/* ---------------------------------------------------------------------
 *
 * Copyright (C) 2020 - 2022 by the deal.II authors
 *
 * This file is part of the deal.II library.
 *
 * The deal.II library is free software; you can use it, redistribute
 * it, and/or modify it under the terms of the GNU Lesser General
 * Public License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 * The full text of the license can be found in the file LICENSE.md at
 * the top level directory of deal.II.
 *
 * ---------------------------------------------------------------------

 *
 * Author: Martin Kronbichler, Peter Munch, David Schneider, 2020
 */

// @sect3{Parameters and utility functions}

// The same includes as in step-67:
#include <deal.II/base/conditional_ostream.h>
#include <deal.II/base/function.h>
#include <deal.II/base/logstream.h>
#include <deal.II/base/time_stepping.h>
#include <deal.II/base/timer.h>
#include <deal.II/base/utilities.h>
#include <deal.II/base/vectorization.h>

#include <deal.II/distributed/tria.h>

#include <deal.II/dofs/dof_handler.h>

#include <deal.II/fe/fe_dgq.h>
#include <deal.II/fe/fe_system.h>

#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/tria.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>

#include <deal.II/lac/affine_constraints.h>
#include <deal.II/lac/la_parallel_vector.h>

#include <deal.II/matrix_free/fe_evaluation.h>
#include <deal.II/matrix_free/matrix_free.h>
#include <deal.II/matrix_free/operators.h>

#include <deal.II/numerics/data_out.h>

#include <fstream>
#include <iomanip>
#include <iostream>

// A new include for categorizing of cells according to their boundary IDs:
#include <deal.II/matrix_free/tools.h>



namespace Euler_DG
{
  using namespace dealii;

  // The same input parameters as in step-67:
  constexpr unsigned int testcase             = 1;
  constexpr unsigned int dimension            = 2;
  constexpr unsigned int n_global_refinements = 2;
  constexpr unsigned int fe_degree            = 5;
  constexpr unsigned int n_q_points_1d        = fe_degree + 2;

  // This parameter specifies the size of the shared-memory group. Currently,
  // only the values 1 and numbers::invalid_unsigned_int is possible, leading
  // to the options that the memory features can be turned off or all processes
  // having access to the same shared-memory domain are grouped together.
  constexpr unsigned int group_size = numbers::invalid_unsigned_int;

  using Number = double;

  // Here, the type of the data structure is chosen for vectorization. In the
  // default case, VectorizedArray<Number> is used, i.e., the highest
  // instruction-set-architecture extension available on the given hardware with
  // the maximum number of vector lanes is used. However, one might reduce
  // the number of filled lanes, e.g., by writing
  // <code>using VectorizedArrayType = VectorizedArray<Number, 4></code> to only
  // process 4 cells.
  using VectorizedArrayType = VectorizedArray<Number>;

  // The following parameters have not changed:
  constexpr double gamma       = 1.4;
  constexpr double final_time  = testcase == 0 ? 10 : 2.0;
  constexpr double output_tick = testcase == 0 ? 1 : 0.05;

  const double courant_number = 0.15 / std::pow(fe_degree, 1.5);

  // Specify max number of time steps useful for performance studies.
  constexpr unsigned int max_time_steps = numbers::invalid_unsigned_int;

  // Runge-Kutta-related functions copied from step-67 and slightly modified
  // with the purpose to minimize global vector access:
  enum LowStorageRungeKuttaScheme
  {
    stage_3_order_3,
    stage_5_order_4,
    stage_7_order_4,
    stage_9_order_5,
  };
  constexpr LowStorageRungeKuttaScheme lsrk_scheme = stage_5_order_4;



  class LowStorageRungeKuttaIntegrator
  {
  public:
    LowStorageRungeKuttaIntegrator(const LowStorageRungeKuttaScheme scheme)
    {
      TimeStepping::runge_kutta_method lsrk;
      switch (scheme)
        {
          case stage_3_order_3:
            lsrk = TimeStepping::LOW_STORAGE_RK_STAGE3_ORDER3;
            break;
          case stage_5_order_4:
            lsrk = TimeStepping::LOW_STORAGE_RK_STAGE5_ORDER4;
            break;
          case stage_7_order_4:
            lsrk = TimeStepping::LOW_STORAGE_RK_STAGE7_ORDER4;
            break;
          case stage_9_order_5:
            lsrk = TimeStepping::LOW_STORAGE_RK_STAGE9_ORDER5;
            break;

          default:
            AssertThrow(false, ExcNotImplemented());
        }
      TimeStepping::LowStorageRungeKutta<
        LinearAlgebra::distributed::Vector<Number>>
                          rk_integrator(lsrk);
      std::vector<double> ci; // not used
      rk_integrator.get_coefficients(ai, bi, ci);
    }

    unsigned int n_stages() const
    {
      return bi.size();
    }

    template <typename VectorType, typename Operator>
    void perform_time_step(const Operator &pde_operator,
                           const double    current_time,
                           const double    time_step,
                           VectorType &    solution,
                           VectorType &    vec_ri,
                           VectorType &    vec_ki) const
    {
      AssertDimension(ai.size() + 1, bi.size());

      vec_ki.swap(solution);

      double sum_previous_bi = 0;
      for (unsigned int stage = 0; stage < bi.size(); ++stage)
        {
          const double c_i = stage == 0 ? 0 : sum_previous_bi + ai[stage - 1];

          pde_operator.perform_stage(stage,
                                     current_time + c_i * time_step,
                                     bi[stage] * time_step,
                                     (stage == bi.size() - 1 ?
                                        0 :
                                        ai[stage] * time_step),
                                     (stage % 2 == 0 ? vec_ki : vec_ri),
                                     (stage % 2 == 0 ? vec_ri : vec_ki),
                                     solution);

          if (stage > 0)
            sum_previous_bi += bi[stage - 1];
        }
    }

  private:
    std::vector<double> bi;
    std::vector<double> ai;
  };


  // Euler-specific utility functions from step-67:
  enum EulerNumericalFlux
  {
    lax_friedrichs_modified,
    harten_lax_vanleer,
  };
  constexpr EulerNumericalFlux numerical_flux_type = lax_friedrichs_modified;



  template <int dim>
  class ExactSolution : public Function<dim>
  {
  public:
    ExactSolution(const double time)
      : Function<dim>(dim + 2, time)
    {}

    virtual double value(const Point<dim> & p,
                         const unsigned int component = 0) const override;
  };



  template <int dim>
  double ExactSolution<dim>::value(const Point<dim> & x,
                                   const unsigned int component) const
  {
    const double t = this->get_time();

    switch (testcase)
      {
        case 0:
          {
            Assert(dim == 2, ExcNotImplemented());
            const double beta = 5;

            Point<dim> x0;
            x0[0] = 5.;
            const double radius_sqr =
              (x - x0).norm_square() - 2. * (x[0] - x0[0]) * t + t * t;
            const double factor =
              beta / (numbers::PI * 2) * std::exp(1. - radius_sqr);
            const double density_log = std::log2(
              std::abs(1. - (gamma - 1.) / gamma * 0.25 * factor * factor));
            const double density = std::exp2(density_log * (1. / (gamma - 1.)));
            const double u       = 1. - factor * (x[1] - x0[1]);
            const double v       = factor * (x[0] - t - x0[0]);

            if (component == 0)
              return density;
            else if (component == 1)
              return density * u;
            else if (component == 2)
              return density * v;
            else
              {
                const double pressure =
                  std::exp2(density_log * (gamma / (gamma - 1.)));
                return pressure / (gamma - 1.) +
                       0.5 * (density * u * u + density * v * v);
              }
          }

        case 1:
          {
            if (component == 0)
              return 1.;
            else if (component == 1)
              return 0.4;
            else if (component == dim + 1)
              return 3.097857142857143;
            else
              return 0.;
          }

        default:
          Assert(false, ExcNotImplemented());
          return 0.;
      }
  }



  template <int dim, typename Number>
  inline DEAL_II_ALWAYS_INLINE //
    Tensor<1, dim, Number>
    euler_velocity(const Tensor<1, dim + 2, Number> &conserved_variables)
  {
    const Number inverse_density = Number(1.) / conserved_variables[0];

    Tensor<1, dim, Number> velocity;
    for (unsigned int d = 0; d < dim; ++d)
      velocity[d] = conserved_variables[1 + d] * inverse_density;

    return velocity;
  }

  template <int dim, typename Number>
  inline DEAL_II_ALWAYS_INLINE //
    Number
    euler_pressure(const Tensor<1, dim + 2, Number> &conserved_variables)
  {
    const Tensor<1, dim, Number> velocity =
      euler_velocity<dim>(conserved_variables);

    Number rho_u_dot_u = conserved_variables[1] * velocity[0];
    for (unsigned int d = 1; d < dim; ++d)
      rho_u_dot_u += conserved_variables[1 + d] * velocity[d];

    return (gamma - 1.) * (conserved_variables[dim + 1] - 0.5 * rho_u_dot_u);
  }

  template <int dim, typename Number>
  inline DEAL_II_ALWAYS_INLINE //
    Tensor<1, dim + 2, Tensor<1, dim, Number>>
    euler_flux(const Tensor<1, dim + 2, Number> &conserved_variables)
  {
    const Tensor<1, dim, Number> velocity =
      euler_velocity<dim>(conserved_variables);
    const Number pressure = euler_pressure<dim>(conserved_variables);

    Tensor<1, dim + 2, Tensor<1, dim, Number>> flux;
    for (unsigned int d = 0; d < dim; ++d)
      {
        flux[0][d] = conserved_variables[1 + d];
        for (unsigned int e = 0; e < dim; ++e)
          flux[e + 1][d] = conserved_variables[e + 1] * velocity[d];
        flux[d + 1][d] += pressure;
        flux[dim + 1][d] =
          velocity[d] * (conserved_variables[dim + 1] + pressure);
      }

    return flux;
  }

  template <int n_components, int dim, typename Number>
  inline DEAL_II_ALWAYS_INLINE //
    Tensor<1, n_components, Number>
    operator*(const Tensor<1, n_components, Tensor<1, dim, Number>> &matrix,
              const Tensor<1, dim, Number> &                         vector)
  {
    Tensor<1, n_components, Number> result;
    for (unsigned int d = 0; d < n_components; ++d)
      result[d] = matrix[d] * vector;
    return result;
  }

  template <int dim, typename Number>
  inline DEAL_II_ALWAYS_INLINE //
    Tensor<1, dim + 2, Number>
    euler_numerical_flux(const Tensor<1, dim + 2, Number> &u_m,
                         const Tensor<1, dim + 2, Number> &u_p,
                         const Tensor<1, dim, Number> &    normal)
  {
    const auto velocity_m = euler_velocity<dim>(u_m);
    const auto velocity_p = euler_velocity<dim>(u_p);

    const auto pressure_m = euler_pressure<dim>(u_m);
    const auto pressure_p = euler_pressure<dim>(u_p);

    const auto flux_m = euler_flux<dim>(u_m);
    const auto flux_p = euler_flux<dim>(u_p);

    switch (numerical_flux_type)
      {
        case lax_friedrichs_modified:
          {
            const auto lambda =
              0.5 * std::sqrt(std::max(velocity_p.norm_square() +
                                         gamma * pressure_p * (1. / u_p[0]),
                                       velocity_m.norm_square() +
                                         gamma * pressure_m * (1. / u_m[0])));

            return 0.5 * (flux_m * normal + flux_p * normal) +
                   0.5 * lambda * (u_m - u_p);
          }

        case harten_lax_vanleer:
          {
            const auto avg_velocity_normal =
              0.5 * ((velocity_m + velocity_p) * normal);
            const auto   avg_c = std::sqrt(std::abs(
              0.5 * gamma *
              (pressure_p * (1. / u_p[0]) + pressure_m * (1. / u_m[0]))));
            const Number s_pos =
              std::max(Number(), avg_velocity_normal + avg_c);
            const Number s_neg =
              std::min(Number(), avg_velocity_normal - avg_c);
            const Number inverse_s = Number(1.) / (s_pos - s_neg);

            return inverse_s *
                   ((s_pos * (flux_m * normal) - s_neg * (flux_p * normal)) -
                    s_pos * s_neg * (u_m - u_p));
          }

        default:
          {
            Assert(false, ExcNotImplemented());
            return {};
          }
      }
  }



  // General-purpose utility functions from step-67:
  template <int dim, typename VectorizedArrayType>
  VectorizedArrayType
  evaluate_function(const Function<dim> &                  function,
                    const Point<dim, VectorizedArrayType> &p_vectorized,
                    const unsigned int                     component)
  {
    VectorizedArrayType result;
    for (unsigned int v = 0; v < VectorizedArrayType::size(); ++v)
      {
        Point<dim> p;
        for (unsigned int d = 0; d < dim; ++d)
          p[d] = p_vectorized[d][v];
        result[v] = function.value(p, component);
      }
    return result;
  }


  template <int dim, typename VectorizedArrayType, int n_components = dim + 2>
  Tensor<1, n_components, VectorizedArrayType>
  evaluate_function(const Function<dim> &                  function,
                    const Point<dim, VectorizedArrayType> &p_vectorized)
  {
    AssertDimension(function.n_components, n_components);
    Tensor<1, n_components, VectorizedArrayType> result;
    for (unsigned int v = 0; v < VectorizedArrayType::size(); ++v)
      {
        Point<dim> p;
        for (unsigned int d = 0; d < dim; ++d)
          p[d] = p_vectorized[d][v];
        for (unsigned int d = 0; d < n_components; ++d)
          result[d][v] = function.value(p, d);
      }
    return result;
  }


  // @sect3{Euler operator using a cell-centric loop and MPI-3.0 shared memory}

  // Euler operator from step-67 with some changes as detailed below:
  template <int dim, int degree, int n_points_1d>
  class EulerOperator
  {
  public:
    static constexpr unsigned int n_quadrature_points_1d = n_points_1d;

    EulerOperator(TimerOutput &timer_output);

    ~EulerOperator();

    void reinit(const Mapping<dim> &   mapping,
                const DoFHandler<dim> &dof_handler);

    void set_inflow_boundary(const types::boundary_id       boundary_id,
                             std::unique_ptr<Function<dim>> inflow_function);

    void set_subsonic_outflow_boundary(
      const types::boundary_id       boundary_id,
      std::unique_ptr<Function<dim>> outflow_energy);

    void set_wall_boundary(const types::boundary_id boundary_id);

    void set_body_force(std::unique_ptr<Function<dim>> body_force);

    void
    perform_stage(const unsigned int                                stage,
                  const Number                                      cur_time,
                  const Number                                      bi,
                  const Number                                      ai,
                  const LinearAlgebra::distributed::Vector<Number> &current_ri,
                  LinearAlgebra::distributed::Vector<Number> &      vec_ki,
                  LinearAlgebra::distributed::Vector<Number> &solution) const;

    void project(const Function<dim> &                       function,
                 LinearAlgebra::distributed::Vector<Number> &solution) const;

    std::array<double, 3> compute_errors(
      const Function<dim> &                             function,
      const LinearAlgebra::distributed::Vector<Number> &solution) const;

    double compute_cell_transport_speed(
      const LinearAlgebra::distributed::Vector<Number> &solution) const;

    void
    initialize_vector(LinearAlgebra::distributed::Vector<Number> &vector) const;

  private:
    // Instance of SubCommunicatorWrapper containing the sub-communicator, which
    // we need to pass to MatrixFree::reinit() to be able to exploit MPI-3.0
    // shared-memory capabilities:
    MPI_Comm subcommunicator;

    MatrixFree<dim, Number, VectorizedArrayType> data;

    TimerOutput &timer;

    std::map<types::boundary_id, std::unique_ptr<Function<dim>>>
      inflow_boundaries;
    std::map<types::boundary_id, std::unique_ptr<Function<dim>>>
                                   subsonic_outflow_boundaries;
    std::set<types::boundary_id>   wall_boundaries;
    std::unique_ptr<Function<dim>> body_force;
  };



  // New constructor, which creates a sub-communicator. The user can specify
  // the size of the sub-communicator via the global parameter group_size. If
  // the size is set to -1, all MPI processes of a
  // shared-memory domain are combined to a group. The specified size is
  // decisive for the benefit of the shared-memory capabilities of MatrixFree
  // and, therefore, setting the <code>size</code> to <code>-1</code> is a
  // reasonable choice. By setting, the size to <code>1</code> users explicitly
  // disable the MPI-3.0 shared-memory features of MatrixFree and rely
  // completely on MPI-2.0 features, like <code>MPI_Isend</code> and
  // <code>MPI_Irecv</code>.
  template <int dim, int degree, int n_points_1d>
  EulerOperator<dim, degree, n_points_1d>::EulerOperator(TimerOutput &timer)
    : timer(timer)
  {
#ifdef DEAL_II_WITH_MPI
    if (group_size == 1)
      {
        this->subcommunicator = MPI_COMM_SELF;
      }
    else if (group_size == numbers::invalid_unsigned_int)
      {
        const auto rank = Utilities::MPI::this_mpi_process(MPI_COMM_WORLD);

        MPI_Comm_split_type(MPI_COMM_WORLD,
                            MPI_COMM_TYPE_SHARED,
                            rank,
                            MPI_INFO_NULL,
                            &subcommunicator);
      }
    else
      {
        Assert(false, ExcNotImplemented());
      }
#else
    (void)subcommunicator;
    (void)group_size;
    this->subcommunicator = MPI_COMM_SELF;
#endif
  }


  // New destructor responsible for freeing of the sub-communicator.
  template <int dim, int degree, int n_points_1d>
  EulerOperator<dim, degree, n_points_1d>::~EulerOperator()
  {
#ifdef DEAL_II_WITH_MPI
    if (this->subcommunicator != MPI_COMM_SELF)
      MPI_Comm_free(&subcommunicator);
#endif
  }


  // Modified reinit() function to setup the internal data structures in
  // MatrixFree in a way that it is usable by the cell-centric loops and
  // the MPI-3.0 shared-memory capabilities are used:
  template <int dim, int degree, int n_points_1d>
  void EulerOperator<dim, degree, n_points_1d>::reinit(
    const Mapping<dim> &   mapping,
    const DoFHandler<dim> &dof_handler)
  {
    const std::vector<const DoFHandler<dim> *> dof_handlers = {&dof_handler};
    const AffineConstraints<double>            dummy;
    const std::vector<const AffineConstraints<double> *> constraints = {&dummy};
    const std::vector<Quadrature<1>> quadratures = {QGauss<1>(n_q_points_1d),
                                                    QGauss<1>(fe_degree + 1)};

    typename MatrixFree<dim, Number, VectorizedArrayType>::AdditionalData
      additional_data;
    additional_data.mapping_update_flags =
      (update_gradients | update_JxW_values | update_quadrature_points |
       update_values);
    additional_data.mapping_update_flags_inner_faces =
      (update_JxW_values | update_quadrature_points | update_normal_vectors |
       update_values);
    additional_data.mapping_update_flags_boundary_faces =
      (update_JxW_values | update_quadrature_points | update_normal_vectors |
       update_values);
    additional_data.tasks_parallel_scheme =
      MatrixFree<dim, Number, VectorizedArrayType>::AdditionalData::none;

    // Categorize cells so that all lanes have the same boundary IDs for each
    // face. This is strictly not necessary, however, allows to write simpler
    // code in EulerOperator::perform_stage() without masking, since it is
    // guaranteed that all cells grouped together (in a VectorizedArray)
    // have to perform exactly the same operation also on the faces.
    MatrixFreeTools::categorize_by_boundary_ids(dof_handler.get_triangulation(),
                                                additional_data);

    // Enable MPI-3.0 shared-memory capabilities within MatrixFree by providing
    // the sub-communicator:
    additional_data.communicator_sm = subcommunicator;

    data.reinit(
      mapping, dof_handlers, constraints, quadratures, additional_data);
  }


  // The following function does an entire stage of a Runge--Kutta update and is
  // - alongside the slightly modified setup - the heart of this tutorial
  // compared to step-67.
  //
  // In contrast to step-67, we are not executing the advection step
  // (using MatrixFree::loop()) and the inverse mass-matrix step
  // (using MatrixFree::cell_loop()) in sequence, but evaluate everything in
  // one go inside of MatrixFree::loop_cell_centric(). This function expects
  // a single function that is executed on each locally-owned (macro) cell as
  // parameter so that we need to loop over all faces of that cell and perform
  // needed integration steps on our own.
  //
  // The following function contains to a large extent copies of the following
  // functions from step-67 so that comments related the evaluation of the weak
  // form are skipped here:
  // - <code>EulerDG::EulerOperator::local_apply_cell</code>
  // - <code>EulerDG::EulerOperator::local_apply_face</code>
  // - <code>EulerDG::EulerOperator::local_apply_boundary_face</code>
  // - <code>EulerDG::EulerOperator::local_apply_inverse_mass_matrix</code>
  template <int dim, int degree, int n_points_1d>
  void EulerOperator<dim, degree, n_points_1d>::perform_stage(
    const unsigned int                                stage,
    const Number                                      current_time,
    const Number                                      bi,
    const Number                                      ai,
    const LinearAlgebra::distributed::Vector<Number> &current_ri,
    LinearAlgebra::distributed::Vector<Number> &      vec_ki,
    LinearAlgebra::distributed::Vector<Number> &      solution) const
  {
    for (auto &i : inflow_boundaries)
      i.second->set_time(current_time);
    for (auto &i : subsonic_outflow_boundaries)
      i.second->set_time(current_time);

    // Run a cell-centric loop by calling MatrixFree::loop_cell_centric() and
    // providing a lambda containing the effects of the cell, face and
    // boundary-face integrals:
    data.template loop_cell_centric<LinearAlgebra::distributed::Vector<Number>,
                                    LinearAlgebra::distributed::Vector<Number>>(
      [&](const auto &data, auto &dst, const auto &src, const auto cell_range) {
        using FECellIntegral = FEEvaluation<dim,
                                            degree,
                                            n_points_1d,
                                            dim + 2,
                                            Number,
                                            VectorizedArrayType>;
        using FEFaceIntegral = FEFaceEvaluation<dim,
                                                degree,
                                                n_points_1d,
                                                dim + 2,
                                                Number,
                                                VectorizedArrayType>;

        FECellIntegral phi(data);
        FECellIntegral phi_temp(data);
        FEFaceIntegral phi_m(data, true);
        FEFaceIntegral phi_p(data, false);

        Tensor<1, dim, VectorizedArrayType>     constant_body_force;
        const Functions::ConstantFunction<dim> *constant_function =
          dynamic_cast<Functions::ConstantFunction<dim> *>(body_force.get());

        if (constant_function)
          constant_body_force =
            evaluate_function<dim, VectorizedArrayType, dim>(
              *constant_function, Point<dim, VectorizedArrayType>());

        const dealii::internal::EvaluatorTensorProduct<
          dealii::internal::EvaluatorVariant::evaluate_evenodd,
          dim,
          n_points_1d,
          n_points_1d,
          VectorizedArrayType>
          eval(AlignedVector<VectorizedArrayType>(),
               data.get_shape_info().data[0].shape_gradients_collocation_eo,
               AlignedVector<VectorizedArrayType>());

        AlignedVector<VectorizedArrayType> buffer(phi.static_n_q_points *
                                                  phi.n_components);

        // Loop over all cell batches:
        for (unsigned int cell = cell_range.first; cell < cell_range.second;
             ++cell)
          {
            phi.reinit(cell);

            if (ai != Number())
              phi_temp.reinit(cell);

            // Read values from global vector and compute the values at the
            // quadrature points:
            if (ai != Number() && stage == 0)
              {
                phi.read_dof_values(src);

                for (unsigned int i = 0;
                     i < phi.static_dofs_per_component * (dim + 2);
                     ++i)
                  phi_temp.begin_dof_values()[i] = phi.begin_dof_values()[i];

                phi.evaluate(EvaluationFlags::values);
              }
            else
              {
                phi.gather_evaluate(src, EvaluationFlags::values);
              }

            // Buffer the computed values at the quadrature points, since
            // these are overridden by FEEvaluation::submit_value() in the next
            // step, however, are needed later on for the face integrals:
            for (unsigned int i = 0; i < phi.static_n_q_points * (dim + 2); ++i)
              buffer[i] = phi.begin_values()[i];

            // Apply the cell integral at the cell quadrature points. See also
            // the function <code>EulerOperator::local_apply_cell()</code> from
            // step-67:
            for (unsigned int q = 0; q < phi.n_q_points; ++q)
              {
                const auto w_q = phi.get_value(q);
                phi.submit_gradient(euler_flux<dim>(w_q), q);
                if (body_force.get() != nullptr)
                  {
                    const Tensor<1, dim, VectorizedArrayType> force =
                      constant_function ?
                        constant_body_force :
                        evaluate_function<dim, VectorizedArrayType, dim>(
                          *body_force, phi.quadrature_point(q));

                    Tensor<1, dim + 2, VectorizedArrayType> forcing;
                    for (unsigned int d = 0; d < dim; ++d)
                      forcing[d + 1] = w_q[0] * force[d];
                    for (unsigned int d = 0; d < dim; ++d)
                      forcing[dim + 1] += force[d] * w_q[d + 1];

                    phi.submit_value(forcing, q);
                  }
              }

            // Test with the gradient of the test functions in the quadrature
            // points. We skip the interpolation back to the support points
            // of the element, since we first collect all contributions in the
            // cell quadrature points and only perform the interpolation back
            // as the final step.
            {
              auto *values_ptr   = phi.begin_values();
              auto *gradient_ptr = phi.begin_gradients();

              for (unsigned int c = 0; c < dim + 2; ++c)
                {
                  if (dim >= 1 && body_force.get() == nullptr)
                    eval.template gradients<0, false, false>(
                      gradient_ptr + phi.static_n_q_points * 0, values_ptr);
                  else if (dim >= 1)
                    eval.template gradients<0, false, true>(
                      gradient_ptr + phi.static_n_q_points * 0, values_ptr);
                  if (dim >= 2)
                    eval.template gradients<1, false, true>(
                      gradient_ptr + phi.static_n_q_points * 1, values_ptr);
                  if (dim >= 3)
                    eval.template gradients<2, false, true>(
                      gradient_ptr + phi.static_n_q_points * 2, values_ptr);

                  values_ptr += phi.static_n_q_points;
                  gradient_ptr += phi.static_n_q_points * dim;
                }
            }

            // Loop over all faces of the current cell:
            for (unsigned int face = 0;
                 face < GeometryInfo<dim>::faces_per_cell;
                 ++face)
              {
                // Determine the boundary ID of the current face. Since we have
                // set up MatrixFree in a way that all filled lanes have
                // guaranteed the same boundary ID, we can select the
                // boundary ID of the first lane.
                const auto boundary_ids =
                  data.get_faces_by_cells_boundary_id(cell, face);

                Assert(std::equal(boundary_ids.begin(),
                                  boundary_ids.begin() +
                                    data.n_active_entries_per_cell_batch(cell),
                                  boundary_ids.begin()),
                       ExcMessage("Boundary IDs of lanes differ."));

                const auto boundary_id = boundary_ids[0];

                phi_m.reinit(cell, face);

                // Interpolate the values from the cell quadrature points to the
                // quadrature points of the current face via a simple 1D
                // interpolation:
                internal::FEFaceNormalEvaluationImpl<dim,
                                                     n_points_1d - 1,
                                                     VectorizedArrayType>::
                  template interpolate_quadrature<true, false>(
                    dim + 2,
                    EvaluationFlags::values,
                    data.get_shape_info(),
                    buffer.data(),
                    phi_m.begin_values(),
                    face);

                // Check if the face is an internal or a boundary face and
                // select a different code path based on this information:
                if (boundary_id == numbers::internal_face_boundary_id)
                  {
                    // Process and internal face. The following lines of code
                    // are a copy of the function
                    // <code>EulerDG::EulerOperator::local_apply_face</code>
                    // from step-67:
                    phi_p.reinit(cell, face);
                    phi_p.gather_evaluate(src, EvaluationFlags::values);

                    for (unsigned int q = 0; q < phi_m.n_q_points; ++q)
                      {
                        const auto numerical_flux =
                          euler_numerical_flux<dim>(phi_m.get_value(q),
                                                    phi_p.get_value(q),
                                                    phi_m.get_normal_vector(q));
                        phi_m.submit_value(-numerical_flux, q);
                      }
                  }
                else
                  {
                    // Process a boundary face. These following lines of code
                    // are a copy of the function
                    // <code>EulerDG::EulerOperator::local_apply_boundary_face</code>
                    // from step-67:
                    for (unsigned int q = 0; q < phi_m.n_q_points; ++q)
                      {
                        const auto w_m    = phi_m.get_value(q);
                        const auto normal = phi_m.get_normal_vector(q);

                        auto rho_u_dot_n = w_m[1] * normal[0];
                        for (unsigned int d = 1; d < dim; ++d)
                          rho_u_dot_n += w_m[1 + d] * normal[d];

                        bool at_outflow = false;

                        Tensor<1, dim + 2, VectorizedArrayType> w_p;

                        if (wall_boundaries.find(boundary_id) !=
                            wall_boundaries.end())
                          {
                            w_p[0] = w_m[0];
                            for (unsigned int d = 0; d < dim; ++d)
                              w_p[d + 1] =
                                w_m[d + 1] - 2. * rho_u_dot_n * normal[d];
                            w_p[dim + 1] = w_m[dim + 1];
                          }
                        else if (inflow_boundaries.find(boundary_id) !=
                                 inflow_boundaries.end())
                          w_p = evaluate_function(
                            *inflow_boundaries.find(boundary_id)->second,
                            phi_m.quadrature_point(q));
                        else if (subsonic_outflow_boundaries.find(
                                   boundary_id) !=
                                 subsonic_outflow_boundaries.end())
                          {
                            w_p = w_m;
                            w_p[dim + 1] =
                              evaluate_function(*subsonic_outflow_boundaries
                                                   .find(boundary_id)
                                                   ->second,
                                                phi_m.quadrature_point(q),
                                                dim + 1);
                            at_outflow = true;
                          }
                        else
                          AssertThrow(false,
                                      ExcMessage(
                                        "Unknown boundary id, did "
                                        "you set a boundary condition for "
                                        "this part of the domain boundary?"));

                        auto flux = euler_numerical_flux<dim>(w_m, w_p, normal);

                        if (at_outflow)
                          for (unsigned int v = 0;
                               v < VectorizedArrayType::size();
                               ++v)
                            {
                              if (rho_u_dot_n[v] < -1e-12)
                                for (unsigned int d = 0; d < dim; ++d)
                                  flux[d + 1][v] = 0.;
                            }

                        phi_m.submit_value(-flux, q);
                      }
                  }

                // Evaluate local integrals related to cell by quadrature and
                // add into cell contribution via a simple 1D interpolation:
                internal::FEFaceNormalEvaluationImpl<dim,
                                                     n_points_1d - 1,
                                                     VectorizedArrayType>::
                  template interpolate_quadrature<false, true>(
                    dim + 2,
                    EvaluationFlags::values,
                    data.get_shape_info(),
                    phi_m.begin_values(),
                    phi.begin_values(),
                    face);
              }

            // Apply inverse mass matrix in the cell quadrature points. See
            // also the function
            // <code>EulerDG::EulerOperator::local_apply_inverse_mass_matrix()</code>
            // from step-67:
            for (unsigned int q = 0; q < phi.static_n_q_points; ++q)
              {
                const auto factor = VectorizedArrayType(1.0) / phi.JxW(q);
                for (unsigned int c = 0; c < dim + 2; ++c)
                  phi.begin_values()[c * phi.static_n_q_points + q] =
                    phi.begin_values()[c * phi.static_n_q_points + q] * factor;
              }

            // Transform values from collocation space to the original
            // Gauss-Lobatto space:
            internal::FEEvaluationImplBasisChange<
              dealii::internal::EvaluatorVariant::evaluate_evenodd,
              internal::EvaluatorQuantity::hessian,
              dim,
              degree + 1,
              n_points_1d,
              VectorizedArrayType,
              VectorizedArrayType>::do_backward(dim + 2,
                                                data.get_shape_info()
                                                  .data[0]
                                                  .inverse_shape_values_eo,
                                                false,
                                                phi.begin_values(),
                                                phi.begin_dof_values());

            // Perform Runge-Kutta update and write results back to global
            // vectors:
            if (ai == Number())
              {
                for (unsigned int q = 0; q < phi.static_dofs_per_cell; ++q)
                  phi.begin_dof_values()[q] = bi * phi.begin_dof_values()[q];
                phi.distribute_local_to_global(solution);
              }
            else
              {
                if (stage != 0)
                  phi_temp.read_dof_values(solution);

                for (unsigned int q = 0; q < phi.static_dofs_per_cell; ++q)
                  {
                    const auto K_i = phi.begin_dof_values()[q];

                    phi.begin_dof_values()[q] =
                      phi_temp.begin_dof_values()[q] + (ai * K_i);

                    phi_temp.begin_dof_values()[q] += bi * K_i;
                  }
                phi.set_dof_values(dst);
                phi_temp.set_dof_values(solution);
              }
          }
      },
      vec_ki,
      current_ri,
      true,
      MatrixFree<dim, Number, VectorizedArrayType>::DataAccessOnFaces::values);
  }



  // From here, the code of step-67 has not changed.
  template <int dim, int degree, int n_points_1d>
  void EulerOperator<dim, degree, n_points_1d>::initialize_vector(
    LinearAlgebra::distributed::Vector<Number> &vector) const
  {
    data.initialize_dof_vector(vector);
  }



  template <int dim, int degree, int n_points_1d>
  void EulerOperator<dim, degree, n_points_1d>::set_inflow_boundary(
    const types::boundary_id       boundary_id,
    std::unique_ptr<Function<dim>> inflow_function)
  {
    AssertThrow(subsonic_outflow_boundaries.find(boundary_id) ==
                    subsonic_outflow_boundaries.end() &&
                  wall_boundaries.find(boundary_id) == wall_boundaries.end(),
                ExcMessage("You already set the boundary with id " +
                           std::to_string(static_cast<int>(boundary_id)) +
                           " to another type of boundary before now setting " +
                           "it as inflow"));
    AssertThrow(inflow_function->n_components == dim + 2,
                ExcMessage("Expected function with dim+2 components"));

    inflow_boundaries[boundary_id] = std::move(inflow_function);
  }



  template <int dim, int degree, int n_points_1d>
  void EulerOperator<dim, degree, n_points_1d>::set_subsonic_outflow_boundary(
    const types::boundary_id       boundary_id,
    std::unique_ptr<Function<dim>> outflow_function)
  {
    AssertThrow(inflow_boundaries.find(boundary_id) ==
                    inflow_boundaries.end() &&
                  wall_boundaries.find(boundary_id) == wall_boundaries.end(),
                ExcMessage("You already set the boundary with id " +
                           std::to_string(static_cast<int>(boundary_id)) +
                           " to another type of boundary before now setting " +
                           "it as subsonic outflow"));
    AssertThrow(outflow_function->n_components == dim + 2,
                ExcMessage("Expected function with dim+2 components"));

    subsonic_outflow_boundaries[boundary_id] = std::move(outflow_function);
  }



  template <int dim, int degree, int n_points_1d>
  void EulerOperator<dim, degree, n_points_1d>::set_wall_boundary(
    const types::boundary_id boundary_id)
  {
    AssertThrow(inflow_boundaries.find(boundary_id) ==
                    inflow_boundaries.end() &&
                  subsonic_outflow_boundaries.find(boundary_id) ==
                    subsonic_outflow_boundaries.end(),
                ExcMessage("You already set the boundary with id " +
                           std::to_string(static_cast<int>(boundary_id)) +
                           " to another type of boundary before now setting " +
                           "it as wall boundary"));

    wall_boundaries.insert(boundary_id);
  }



  template <int dim, int degree, int n_points_1d>
  void EulerOperator<dim, degree, n_points_1d>::set_body_force(
    std::unique_ptr<Function<dim>> body_force)
  {
    AssertDimension(body_force->n_components, dim);

    this->body_force = std::move(body_force);
  }



  template <int dim, int degree, int n_points_1d>
  void EulerOperator<dim, degree, n_points_1d>::project(
    const Function<dim> &                       function,
    LinearAlgebra::distributed::Vector<Number> &solution) const
  {
    FEEvaluation<dim, degree, degree + 1, dim + 2, Number, VectorizedArrayType>
      phi(data, 0, 1);
    MatrixFreeOperators::CellwiseInverseMassMatrix<dim,
                                                   degree,
                                                   dim + 2,
                                                   Number,
                                                   VectorizedArrayType>
      inverse(phi);
    solution.zero_out_ghost_values();
    for (unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
      {
        phi.reinit(cell);
        for (unsigned int q = 0; q < phi.n_q_points; ++q)
          phi.submit_dof_value(evaluate_function(function,
                                                 phi.quadrature_point(q)),
                               q);
        inverse.transform_from_q_points_to_basis(dim + 2,
                                                 phi.begin_dof_values(),
                                                 phi.begin_dof_values());
        phi.set_dof_values(solution);
      }
  }



  template <int dim, int degree, int n_points_1d>
  std::array<double, 3> EulerOperator<dim, degree, n_points_1d>::compute_errors(
    const Function<dim> &                             function,
    const LinearAlgebra::distributed::Vector<Number> &solution) const
  {
    TimerOutput::Scope t(timer, "compute errors");
    double             errors_squared[3] = {};
    FEEvaluation<dim, degree, n_points_1d, dim + 2, Number, VectorizedArrayType>
      phi(data, 0, 0);

    for (unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
      {
        phi.reinit(cell);
        phi.gather_evaluate(solution, EvaluationFlags::values);
        VectorizedArrayType local_errors_squared[3] = {};
        for (unsigned int q = 0; q < phi.n_q_points; ++q)
          {
            const auto error =
              evaluate_function(function, phi.quadrature_point(q)) -
              phi.get_value(q);
            const auto JxW = phi.JxW(q);

            local_errors_squared[0] += error[0] * error[0] * JxW;
            for (unsigned int d = 0; d < dim; ++d)
              local_errors_squared[1] += (error[d + 1] * error[d + 1]) * JxW;
            local_errors_squared[2] += (error[dim + 1] * error[dim + 1]) * JxW;
          }
        for (unsigned int v = 0; v < data.n_active_entries_per_cell_batch(cell);
             ++v)
          for (unsigned int d = 0; d < 3; ++d)
            errors_squared[d] += local_errors_squared[d][v];
      }

    Utilities::MPI::sum(errors_squared, MPI_COMM_WORLD, errors_squared);

    std::array<double, 3> errors;
    for (unsigned int d = 0; d < 3; ++d)
      errors[d] = std::sqrt(errors_squared[d]);

    return errors;
  }



  template <int dim, int degree, int n_points_1d>
  double EulerOperator<dim, degree, n_points_1d>::compute_cell_transport_speed(
    const LinearAlgebra::distributed::Vector<Number> &solution) const
  {
    TimerOutput::Scope t(timer, "compute transport speed");
    Number             max_transport = 0;
    FEEvaluation<dim, degree, degree + 1, dim + 2, Number, VectorizedArrayType>
      phi(data, 0, 1);

    for (unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
      {
        phi.reinit(cell);
        phi.gather_evaluate(solution, EvaluationFlags::values);
        VectorizedArrayType local_max = 0.;
        for (unsigned int q = 0; q < phi.n_q_points; ++q)
          {
            const auto solution = phi.get_value(q);
            const auto velocity = euler_velocity<dim>(solution);
            const auto pressure = euler_pressure<dim>(solution);

            const auto          inverse_jacobian = phi.inverse_jacobian(q);
            const auto          convective_speed = inverse_jacobian * velocity;
            VectorizedArrayType convective_limit = 0.;
            for (unsigned int d = 0; d < dim; ++d)
              convective_limit =
                std::max(convective_limit, std::abs(convective_speed[d]));

            const auto speed_of_sound =
              std::sqrt(gamma * pressure * (1. / solution[0]));

            Tensor<1, dim, VectorizedArrayType> eigenvector;
            for (unsigned int d = 0; d < dim; ++d)
              eigenvector[d] = 1.;
            for (unsigned int i = 0; i < 5; ++i)
              {
                eigenvector = transpose(inverse_jacobian) *
                              (inverse_jacobian * eigenvector);
                VectorizedArrayType eigenvector_norm = 0.;
                for (unsigned int d = 0; d < dim; ++d)
                  eigenvector_norm =
                    std::max(eigenvector_norm, std::abs(eigenvector[d]));
                eigenvector /= eigenvector_norm;
              }
            const auto jac_times_ev   = inverse_jacobian * eigenvector;
            const auto max_eigenvalue = std::sqrt(
              (jac_times_ev * jac_times_ev) / (eigenvector * eigenvector));
            local_max =
              std::max(local_max,
                       max_eigenvalue * speed_of_sound + convective_limit);
          }

        for (unsigned int v = 0; v < data.n_active_entries_per_cell_batch(cell);
             ++v)
          for (unsigned int d = 0; d < 3; ++d)
            max_transport = std::max(max_transport, local_max[v]);
      }

    max_transport = Utilities::MPI::max(max_transport, MPI_COMM_WORLD);

    return max_transport;
  }



  template <int dim>
  class EulerProblem
  {
  public:
    EulerProblem();

    void run();

  private:
    void make_grid_and_dofs();

    void output_results(const unsigned int result_number);

    LinearAlgebra::distributed::Vector<Number> solution;

    ConditionalOStream pcout;

#ifdef DEAL_II_WITH_P4EST
    parallel::distributed::Triangulation<dim> triangulation;
#else
    Triangulation<dim> triangulation;
#endif

    FESystem<dim>   fe;
    MappingQ<dim>   mapping;
    DoFHandler<dim> dof_handler;

    TimerOutput timer;

    EulerOperator<dim, fe_degree, n_q_points_1d> euler_operator;

    double time, time_step;

    class Postprocessor : public DataPostprocessor<dim>
    {
    public:
      Postprocessor();

      virtual void evaluate_vector_field(
        const DataPostprocessorInputs::Vector<dim> &inputs,
        std::vector<Vector<double>> &computed_quantities) const override;

      virtual std::vector<std::string> get_names() const override;

      virtual std::vector<
        DataComponentInterpretation::DataComponentInterpretation>
      get_data_component_interpretation() const override;

      virtual UpdateFlags get_needed_update_flags() const override;

    private:
      const bool do_schlieren_plot;
    };
  };



  template <int dim>
  EulerProblem<dim>::Postprocessor::Postprocessor()
    : do_schlieren_plot(dim == 2)
  {}



  template <int dim>
  void EulerProblem<dim>::Postprocessor::evaluate_vector_field(
    const DataPostprocessorInputs::Vector<dim> &inputs,
    std::vector<Vector<double>> &               computed_quantities) const
  {
    const unsigned int n_evaluation_points = inputs.solution_values.size();

    if (do_schlieren_plot == true)
      Assert(inputs.solution_gradients.size() == n_evaluation_points,
             ExcInternalError());

    Assert(computed_quantities.size() == n_evaluation_points,
           ExcInternalError());
    Assert(inputs.solution_values[0].size() == dim + 2, ExcInternalError());
    Assert(computed_quantities[0].size() ==
             dim + 2 + (do_schlieren_plot == true ? 1 : 0),
           ExcInternalError());

    for (unsigned int q = 0; q < n_evaluation_points; ++q)
      {
        Tensor<1, dim + 2> solution;
        for (unsigned int d = 0; d < dim + 2; ++d)
          solution[d] = inputs.solution_values[q](d);

        const double         density  = solution[0];
        const Tensor<1, dim> velocity = euler_velocity<dim>(solution);
        const double         pressure = euler_pressure<dim>(solution);

        for (unsigned int d = 0; d < dim; ++d)
          computed_quantities[q](d) = velocity[d];
        computed_quantities[q](dim)     = pressure;
        computed_quantities[q](dim + 1) = std::sqrt(gamma * pressure / density);

        if (do_schlieren_plot == true)
          computed_quantities[q](dim + 2) =
            inputs.solution_gradients[q][0] * inputs.solution_gradients[q][0];
      }
  }



  template <int dim>
  std::vector<std::string> EulerProblem<dim>::Postprocessor::get_names() const
  {
    std::vector<std::string> names;
    for (unsigned int d = 0; d < dim; ++d)
      names.emplace_back("velocity");
    names.emplace_back("pressure");
    names.emplace_back("speed_of_sound");

    if (do_schlieren_plot == true)
      names.emplace_back("schlieren_plot");

    return names;
  }



  template <int dim>
  std::vector<DataComponentInterpretation::DataComponentInterpretation>
  EulerProblem<dim>::Postprocessor::get_data_component_interpretation() const
  {
    std::vector<DataComponentInterpretation::DataComponentInterpretation>
      interpretation;
    for (unsigned int d = 0; d < dim; ++d)
      interpretation.push_back(
        DataComponentInterpretation::component_is_part_of_vector);
    interpretation.push_back(DataComponentInterpretation::component_is_scalar);
    interpretation.push_back(DataComponentInterpretation::component_is_scalar);

    if (do_schlieren_plot == true)
      interpretation.push_back(
        DataComponentInterpretation::component_is_scalar);

    return interpretation;
  }



  template <int dim>
  UpdateFlags EulerProblem<dim>::Postprocessor::get_needed_update_flags() const
  {
    if (do_schlieren_plot == true)
      return update_values | update_gradients;
    else
      return update_values;
  }



  template <int dim>
  EulerProblem<dim>::EulerProblem()
    : pcout(std::cout, Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
#ifdef DEAL_II_WITH_P4EST
    , triangulation(MPI_COMM_WORLD)
#endif
    , fe(FE_DGQ<dim>(fe_degree), dim + 2)
    , mapping(fe_degree)
    , dof_handler(triangulation)
    , timer(pcout, TimerOutput::never, TimerOutput::wall_times)
    , euler_operator(timer)
    , time(0)
    , time_step(0)
  {}



  template <int dim>
  void EulerProblem<dim>::make_grid_and_dofs()
  {
    switch (testcase)
      {
        case 0:
          {
            Point<dim> lower_left;
            for (unsigned int d = 1; d < dim; ++d)
              lower_left[d] = -5;

            Point<dim> upper_right;
            upper_right[0] = 10;
            for (unsigned int d = 1; d < dim; ++d)
              upper_right[d] = 5;

            GridGenerator::hyper_rectangle(triangulation,
                                           lower_left,
                                           upper_right);
            triangulation.refine_global(2);

            euler_operator.set_inflow_boundary(
              0, std::make_unique<ExactSolution<dim>>(0));

            break;
          }

        case 1:
          {
            GridGenerator::channel_with_cylinder(
              triangulation, 0.03, 1, 0, true);

            euler_operator.set_inflow_boundary(
              0, std::make_unique<ExactSolution<dim>>(0));
            euler_operator.set_subsonic_outflow_boundary(
              1, std::make_unique<ExactSolution<dim>>(0));

            euler_operator.set_wall_boundary(2);
            euler_operator.set_wall_boundary(3);

            if (dim == 3)
              euler_operator.set_body_force(
                std::make_unique<Functions::ConstantFunction<dim>>(
                  std::vector<double>({0., 0., -0.2})));

            break;
          }

        default:
          Assert(false, ExcNotImplemented());
      }

    triangulation.refine_global(n_global_refinements);

    dof_handler.distribute_dofs(fe);

    euler_operator.reinit(mapping, dof_handler);
    euler_operator.initialize_vector(solution);

    std::locale s = pcout.get_stream().getloc();
    pcout.get_stream().imbue(std::locale(""));
    pcout << "Number of degrees of freedom: " << dof_handler.n_dofs()
          << " ( = " << (dim + 2) << " [vars] x "
          << triangulation.n_global_active_cells() << " [cells] x "
          << Utilities::pow(fe_degree + 1, dim) << " [dofs/cell/var] )"
          << std::endl;
    pcout.get_stream().imbue(s);
  }



  template <int dim>
  void EulerProblem<dim>::output_results(const unsigned int result_number)
  {
    const std::array<double, 3> errors =
      euler_operator.compute_errors(ExactSolution<dim>(time), solution);
    const std::string quantity_name = testcase == 0 ? "error" : "norm";

    pcout << "Time:" << std::setw(8) << std::setprecision(3) << time
          << ", dt: " << std::setw(8) << std::setprecision(2) << time_step
          << ", " << quantity_name << " rho: " << std::setprecision(4)
          << std::setw(10) << errors[0] << ", rho * u: " << std::setprecision(4)
          << std::setw(10) << errors[1] << ", energy:" << std::setprecision(4)
          << std::setw(10) << errors[2] << std::endl;

    {
      TimerOutput::Scope t(timer, "output");

      Postprocessor postprocessor;
      DataOut<dim>  data_out;

      DataOutBase::VtkFlags flags;
      flags.write_higher_order_cells = true;
      data_out.set_flags(flags);

      data_out.attach_dof_handler(dof_handler);
      {
        std::vector<std::string> names;
        names.emplace_back("density");
        for (unsigned int d = 0; d < dim; ++d)
          names.emplace_back("momentum");
        names.emplace_back("energy");

        std::vector<DataComponentInterpretation::DataComponentInterpretation>
          interpretation;
        interpretation.push_back(
          DataComponentInterpretation::component_is_scalar);
        for (unsigned int d = 0; d < dim; ++d)
          interpretation.push_back(
            DataComponentInterpretation::component_is_part_of_vector);
        interpretation.push_back(
          DataComponentInterpretation::component_is_scalar);

        data_out.add_data_vector(dof_handler, solution, names, interpretation);
      }
      data_out.add_data_vector(solution, postprocessor);

      LinearAlgebra::distributed::Vector<Number> reference;
      if (testcase == 0 && dim == 2)
        {
          reference.reinit(solution);
          euler_operator.project(ExactSolution<dim>(time), reference);
          reference.sadd(-1., 1, solution);
          std::vector<std::string> names;
          names.emplace_back("error_density");
          for (unsigned int d = 0; d < dim; ++d)
            names.emplace_back("error_momentum");
          names.emplace_back("error_energy");

          std::vector<DataComponentInterpretation::DataComponentInterpretation>
            interpretation;
          interpretation.push_back(
            DataComponentInterpretation::component_is_scalar);
          for (unsigned int d = 0; d < dim; ++d)
            interpretation.push_back(
              DataComponentInterpretation::component_is_part_of_vector);
          interpretation.push_back(
            DataComponentInterpretation::component_is_scalar);

          data_out.add_data_vector(dof_handler,
                                   reference,
                                   names,
                                   interpretation);
        }

      Vector<double> mpi_owner(triangulation.n_active_cells());
      mpi_owner = Utilities::MPI::this_mpi_process(MPI_COMM_WORLD);
      data_out.add_data_vector(mpi_owner, "owner");

      data_out.build_patches(mapping,
                             fe.degree,
                             DataOut<dim>::curved_inner_cells);

      const std::string filename =
        "solution_" + Utilities::int_to_string(result_number, 3) + ".vtu";
      data_out.write_vtu_in_parallel(filename, MPI_COMM_WORLD);
    }
  }



  template <int dim>
  void EulerProblem<dim>::run()
  {
    {
      const unsigned int n_vect_number = VectorizedArrayType::size();
      const unsigned int n_vect_bits   = 8 * sizeof(Number) * n_vect_number;

      pcout << "Running with "
            << Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD)
            << " MPI processes" << std::endl;
      pcout << "Vectorization over " << n_vect_number << ' '
            << (std::is_same<Number, double>::value ? "doubles" : "floats")
            << " = " << n_vect_bits << " bits ("
            << Utilities::System::get_current_vectorization_level() << ')'
            << std::endl;
    }

    make_grid_and_dofs();

    const LowStorageRungeKuttaIntegrator integrator(lsrk_scheme);

    LinearAlgebra::distributed::Vector<Number> rk_register_1;
    LinearAlgebra::distributed::Vector<Number> rk_register_2;
    rk_register_1.reinit(solution);
    rk_register_2.reinit(solution);

    euler_operator.project(ExactSolution<dim>(time), solution);


    double min_vertex_distance = std::numeric_limits<double>::max();
    for (const auto &cell : triangulation.active_cell_iterators())
      if (cell->is_locally_owned())
        min_vertex_distance =
          std::min(min_vertex_distance, cell->minimum_vertex_distance());
    min_vertex_distance =
      Utilities::MPI::min(min_vertex_distance, MPI_COMM_WORLD);

    time_step = courant_number * integrator.n_stages() /
                euler_operator.compute_cell_transport_speed(solution);
    pcout << "Time step size: " << time_step
          << ", minimal h: " << min_vertex_distance
          << ", initial transport scaling: "
          << 1. / euler_operator.compute_cell_transport_speed(solution)
          << std::endl
          << std::endl;

    output_results(0);

    unsigned int timestep_number = 0;

    while (time < final_time - 1e-12 && timestep_number < max_time_steps)
      {
        ++timestep_number;
        if (timestep_number % 5 == 0)
          time_step =
            courant_number * integrator.n_stages() /
            Utilities::truncate_to_n_digits(
              euler_operator.compute_cell_transport_speed(solution), 3);

        {
          TimerOutput::Scope t(timer, "rk time stepping total");
          integrator.perform_time_step(euler_operator,
                                       time,
                                       time_step,
                                       solution,
                                       rk_register_1,
                                       rk_register_2);
        }

        time += time_step;

        if (static_cast<int>(time / output_tick) !=
              static_cast<int>((time - time_step) / output_tick) ||
            time >= final_time - 1e-12)
          output_results(
            static_cast<unsigned int>(std::round(time / output_tick)));
      }

    timer.print_wall_time_statistics(MPI_COMM_WORLD);
    pcout << std::endl;
  }

} // namespace Euler_DG


int main(int argc, char **argv)
{
  using namespace Euler_DG;
  using namespace dealii;

  Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);

  try
    {
      deallog.depth_console(0);

      EulerProblem<dimension> euler_problem;
      euler_problem.run();
    }
  catch (std::exception &exc)
    {
      std::cerr << std::endl
                << std::endl
                << "----------------------------------------------------"
                << std::endl;
      std::cerr << "Exception on processing: " << std::endl
                << exc.what() << std::endl
                << "Aborting!" << std::endl
                << "----------------------------------------------------"
                << std::endl;

      return 1;
    }
  catch (...)
    {
      std::cerr << std::endl
                << std::endl
                << "----------------------------------------------------"
                << std::endl;
      std::cerr << "Unknown exception!" << std::endl
                << "Aborting!" << std::endl
                << "----------------------------------------------------"
                << std::endl;
      return 1;
    }

  return 0;
}
