Tempus_BackwardEulerTest.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 "Teuchos_DefaultComm.hpp"
 
#include "Tempus_config.hpp"
#include "Tempus_IntegratorBasic.hpp"
#include "Tempus_StepperBackwardEuler.hpp"
 
#include "../TestModels/SinCosModel.hpp"
#include "../TestModels/CDR_Model.hpp"
#include "../TestModels/VanDerPolModel.hpp"
#include "../TestUtils/Tempus_ConvergenceTestUtils.hpp"
 
#include "Stratimikos_DefaultLinearSolverBuilder.hpp"
#include "Thyra_LinearOpWithSolveFactoryHelpers.hpp"
 
#ifdef Tempus_ENABLE_MPI
#include "Epetra_MpiComm.h"
#else
#include "Epetra_SerialComm.h"
#endif
 
#include <vector>
#include <fstream>
#include <sstream>
#include <limits>
 
namespace Tempus_Test {
 
using Teuchos::RCP;
using Teuchos::rcp;
using Teuchos::rcp_const_cast;
using Teuchos::ParameterList;
using Teuchos::sublist;
using Teuchos::getParametersFromXmlFile;
 
using Tempus::IntegratorBasic;
using Tempus::SolutionHistory;
using Tempus::SolutionState;
 
 
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(BackwardEuler, ParameterList)
{
  // Read params from .xml file
  RCP<ParameterList> pList =
    getParametersFromXmlFile("Tempus_BackwardEuler_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);
 
  // Test constructor IntegratorBasic(tempusPL, model)
  {
    RCP<Tempus::IntegratorBasic<double> > integrator =
      Tempus::integratorBasic<double>(tempusPL, model);
 
    RCP<ParameterList> stepperPL = sublist(tempusPL, "Default Stepper", true);
    // Match Predictor for comparison
    stepperPL->set("Predictor Stepper Type", "None");
    RCP<const ParameterList> defaultPL =
      integrator->getStepper()->getValidParameters();
 
    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, "Backward Euler");
 
    RCP<ParameterList> stepperPL = sublist(tempusPL, "Default Stepper", true);
    RCP<const ParameterList> defaultPL =
      integrator->getStepper()->getValidParameters();
 
    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)
  }
}
 
 
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(BackwardEuler, ConstructingFromDefaults)
{
  double dt = 0.1;
 
  // Read params from .xml file
  RCP<ParameterList> pList =
    getParametersFromXmlFile("Tempus_BackwardEuler_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::StepperBackwardEuler<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 = Tempus::createSolutionStateX(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 = BackwardEuler" << 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), 0.798923, 1.0e-4 );
  TEST_FLOATING_EQUALITY(get_ele(*(x), 1), 0.516729, 1.0e-4 );
}
 
 
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(BackwardEuler, SinCos)
{
  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 = 7;
  double dt = 0.2;
  double time = 0.0;
  for (int n=0; n<nTimeStepSizes; n++) {
 
    // Read params from .xml file
    RCP<ParameterList> pList =
      getParametersFromXmlFile("Tempus_BackwardEuler_SinCos.xml");
 
    //std::ofstream ftmp("PL.txt");
    //pList->print(ftmp);
    //ftmp.close();
 
    // 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));
 
    dt /= 2;
 
    // Setup the Integrator and reset initial time step
    RCP<ParameterList> pl = sublist(pList, "Tempus", true);
    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");
    TEST_FLOATING_EQUALITY(time, timeFinal, 1.0e-14);
 
    // Plot sample solution and exact solution
    if (n == 0) {
      RCP<const SolutionHistory<double> > solutionHistory =
        integrator->getSolutionHistory();
      writeSolution("Tempus_BackwardEuler_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 = Tempus::createSolutionStateX(
          rcp_const_cast<Thyra::VectorBase<double> > (
            model->getExactSolution(time_i).get_x()),
          rcp_const_cast<Thyra::VectorBase<double> > (
            model->getExactSolution(time_i).get_x_dot()));
        state->setTime((*solutionHistory)[i]->getTime());
        solnHistExact->addState(state);
      }
      writeSolution("Tempus_BackwardEuler_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_BackwardEuler_SinCos-Error.dat",
                  stepper, StepSize,
                  solutions,    xErrorNorm,    xSlope,
                  solutionsDot, xDotErrorNorm, xDotSlope);
 
  TEST_FLOATING_EQUALITY( xSlope,               order, 0.01   );
  TEST_FLOATING_EQUALITY( xErrorNorm[0],    0.0486418, 1.0e-4 );
  TEST_FLOATING_EQUALITY( xDotSlope,            order, 0.01   );
  TEST_FLOATING_EQUALITY( xDotErrorNorm[0], 0.0486418, 1.0e-4 );
 
  Teuchos::TimeMonitor::summarize();
}
 
 
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(BackwardEuler, CDR)
{
  // Create a communicator for Epetra objects
  RCP<Epetra_Comm> comm;
#ifdef Tempus_ENABLE_MPI
  comm = rcp(new Epetra_MpiComm(MPI_COMM_WORLD));
#else
  comm = rcp(new Epetra_SerialComm);
#endif
 
  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 = 5;
  double dt = 0.2;
  for (int n=0; n<nTimeStepSizes; n++) {
 
    // Read params from .xml file
    RCP<ParameterList> pList =
      getParametersFromXmlFile("Tempus_BackwardEuler_CDR.xml");
 
    // Create CDR Model
    RCP<ParameterList> model_pl = sublist(pList, "CDR Model", true);
    const int num_elements = model_pl->get<int>("num elements");
    const double left_end = model_pl->get<double>("left end");
    const double right_end = model_pl->get<double>("right end");
    const double a_convection = model_pl->get<double>("a (convection)");
    const double k_source = model_pl->get<double>("k (source)");
 
    auto model = rcp(new Tempus_Test::CDR_Model<double>(comm,
                                                        num_elements,
                                                        left_end,
                                                        right_end,
                                                        a_convection,
                                                        k_source));
 
    // Set the factory
    ::Stratimikos::DefaultLinearSolverBuilder builder;
 
    auto p = rcp(new ParameterList);
    p->set("Linear Solver Type", "Belos");
    p->set("Preconditioner Type", "None");
    builder.setParameterList(p);
 
    RCP< ::Thyra::LinearOpWithSolveFactoryBase<double> >
      lowsFactory = builder.createLinearSolveStrategy("");
 
    model->set_W_factory(lowsFactory);
 
    // Set the step size
    dt /= 2;
 
    // Setup the Integrator and reset initial time step
    RCP<ParameterList> pl = sublist(pList, "Tempus", true);
    pl->sublist("Demo 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'
    double time = integrator->getTime();
    double timeFinal =pl->sublist("Demo 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 == nTimeStepSizes-1) && (comm->NumProc() == 1)) {
      std::ofstream ftmp("Tempus_BackwardEuler_CDR.dat");
      ftmp << "TITLE=\"Backward Euler Solution to CDR\"\n"
           << "VARIABLES=\"z\",\"T\"\n";
      const double dx = std::fabs(left_end-right_end) /
                        static_cast<double>(num_elements);
      RCP<const SolutionHistory<double> > solutionHistory =
        integrator->getSolutionHistory();
      int nStates = solutionHistory->getNumStates();
      for (int i=0; i<nStates; i++) {
        RCP<const SolutionState<double> > solutionState = (*solutionHistory)[i];
        RCP<const Thyra::VectorBase<double> > x = solutionState->getX();
        double ttime = solutionState->getTime();
        ftmp << "ZONE T=\"Time="<<ttime<<"\", I="
             <<num_elements+1<<", F=BLOCK\n";
        for (int j = 0; j < num_elements+1; j++) {
          const double x_coord = left_end + static_cast<double>(j) * dx;
          ftmp << x_coord << "   ";
        }
        ftmp << std::endl;
        for (int j=0; j<num_elements+1; j++) ftmp << get_ele(*x, j) << "   ";
        ftmp << std::endl;
      }
      ftmp.close();
    }
  }
 
  // Check the order and intercept
  double xSlope = 0.0;
  double xDotSlope = 0.0;
  RCP<Tempus::Stepper<double> > stepper = integrator->getStepper();
  writeOrderError("Tempus_BackwardEuler_CDR-Error.dat",
                  stepper, StepSize,
                  solutions,    xErrorNorm,    xSlope,
                  solutionsDot, xDotErrorNorm, xDotSlope);
 
  TEST_FLOATING_EQUALITY( xSlope,            1.32213, 0.01   );
  TEST_FLOATING_EQUALITY( xErrorNorm[0],    0.116919, 1.0e-4 );
  TEST_FLOATING_EQUALITY( xDotSlope,         1.32052, 0.01 );
  TEST_FLOATING_EQUALITY( xDotErrorNorm[0], 0.449888, 1.0e-4 );
  // At small dt, slopes should be equal to order.
  //double order = stepper->getOrder();
  //TEST_FLOATING_EQUALITY( xSlope,              order, 0.01   );
  //TEST_FLOATING_EQUALITY( xDotSlope,           order, 0.01 );
 
  // Write fine mesh solution at final time
  // This only works for ONE MPI process
  if (comm->NumProc() == 1) {
    RCP<ParameterList> pList =
      getParametersFromXmlFile("Tempus_BackwardEuler_CDR.xml");
    RCP<ParameterList> model_pl = sublist(pList, "CDR Model", true);
    const int num_elements = model_pl->get<int>("num elements");
    const double left_end = model_pl->get<double>("left end");
    const double right_end = model_pl->get<double>("right end");
 
    const Thyra::VectorBase<double>& x = *(solutions[solutions.size()-1]);
 
    std::ofstream ftmp("Tempus_BackwardEuler_CDR-Solution.dat");
    for (int n = 0; n < num_elements+1; n++) {
      const double dx = std::fabs(left_end-right_end) /
                        static_cast<double>(num_elements);
      const double x_coord = left_end + static_cast<double>(n) * dx;
      ftmp << x_coord << "   " <<  Thyra::get_ele(x,n) << std::endl;
    }
    ftmp.close();
  }
 
  Teuchos::TimeMonitor::summarize();
}
 
 
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(BackwardEuler, VanDerPol)
{
  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 = 4;
  double dt = 0.05;
  for (int n=0; n<nTimeStepSizes; n++) {
 
    // Read params from .xml file
    RCP<ParameterList> pList =
      getParametersFromXmlFile("Tempus_BackwardEuler_VanDerPol.xml");
 
    // Setup the VanDerPolModel
    RCP<ParameterList> vdpm_pl = sublist(pList, "VanDerPolModel", true);
    auto model = rcp(new VanDerPolModel<double>(vdpm_pl));
 
    // Set the step size
    dt /= 2;
    if (n == nTimeStepSizes-1) dt /= 10.0;
 
    // Setup the Integrator and reset initial time step
    RCP<ParameterList> pl = sublist(pList, "Tempus", true);
    pl->sublist("Demo 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'
    double time = integrator->getTime();
    double timeFinal =pl->sublist("Demo 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_BackwardEuler_VanDerPol-Ref.dat";
      if (n == 0) fname = "Tempus_BackwardEuler_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_BackwardEuler_VanDerPol-Error.dat",
                  stepper, StepSize,
                  solutions,    xErrorNorm,    xSlope,
                  solutionsDot, xDotErrorNorm, xDotSlope);
 
  TEST_FLOATING_EQUALITY( xSlope,            order, 0.10   );
  TEST_FLOATING_EQUALITY( xErrorNorm[0],  0.571031, 1.0e-4 );
  TEST_FLOATING_EQUALITY( xDotSlope,       1.74898, 0.10   );
  TEST_FLOATING_EQUALITY( xDotErrorNorm[0], 1.0038, 1.0e-4 );
  // At small dt, slopes should be equal to order.
  //TEST_FLOATING_EQUALITY( xDotSlope,       order, 0.01 );
 
  Teuchos::TimeMonitor::summarize();
}
 
 
// ************************************************************
// ************************************************************
TEUCHOS_UNIT_TEST(BackwardEuler, OptInterface)
{
  // Read params from .xml file
  RCP<ParameterList> pList =
    getParametersFromXmlFile("Tempus_BackwardEuler_SinCos.xml");
 
  // Setup the SinCosModel
  RCP<ParameterList> scm_pl = sublist(pList, "SinCosModel", true);
  auto model = rcp(new SinCosModel<double>(scm_pl));
 
  // Setup the Integrator
  RCP<ParameterList> pl = sublist(pList, "Tempus", true);
  RCP<Tempus::IntegratorBasic<double> >integrator =
    Tempus::integratorBasic<double>(pl, model);
 
  // Integrate to timeMax
  bool integratorStatus = integrator->advanceTime();
  TEST_ASSERT(integratorStatus);
 
  // Get solution history
  RCP<const SolutionHistory<double> > solutionHistory =
    integrator->getSolutionHistory();
 
  // Get the stepper and cast to optimization interface
  RCP<Tempus::Stepper<double> > stepper = integrator->getStepper();
  RCP<Tempus::StepperOptimizationInterface<double> > opt_stepper =
    Teuchos::rcp_dynamic_cast< Tempus::StepperOptimizationInterface<double> >(
      stepper, true);
 
  // Check stencil length
  TEST_EQUALITY( opt_stepper->stencilLength(), 2);
 
  // Create needed vectors/multivectors
  Teuchos::Array< RCP<const Thyra::VectorBase<double> > > x(2);
  Teuchos::Array<double> t(2);
  RCP< const Thyra::VectorBase<double> > p =
    model->getNominalValues().get_p(0);
  RCP< Thyra::VectorBase<double> > x_dot =
    Thyra::createMember(model->get_x_space());
  RCP< Thyra::VectorBase<double> > f =
    Thyra::createMember(model->get_f_space());
  RCP< Thyra::VectorBase<double> > f2 =
    Thyra::createMember(model->get_f_space());
  RCP< Thyra::LinearOpBase<double> > dfdx =
    model->create_W_op();
  RCP< Thyra::LinearOpBase<double> > dfdx2 =
    model->create_W_op();
  RCP< Thyra::MultiVectorBase<double> > dfdx_mv =
    Teuchos::rcp_dynamic_cast< Thyra::MultiVectorBase<double> >(dfdx,true);
  RCP< Thyra::MultiVectorBase<double> > dfdx_mv2 =
    Teuchos::rcp_dynamic_cast< Thyra::MultiVectorBase<double> >(dfdx2,true);
  const int num_p = p->range()->dim();
  RCP< Thyra::MultiVectorBase<double> > dfdp =
    Thyra::createMembers(model->get_f_space(), num_p);
  RCP< Thyra::MultiVectorBase<double> > dfdp2 =
    Thyra::createMembers(model->get_f_space(), num_p);
  RCP< Thyra::LinearOpWithSolveBase<double> > W =
    model->create_W();
  RCP< Thyra::LinearOpWithSolveBase<double> > W2 =
    model->create_W();
  RCP< Thyra::MultiVectorBase<double> > tmp =
    Thyra::createMembers(model->get_x_space(), num_p);
  RCP< Thyra::MultiVectorBase<double> > tmp2 =
    Thyra::createMembers(model->get_x_space(), num_p);
  std::vector<double> nrms(num_p);
  double err;
 
  // Loop over states, checking residuals and derivatives
  const int n = solutionHistory->getNumStates();
  for (int i=1; i<n; ++i) {
    RCP<const SolutionState<double> > state = (*solutionHistory)[i];
    RCP<const SolutionState<double> > prev_state = (*solutionHistory)[i-1];
 
    // Fill x, t stencils
    x[0] = state->getX();
    x[1] = prev_state->getX();
    t[0] = state->getTime();
    t[1] = prev_state->getTime();
 
    // Compute x_dot
    const double dt = t[0]-t[1];
    Thyra::V_StVpStV(x_dot.ptr(), 1.0/dt, *(x[0]), -1.0/dt, *(x[1]));
 
    // Create model inargs
    typedef Thyra::ModelEvaluatorBase MEB;
    MEB::InArgs<double> in_args = model->createInArgs();
    MEB::OutArgs<double> out_args = model->createOutArgs();
    in_args.set_x(x[0]);
    in_args.set_x_dot(x_dot);
    in_args.set_t(t[0]);
    in_args.set_p(0,p);
 
    const double tol = 1.0e-14;
 
    // Check residual
    opt_stepper->computeStepResidual(*f, x, t, *p, 0);
    out_args.set_f(f2);
    model->evalModel(in_args, out_args);
    out_args.set_f(Teuchos::null);
    Thyra::V_VmV(f.ptr(), *f, *f2);
    err = Thyra::norm(*f);
    TEST_FLOATING_EQUALITY(err, 0.0, tol);
 
    // Check df/dx_n
    // df/dx_n = df/dx_dot * dx_dot/dx_n + df/dx_n = 1/dt*df/dx_dot + df/dx_n
    opt_stepper->computeStepJacobian(*dfdx, x, t, *p, 0, 0);
    out_args.set_W_op(dfdx2);
    in_args.set_alpha(1.0/dt);
    in_args.set_beta(1.0);
    model->evalModel(in_args, out_args);
    out_args.set_W_op(Teuchos::null);
    Thyra::V_VmV(dfdx_mv.ptr(), *dfdx_mv, *dfdx_mv2);
    Thyra::norms(*dfdx_mv, Teuchos::arrayViewFromVector(nrms));
    err = 0.0;
    for (auto nrm : nrms) err += nrm;
    TEST_FLOATING_EQUALITY(err, 0.0, tol);
 
    // Check df/dx_{n-1}
    // df/dx_{n-1} = df/dx_dot * dx_dot/dx_{n-1} = -1/dt*df/dx_dot
    opt_stepper->computeStepJacobian(*dfdx, x, t, *p, 0, 1);
    out_args.set_W_op(dfdx2);
    in_args.set_alpha(-1.0/dt);
    in_args.set_beta(0.0);
    model->evalModel(in_args, out_args);
    out_args.set_W_op(Teuchos::null);
    Thyra::V_VmV(dfdx_mv.ptr(), *dfdx_mv, *dfdx_mv2);
    Thyra::norms(*dfdx_mv, Teuchos::arrayViewFromVector(nrms));
    err = 0.0;
    for (auto nrm : nrms) err += nrm;
    TEST_FLOATING_EQUALITY(err, 0.0, tol);
 
    // Check df/dp
    opt_stepper->computeStepParamDeriv(*dfdp, x, t, *p, 0);
    out_args.set_DfDp(
      0, MEB::Derivative<double>(dfdp2, MEB::DERIV_MV_JACOBIAN_FORM));
    model->evalModel(in_args, out_args);
    out_args.set_DfDp(0, MEB::Derivative<double>());
    Thyra::V_VmV(dfdp.ptr(), *dfdp, *dfdp2);
    Thyra::norms(*dfdp, Teuchos::arrayViewFromVector(nrms));
    err = 0.0;
    for (auto nrm : nrms) err += nrm;
    TEST_FLOATING_EQUALITY(err, 0.0, tol);
 
    // Check W
    opt_stepper->computeStepSolver(*W, x, t, *p, 0);
    out_args.set_W(W2);
    in_args.set_alpha(1.0/dt);
    in_args.set_beta(1.0);
    model->evalModel(in_args, out_args);
    out_args.set_W(Teuchos::null);
    // note:  dfdp overwritten above so dfdp != dfdp2
    Thyra::solve(*W, Thyra::NOTRANS, *dfdp2, tmp.ptr());
    Thyra::solve(*W2, Thyra::NOTRANS, *dfdp2, tmp2.ptr());
    Thyra::V_VmV(tmp.ptr(), *tmp, *tmp2);
    Thyra::norms(*tmp, Teuchos::arrayViewFromVector(nrms));
    err = 0.0;
    for (auto nrm : nrms) err += nrm;
    TEST_FLOATING_EQUALITY(err, 0.0, tol);
  }
 
  Teuchos::TimeMonitor::summarize();
}
 
 
} // namespace Tempus_Test