::endl;
  std::cout << "  =========================" << std::endl;
  std::cout << "  Exact solution   : " << get_ele(*(x_exact), 0) << "   "
                                       << get_ele(*(x_exact), 1) << std::endl;
  std::cout << "  Computed solution: " << get_ele(*(x      ), 0) << "   "
                                       << get_ele(*(x      ), 1) << std::endl;
  std::cout << "  Difference       : " << get_ele(*(xdiff  ), 0) << "   "
                                       << get_ele(*(xdiff  ), 1) << std::endl;
  std::cout << "  =========================" << std::endl;
  TEST_FLOATING_EQUALITY(get_ele(*(x), 0), 0.841470, 1.0e-4 );
  TEST_FLOATING_EQUALITY(get_ele(*(x), 1), 0.540304, 1.0e-4 );
}
#endif // TEST_CONSTRUCTING_FROM_DEFAULTS


#ifdef TEST_SINCOS
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(DIRK, SinCos)
{
  std::vector<std::string> RKMethods;
  RKMethods.push_back("RK Backward Euler");
  RKMethods.push_back("IRK 1 Stage Theta Method");
  RKMethods.push_back("Implicit Midpoint");
  RKMethods.push_back("SDIRK 1 Stage 1st order");
  RKMethods.push_back("SDIRK 2 Stage 2nd order");
  RKMethods.push_back("SDIRK 2 Stage 3rd order");
  RKMethods.push_back("EDIRK 2 Stage 3rd order");
  RKMethods.push_back("EDIRK 2 Stage Theta Method");
  RKMethods.push_back("SDIRK 3 Stage 4th order");
  RKMethods.push_back("SDIRK 5 Stage 4th order");
  RKMethods.push_back("SDIRK 5 Stage 5th order");
  RKMethods.push_back("SDIRK 2(1) Pair");

  std::vector<double> RKMethodErrors;
  RKMethodErrors.push_back(0.0124201);
  RKMethodErrors.push_back(5.20785e-05);
  RKMethodErrors.push_back(5.20785e-05);
  RKMethodErrors.push_back(0.0124201);
  RKMethodErrors.push_back(2.52738e-05);
  RKMethodErrors.push_back(1.40223e-06);
  RKMethodErrors.push_back(2.17004e-07);
  RKMethodErrors.push_back(5.20785e-05);
  RKMethodErrors.push_back(6.41463e-08);
  RKMethodErrors.push_back(3.30631e-10);
  RKMethodErrors.push_back(1.35728e-11);
  RKMethodErrors.push_back(0.0001041);

  for(std::vector<std::string>::size_type m = 0; m != RKMethods.size(); m++) {

    std::string RKMethod = RKMethods[m];
    std::replace(RKMethod.begin(), RKMethod.end(), ' ', '_');
    std::replace(RKMethod.begin(), RKMethod.end(), '/', '.');

    RCP<Tempus::IntegratorBasic<double> > integrator;
    std::vector<RCP<Thyra::VectorBase<double>>> solutions;
    std::vector<RCP<Thyra::VectorBase<double>>> solutionsDot;
    std::vector<double> StepSize;
    std::vector<double> xErrorNorm;
    std::vector<double> xDotErrorNorm;

    const int nTimeStepSizes = 2; // 7 for error plots
    double dt = 0.05;
    double time = 0.0;
    for (int n=0; n<nTimeStepSizes; n++) {

      // Read params from .xml file
      RCP<ParameterList> pList =
        getParametersFromXmlFile("Tempus_DIRK_SinCos.xml");

      // Setup the SinCosModel
      RCP<ParameterList> scm_pl = sublist(pList, "SinCosModel", true);
      auto model = rcp(new SinCosModel<double>(scm_pl));

      // Set the Stepper
      RCP<ParameterList> pl = sublist(pList, "Tempus", true);
      pl->sublist("Default Stepper").set("Stepper Type", RKMethods[m]);
      if (RKMethods[m] == "IRK 1 Stage Theta Method" ||
          RKMethods[m] == "Implicit Midpoint" ||
          RKMethods[m] == "EDIRK 2 Stage Theta Method") {
        pl->sublist("Default Stepper").set<double>("theta", 0.5);
      } else if (RKMethods[m] == "SDIRK 2 Stage 2nd order") {
        pl->sublist("Default Stepper").set("gamma", 0.2928932188134524);
      } else if (RKMethods[m] == "SDIRK 2 Stage 3rd order") {
        pl->sublist("Default Stepper").set("3rd Order A-stable", true);
        pl->sublist("Default Stepper").set("2nd Order L-stable", false);
        pl->sublist("Default Stepper").set("gamma", 0.7886751345948128);
      }

      dt /= 2;

      // Setup the Integrator and reset initial time step
      pl->sublist("Default Integrator")
         .sublist("Time Step Control").set("Initial Time Step", dt);
      integrator = Tempus::integratorBasic<double>(pl, model);

      // Initial Conditions
      // During the Integrator construction, the initial SolutionState
      // is set by default to model->getNominalVales().get_x().  However,
      // the application can set it also by integrator->initializeSolutionHistory.
      RCP<Thyra::VectorBase<double> > x0 =
        model->getNominalValues().get_x()->clone_v();
      integrator->initializeSolutionHistory(0.0, x0);

      // Integrate to timeMax
      bool integratorStatus = integrator->advanceTime();
      TEST_ASSERT(integratorStatus)

      // Test if at 'Final Time'
      time = integrator->getTime();
      double timeFinal = pl->sublist("Default Integrator")
        .sublist("Time Step Control").get<double>("Final Time");
      double tol = 100.0 * std::numeric_limits<double>::epsilon();
      TEST_FLOATING_EQUALITY(time, timeFinal, tol);

      // Plot sample solution and exact solution
      if (n == 0) {
        RCP<const SolutionHistory<double> > solutionHistory =
          integrator->getSolutionHistory();
        writeSolution("Tempus_"+RKMethod+"_SinCos.dat", solutionHistory);

        auto solnHistExact = rcp(new Tempus::SolutionHistory<double>());
        for (int i=0; i<solutionHistory->getNumStates(); i++) {
          double time_i = (*solutionHistory)[i]->getTime();
          auto state = rcp(new Tempus::SolutionState<double>(
              model->getExactSolution(time_i).get_x(),
              model->getExactSolution(time_i).get_x_dot()));
          state->setTime((*solutionHistory)[i]->getTime());
          solnHistExact->addState(state);
        }
        writeSolution("Tempus_"+RKMethod+"_SinCos-Ref.dat", solnHistExact);
      }

      // Store off the final solution and step size
      StepSize.push_back(dt);
      auto solution = Thyra::createMember(model->get_x_space());
      Thyra::copy(*(integrator->getX()),solution.ptr());
      solutions.push_back(solution);
      auto solutionDot = Thyra::createMember(model->get_x_space());
      Thyra::copy(*(integrator->getXdot()),solutionDot.ptr());
      solutionsDot.push_back(solutionDot);
      if (n == nTimeStepSizes-1) {  // Add exact solution last in vector.
        StepSize.push_back(0.0);
        auto solutionExact = Thyra::createMember(model->get_x_space());
        Thyra::copy(*(model->getExactSolution(time).get_x()),solutionExact.ptr());
        solutions.push_back(solutionExact);
        auto solutionDotExact = Thyra::createMember(model->get_x_space());
        Thyra::copy(*(model->getExactSolution(time).get_x_dot()),
                    solutionDotExact.ptr());
        solutionsDot.push_back(solutionDotExact);
      }
    }

    // Check the order and intercept
    double xSlope = 0.0;
    double xDotSlope = 0.0;
    RCP<Tempus::Stepper<double> > stepper = integrator->getStepper();
    double order = stepper->getOrder();
    writeOrderError("Tempus_"+RKMethod+"_SinCos-Error.dat",
                    stepper, StepSize,
                    solutions,    xErrorNorm,    xSlope,
                    solutionsDot, xDotErrorNorm, xDotSlope);

    TEST_FLOATING_EQUALITY( xSlope,               order, 0.01   );
    TEST_FLOATING_EQUALITY( xErrorNorm[0], RKMethodErrors[m], 5.0e-4 );
    // xDot not yet available for DIRK methods.
    //TEST_FLOATING_EQUALITY( xDotSlope,            order, 0.01   );
    //TEST_FLOATING_EQUALITY( xDotErrorNorm[0], 0.0486418, 1.0e-4 );

  }
}
#endif // TEST_SINCOS


#ifdef TEST_VANDERPOL
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(DIRK, VanDerPol)
{
  std::vector<std::string> RKMethods;
  RKMethods.push_back("SDIRK 2 Stage 2nd order");

  std::string RKMethod = RKMethods[0];
  std::replace(RKMethod.begin(), RKMethod.end(), ' ', '_');
  std::replace(RKMethod.begin(), RKMethod.end(), '/', '.');

  RCP<Tempus::IntegratorBasic<double> > integrator;
  std::vector<RCP<Thyra::VectorBase<double>>> solutions;
  std::vector<RCP<Thyra::VectorBase<double>>> solutionsDot;
  std::vector<double> StepSize;
  std::vector<double> xErrorNorm;
  std::vector<double> xDotErrorNorm;

  const int nTimeStepSizes = 3;  // 8 for error plot
  double dt = 0.05;
  double time = 0.0;
  for (int n=0; n<nTimeStepSizes; n++) {

    // Read params from .xml file
    RCP<ParameterList> pList =
      getParametersFromXmlFile("Tempus_DIRK_VanDerPol.xml");

    // Setup the VanDerPolModel
    RCP<ParameterList> vdpm_pl = sublist(pList, "VanDerPolModel", true);
    auto model = rcp(new VanDerPolModel<double>(vdpm_pl));

    // Set the Stepper
    RCP<ParameterList> pl = sublist(pList, "Tempus", true);
    pl->sublist("Default Stepper").set("Stepper Type", RKMethods[0]);
    pl->sublist("Default Stepper").set("gamma", 0.2928932188134524);

    // Set the step size
    dt /= 2;
    if (n == nTimeStepSizes-1) dt /= 10.0;

    // Setup the Integrator and reset initial time step
    pl->sublist("Default Integrator")
       .sublist("Time Step Control").set("Initial Time Step", dt);
    integrator = Tempus::integratorBasic<double>(pl, model);

    // Integrate to timeMax
    bool integratorStatus = integrator->advanceTime();
    TEST_ASSERT(integratorStatus)

    // Test if at 'Final Time'
    time = integrator->getTime();
    double timeFinal =pl->sublist("Default Integrator")
      .sublist("Time Step Control").get<double>("Final Time");
    double tol = 100.0 * std::numeric_limits<double>::epsilon();
    TEST_FLOATING_EQUALITY(time, timeFinal, tol);

    // Store off the final solution and step size
    StepSize.push_back(dt);
    auto solution = Thyra::createMember(model->get_x_space());
    Thyra::copy(*(integrator->getX()),solution.ptr());
    solutions.push_back(solution);
    auto solutionDot = Thyra::createMember(model->get_x_space());
    Thyra::copy(*(integrator->getXdot()),solutionDot.ptr());
    solutionsDot.push_back(solutionDot);

    // Output finest temporal solution for plotting
    // This only works for ONE MPI process
    if ((n == 0) or (n == nTimeStepSizes-1)) {
      std::string fname = "Tempus_"+RKMethod+"_VanDerPol-Ref.dat";
      if (n == 0) fname = "Tempus_"+RKMethod+"_VanDerPol.dat";
      RCP<const SolutionHistory<double> > solutionHistory =
        integrator->getSolutionHistory();
      writeSolution(fname, solutionHistory);
    }
  }

  // Check the order and intercept
  double xSlope = 0.0;
  double xDotSlope = 0.0;
  RCP<Tempus::Stepper<double> > stepper = integrator->getStepper();
  double order = stepper->getOrder();
  writeOrderError("Tempus_"+RKMethod+"_VanDerPol-Error.dat",
                  stepper, StepSize,
                  solutions,    xErrorNorm,    xSlope,
                  solutionsDot, xDotErrorNorm, xDotSlope);

  TEST_FLOATING_EQUALITY( xSlope,            order, 0.06   );
  TEST_FLOATING_EQUALITY( xErrorNorm[0], 1.07525e-05, 1.0e-4 );
  // xDot not yet available for DIRK methods.
  //TEST_FLOATING_EQUALITY( xDotSlope,        1.74898, 0.10 );
  //TEST_FLOATING_EQUALITY( xDotErrorNorm[0],  1.0038, 1.0e-4 );

  Teuchos::TimeMonitor::summarize();
}
#endif // TEST_VANDERPOL


#ifdef TEST_EMBEDDED_DIRK
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(DIRK, EmbeddedVanDerPol)
{

   std::vector<std::string> IntegratorList;
   IntegratorList.push_back("Embedded_Integrator_PID");
   IntegratorList.push_back("Embedded_Integrator");

   // the embedded solution will test the following:
   const int refIstep = 217;

   for(auto integratorChoice : IntegratorList){

      std::cout << "Using Integrator: " << integratorChoice << " !!!" << std::endl;

      // Read params from .xml file
      RCP<ParameterList> pList =
         getParametersFromXmlFile("Tempus_DIRK_VanDerPol.xml");


      // Setup the VanDerPolModel
      RCP<ParameterList> vdpm_pl = sublist(pList, "VanDerPolModel", true);
      auto model = rcp(new VanDerPolModel<double>(vdpm_pl));

      // Set the Integrator and Stepper
      RCP<ParameterList> pl = sublist(pList, "Tempus", true);
      pl->set("Integrator Name", integratorChoice);

      // Setup the Integrator
      RCP<Tempus::IntegratorBasic<double> > integrator =
         Tempus::integratorBasic<double>(pl, model);

     const std::string RKMethod_ =
        pl->sublist(integratorChoice).get<std::string>("Stepper Name");

      // Integrate to timeMax
      bool integratorStatus = integrator->advanceTime();
      TEST_ASSERT(integratorStatus);

      // Test if at 'Final Time'
      double time = integrator->getTime();
      double timeFinal = pl->sublist(integratorChoice)
         .sublist("Time Step Control").get<double>("Final Time");
      TEST_FLOATING_EQUALITY(time, timeFinal, 1.0e-14);


      // Numerical reference solution at timeFinal (for \epsilon = 0.1)
      RCP<Thyra::VectorBase<double> > x = integrator->getX();
      RCP<Thyra::VectorBase<double> > xref = x->clone_v();
      Thyra::set_ele(0, -1.5484458614405929, xref.ptr());
      Thyra::set_ele(1,  1.0181127316101317, xref.ptr());

      // Calculate the error
      RCP<Thyra::VectorBase<double> > xdiff = x->clone_v();
      Thyra::V_StVpStV(xdiff.ptr(), 1.0, *xref, -1.0, *(x));
      const double L2norm = Thyra::norm_2(*xdiff);

      // Test number of steps, failures, and accuracy
      if (integratorChoice == "Embedded_Integrator_PID"){
         const double absTol = pl->sublist(integratorChoice).
            sublist("Time Step Control").get<double>("Maximum Absolute Error");
         const double relTol = pl->sublist(integratorChoice).
            sublist("Time Step Control").get<double>("Maximum Relative Error");


         // get the number of time steps and number of step failure
         //const int nFailure_c = integrator->getSolutionHistory()->
         //getCurrentState()->getMetaData()->getNFailures();
         const int iStep = integrator->getSolutionHistory()->
            getCurrentState()->getIndex();
         //const int nFail = integrator->getSolutionHistory()->
         //   getCurrentState()->getMetaData()->getNRunningFailures();

         // Should be close to the prescribed tolerance
         TEST_FLOATING_EQUALITY(std::log10(L2norm),std::log10(absTol), 0.3 );
         TEST_FLOATING_EQUALITY(std::log10(L2norm),std::log10(relTol), 0.3 );
         // test for number of steps
         TEST_COMPARE(iStep, <=, refIstep);
      }

      // Plot sample solution and exact solution
      std::ofstream ftmp("Tempus_"+integratorChoice+RKMethod_+"_VDP_Example.dat");
      RCP<const SolutionHistory<double> > solutionHistory =
         integrator->getSolutionHistory();
      int nStates = solutionHistory->getNumStates();
      //RCP<const Thyra::VectorBase<double> > x_exact_plot;
      for (int i=0; i<nStates; i++) {
         RCP<const SolutionState<double> > solutionState = (*solutionHistory)[i];
         double time_i = solutionState->getTime();
         RCP<const Thyra::VectorBase<double> > x_plot = solutionState->getX();
         //x_exact_plot = model->getExactSolution(time_i).get_x();
         ftmp << time_i << "   "
            << Thyra::get_ele(*(x_plot), 0) << "   "
            << Thyra::get_ele(*(x_plot), 1) << "   " << std::endl;
      }
      ftmp.close();
   }

   Teuchos::TimeMonitor::summarize();
}
#endif


} // namespace Tempus_Test