step-76.h

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true);
    FEFaceEvaluation<dim, degree, degree + 1, 1, Number> phi_p(data, /*is_interior_face=*/false);
 
    for (unsigned int face = range.first; face < range.second; ++face)
      {
        phi_m.reinit(face);
        phi_m.gather_evaluate(src, face_evaluation_flags);
        phi_p.reinit(face);
        phi_p.gather_evaluate(src, face_evaluation_flags);
 
        // some operations on the face quadrature points
 
        phi_m.integrate_scatter(face_evaluation_flags, dst);
        phi_p.integrate_scatter(face_evaluation_flags, dst);
      }
  },
  [&](const auto &data, auto &dst, const auto &src, const auto range) {
    // operation performed boundary faces
 
    FEFaceEvaluation<dim, degree, degree + 1, 1, Number> phi_m(data, /*is_interior_face=*/true);
 
    for (unsigned int face = range.first; face < range.second; ++face)
      {
        phi_m.reinit(face);
        phi_m.gather_evaluate(src, face_evaluation_flags);
 
        // some operations on the face quadrature points
 
        phi_m.integrate_scatter(face_evaluation_flags, dst);
      }
  },
  dst,
  src);
@endcode
 
in the following way:
 
@code
matrix_free.template loop_cell_centric<VectorType, VectorType>(
  [&](const auto &data, auto &dst, const auto &src, const auto range) {
    FEEvaluation<dim, degree, degree + 1, 1, Number>     phi(data);
    FEFaceEvaluation<dim, degree, degree + 1, 1, Number> phi_m(data, /*is_interior_face=*/true);
    FEFaceEvaluation<dim, degree, degree + 1, 1, Number> phi_p(data, /*is_interior_face=*/false);
 
    for (unsigned int cell = range.first; cell < range.second; ++cell)
      {
        phi.reinit(cell);
        phi.gather_evaluate(src, cell_evaluation_flags);
 
        // some operations on the cell quadrature points
 
        phi.integrate_scatter(cell_evaluation_flags, dst);
 
        // loop over all faces of cell
        for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
          {
            if (data.get_faces_by_cells_boundary_id(cell, face)[0] ==
                numbers::internal_face_boundary_id)
              {
                // internal face
                phi_m.reinit(cell, face);
                phi_m.gather_evaluate(src, face_evaluation_flags);
                phi_p.reinit(cell, face);
                phi_p.gather_evaluate(src, face_evaluation_flags);
 
                // some operations on the face quadrature points
 
                phi_m.integrate_scatter(face_evaluation_flags, dst);
              }
            else
              {
                // boundary face
                phi_m.reinit(cell, face);
                phi_m.gather_evaluate(src, face_evaluation_flags);
 
                // some operations on the face quadrature points
 
                phi_m.integrate_scatter(face_evaluation_flags, dst);
              }
          }
      }
  },
  dst,
  src);
@endcode
 
It should be noted that FEFaceEvaluation is initialized now with two numbers,
the cell number and the local face number. The given example only
highlights how to transform face-centric loops into cell-centric loops and
is by no means efficient, since data is read and written multiple times
from and to the global vector as well as computations are performed
redundantly. Below, we will discuss advanced techniques that target these issues.
 
To be able to use MatrixFree::loop_cell_centric(), following flags of MatrixFree::AdditionalData
have to be enabled:
 
@code
typename MatrixFree<dim, Number>::AdditionalData additional_data;
 
// set flags as usual (not shown)
 
additional_data.hold_all_faces_to_owned_cells       = true;
additional_data.mapping_update_flags_faces_by_cells =
  additional_data.mapping_update_flags_inner_faces |
  additional_data.mapping_update_flags_boundary_faces;
 
data.reinit(mapping, dof_handler, constraint, quadrature, additional_data);
@endcode
 
In particular, these flags enable that the internal data structures are set up
for all faces of the cells.
 
Currently, cell-centric loops in deal.II only work for uniformly refined meshes
and if no constraints are applied (which is the standard case DG is normally
used).
 
 
<a name="step_76-ProvidinglambdastoMatrixFreeloops"></a><h3>Providing lambdas to MatrixFree loops</h3>
 
 
The examples given above have already used lambdas, which have been provided to
matrix-free loops. The following short examples present how to transform functions between
a version where a class and a pointer to one of its methods are used and a
variant where lambdas are utilized.
 
In the following code, a class and a pointer to one of its methods, which should
be interpreted as cell integral, are passed to MatrixFree::loop():
 
@code
void
local_apply_cell(const MatrixFree<dim, Number> &              data,
                 VectorType &                                 dst,
                 const VectorType &                           src,
                 const std::pair<unsigned int, unsigned int> &range) const
{
  FEEvaluation<dim, degree, degree + 1, 1, Number> phi(data);
  for (unsigned int cell = range.first; cell < range.second; ++cell)
    {
      phi.reinit(cell);
      phi.gather_evaluate(src, cell_evaluation_flags);
 
      // some operations on the quadrature points
 
      phi.integrate_scatter(cell_evaluation_flags, dst);
    }
}
@endcode
 
@code
matrix_free.cell_loop(&Operator::local_apply_cell, this, dst, src);
@endcode
 
However, it is also possible to pass an anonymous function via a lambda function
with the same result:
 
@code
matrix_free.template cell_loop<VectorType, VectorType>(
  [&](const auto &data, auto &dst, const auto &src, const auto range) {
    FEEvaluation<dim, degree, degree + 1, 1, Number> phi(data);
    for (unsigned int cell = range.first; cell < range.second; ++cell)
      {
        phi.reinit(cell);
        phi.gather_evaluate(src, cell_evaluation_flags);
 
        // some operations on the quadrature points
 
        phi.integrate_scatter(cell_evaluation_flags, dst);
      }
  },
  dst,
  src);
@endcode
 
<a name="step_76-VectorizedArrayType"></a><h3>VectorizedArrayType</h3>
 
 
The class VectorizedArray<Number> is a key component to achieve the high
node-level performance of the matrix-free algorithms in deal.II.
It is a wrapper class around a short vector of @f$n@f$ entries of type Number and
maps arithmetic operations to appropriate single-instruction/multiple-data
(SIMD) concepts by intrinsic functions. The length of the vector can be
queried by VectorizedArray::size() and its underlying number type by
VectorizedArray::value_type.
 
In the default case (<code>VectorizedArray<Number></code>), the vector length is
set at compile time of the library to
match the highest value supported by the given processor architecture.
However, also a second optional template argument can be
specified as <code>VectorizedArray<Number, size></code>, where <code>size</code> explicitly
controls the  vector length within the capabilities of a particular instruction
set. A full list of supported vector lengths is presented in the following table:
 
<table align="center" class="doxtable">
  <tr>
   <th>double</th>
   <th>float</th>
   <th>ISA</th>
  </tr>
  <tr>
   <td><code>VectorizedArray<double, 1></code></td>
   <td><code>VectorizedArray<float, 1></code></td>
   <td>(auto-vectorization)</td>
  </tr>
  <tr>
   <td><code>VectorizedArray<double, 2></code></td>
   <td><code>VectorizedArray<float, 4></code></td>
   <td>SSE2/AltiVec</td>
  </tr>
  <tr>
   <td><code>VectorizedArray<double, 4></code></td>
   <td><code>VectorizedArray<float, 8></code></td>
   <td>AVX/AVX2</td>
  </tr>
  <tr>
   <td><code>VectorizedArray<double, 8></code></td>
   <td><code>VectorizedArray<float, 16></code></td>
   <td>AVX-512</td>
  </tr>
</table>
 
This allows users to select the vector length/ISA and, as a consequence, the
number of cells to be processed at once in matrix-free operator evaluations,
possibly reducing the pressure on the caches, an severe issue for very high
degrees (and dimensions).
 
A possible further reason to reduce the number of filled lanes
is to simplify debugging: instead of having to look at, e.g., 8
cells, one can concentrate on a single cell.
 
The interface of VectorizedArray also enables the replacement by any type with
a matching interface. Specifically, this prepares deal.II for the <code>std::simd</code>
class that is planned to become part of the C++23 standard. The following table
compares the deal.II-specific SIMD classes and the equivalent C++23 classes:
 
 
<table align="center" class="doxtable">
  <tr>
   <th>VectorizedArray (deal.II)</th>
   <th>std::simd (C++23)</th>
  </tr>
  <tr>
   <td><code>VectorizedArray<Number></code></td>
   <td><code>std::experimental::native_simd<Number></code></td>
  </tr>
  <tr>
   <td><code>VectorizedArray<Number, size></code></td>
   <td><code>std::experimental::fixed_size_simd<Number, size></code></td>
  </tr>
</table>
 *
 *
 * <a name="step_76-CommProg"></a>
 * <h1> The commented program</h1>
 * 
 * 
 * <a name="step_76-Parametersandutilityfunctions"></a> 
 * <h3>Parameters and utility functions</h3>
 * 
 
 * 
 * The same includes as in @ref step_67 "step-67":
 * 
 * @code
 *   #include <deal.II/base/conditional_ostream.h>
 *   #include <deal.II/base/function.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>
 *   
 * @endcode
 * 
 * A new include for categorizing of cells according to their boundary IDs:
 * 
 * @code
 *   #include <deal.II/matrix_free/tools.h>
 *   
 *   
 *   
 *   namespace Euler_DG
 *   {
 *     using namespace dealii;
 *   
 * @endcode
 * 
 * The same input parameters as in @ref step_67 "step-67":
 * 
 * @code
 *     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;
 *   
 * @endcode
 * 
 * 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.
 * 
 * @code
 *     constexpr unsigned int group_size = numbers::invalid_unsigned_int;
 *   
 *     using Number = double;
 *   
 * @endcode
 * 
 * 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.
 * 
 * @code
 *     using VectorizedArrayType = VectorizedArray<Number>;
 *   
 * @endcode
 * 
 * The following parameters have not changed:
 * 
 * @code
 *     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);
 *   
 * @endcode
 * 
 * Specify max number of time steps useful for performance studies.
 * 
 * @code
 *     constexpr unsigned int max_time_steps = numbers::invalid_unsigned_int;
 *   
 * @endcode
 * 
 * Runge-Kutta-related functions copied from @ref step_67 "step-67" and slightly modified
 * with the purpose to minimize global vector access:
 * 
 * @code
 *     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;
 *     };
 *   
 *   
 * @endcode
 * 
 * Euler-specific utility functions from @ref step_67 "step-67":
 * 
 * @code
 *     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:
 *             DEAL_II_NOT_IMPLEMENTED();
 *             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:
 *             {
 *               DEAL_II_NOT_IMPLEMENTED();
 *               return {};
 *             }
 *         }
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * General-purpose utility functions from @ref step_67 "step-67":
 * 
 * @code
 *     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;
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_76-EuleroperatorusingacellcentricloopandMPI30sharedmemory"></a> 
 * <h3>Euler operator using a cell-centric loop and MPI-3.0 shared memory</h3>
 * 
 
 * 
 * Euler operator from @ref step_67 "step-67" with some changes as detailed below:
 * 
 * @code
 *     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:
 * @endcode
 * 
 * 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:
 * 
 * @code
 *       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;
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 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>.
 * 
 * @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
 *         {
 *           DEAL_II_NOT_IMPLEMENTED();
 *         }
 *   #else
 *       (void)subcommunicator;
 *       (void)group_size;
 *       this->subcommunicator = MPI_COMM_SELF;
 *   #endif
 *     }
 *   
 *   
 * @endcode
 * 
 * New destructor responsible for freeing of the sub-communicator.
 * 
 * @code
 *     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
 *     }
 *   
 *   
 * @endcode
 * 
 * Modified reinit() function to set up 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:
 * 
 * @code
 *     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;
 *   
 * @endcode
 * 
 * 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.
 * 
 * @code
 *       MatrixFreeTools::categorize_by_boundary_ids(dof_handler.get_triangulation(),
 *                                                   additional_data);
 *   
 * @endcode
 * 
 * Enable MPI-3.0 shared-memory capabilities within MatrixFree by providing
 * the sub-communicator:
 * 
 * @code
 *       additional_data.communicator_sm = subcommunicator;
 *   
 *       data.reinit(
 *         mapping, dof_handlers, constraints, quadratures, additional_data);
 *     }
 *   
 *   
 * @endcode
 * 
 * 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 @ref step_67 "step-67".
 *   
 
 * 
 * In contrast to @ref step_67 "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 @ref step_67 "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>
 * 
 * @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);
 *   
 * @endcode
 * 
 * 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:
 * 
 * @code
 *       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 internal::EvaluatorTensorProduct<
 *             internal::EvaluatorVariant::evaluate_evenodd,
 *             dim,
 *             n_points_1d,
 *             n_points_1d,
 *             VectorizedArrayType,
 *             Number>
 *             eval({},
 *                  data.get_shape_info().data[0].shape_gradients_collocation_eo,
 *                  {});
 *   
 *           AlignedVector<VectorizedArrayType> buffer(phi.static_n_q_points *
 *                                                     phi.n_components);
 *   
 * @endcode
 * 
 * Loop over all cell batches:
 * 
 * @code
 *           for (unsigned int cell = cell_range.first; cell < cell_range.second;
 *                ++cell)
 *             {
 *               phi.reinit(cell);
 *   
 *               if (ai != Number())
 *                 phi_temp.reinit(cell);
 *   
 * @endcode
 * 
 * Read values from global vector and compute the values at the
 * quadrature points:
 * 
 * @code
 *               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);
 *                 }
 *   
 * @endcode
 * 
 * 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:
 * 
 * @code
 *               for (unsigned int i = 0; i < phi.static_n_q_points * (dim + 2); ++i)
 *                 buffer[i] = phi.begin_values()[i];
 *   
 * @endcode
 * 
 * Apply the cell integral at the cell quadrature points. See also
 * the function <code>EulerOperator::local_apply_cell()</code> from
 * @ref step_67 "step-67":
 * 
 * @code
 *               for (const unsigned int q : phi.quadrature_point_indices())
 *                 {
 *                   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);
 *                     }
 *                 }
 *   
 * @endcode
 * 
 * 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.
 * 
 * @code
 *               {
 *                 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, dim>(gradient_ptr,
 *                                                                     values_ptr);
 *                     else if (dim >= 1)
 *                       eval.template gradients<0, false, true, dim>(gradient_ptr,
 *                                                                    values_ptr);
 *                     if (dim >= 2)
 *                       eval.template gradients<1, false, true, dim>(gradient_ptr +
 *                                                                      1,
 *                                                                    values_ptr);
 *                     if (dim >= 3)
 *                       eval.template gradients<2, false, true, dim>(gradient_ptr +
 *                                                                      2,
 *                                                                    values_ptr);
 *   
 *                     values_ptr += phi.static_n_q_points;
 *                     gradient_ptr += phi.static_n_q_points * dim;
 *                   }
 *               }
 *   
 * @endcode
 * 
 * Loop over all faces of the current cell:
 * 
 * @code
 *               for (unsigned int face = 0;
 *                    face < GeometryInfo<dim>::faces_per_cell;
 *                    ++face)
 *                 {
 * @endcode
 * 
 * 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.
 * 
 * @code
 *                   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);
 *   
 * @endcode
 * 
 * Interpolate the values from the cell quadrature points to the
 * quadrature points of the current face via a simple 1d
 * interpolation:
 * 
 * @code
 *                   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);
 *   
 * @endcode
 * 
 * Check if the face is an internal or a boundary face and
 * select a different code path based on this information:
 * 
 * @code
 *                   if (boundary_id == numbers::internal_face_boundary_id)
 *                     {
 * @endcode
 * 
 * Process and internal face. The following lines of code
 * are a copy of the function
 * <code>EulerDG::EulerOperator::local_apply_face</code>
 * from @ref step_67 "step-67":
 * 
 * @code
 *                       phi_p.reinit(cell, face);
 *                       phi_p.gather_evaluate(src, EvaluationFlags::values);
 *   
 *                       for (const unsigned int q :
 *                            phi_m.quadrature_point_indices())
 *                         {
 *                           const auto numerical_flux =
 *                             euler_numerical_flux<dim>(phi_m.get_value(q),
 *                                                       phi_p.get_value(q),
 *                                                       phi_m.normal_vector(q));
 *                           phi_m.submit_value(-numerical_flux, q);
 *                         }
 *                     }
 *                   else
 *                     {
 * @endcode
 * 
 * Process a boundary face. These following lines of code
 * are a copy of the function
 * <code>EulerDG::EulerOperator::local_apply_boundary_face</code>
 * from @ref step_67 "step-67":
 * 
 * @code
 *                       for (const unsigned int q :
 *                            phi_m.quadrature_point_indices())
 *                         {
 *                           const auto w_m    = phi_m.get_value(q);
 *                           const auto normal = phi_m.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);
 *                         }
 *                     }
 *   
 * @endcode
 * 
 * Evaluate local integrals related to cell by quadrature and
 * add into cell contribution via a simple 1d interpolation:
 * 
 * @code
 *                   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);
 *                 }
 *   
 * @endcode
 * 
 * Apply inverse mass matrix in the cell quadrature points. See
 * also the function
 * <code>EulerDG::EulerOperator::local_apply_inverse_mass_matrix()</code>
 * from @ref step_67 "step-67":
 * 
 * @code
 *               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;
 *                 }
 *   
 * @endcode
 * 
 * Transform values from collocation space to the original
 * Gauss-Lobatto space:
 * 
 * @code
 *               internal::FEEvaluationImplBasisChange<
 *                 internal::EvaluatorVariant::evaluate_evenodd,
 *                 internal::EvaluatorQuantity::hessian,
 *                 dim,
 *                 degree + 1,
 *                 n_points_1d>::do_backward(dim + 2,
 *                                           data.get_shape_info()
 *                                             .data[0]
 *                                             .inverse_shape_values_eo,
 *                                           false,
 *                                           phi.begin_values(),
 *                                           phi.begin_dof_values());
 *   
 * @endcode
 * 
 * Perform Runge-Kutta update and write results back to global
 * vectors:
 * 
 * @code
 *               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);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * From here, the code of @ref step_67 "step-67" has not changed.
 * 
 * @code
 *     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 (const unsigned int q : phi.quadrature_point_indices())
 *             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 (const unsigned int q : phi.quadrature_point_indices())
 *             {
 *               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 (const unsigned int q : phi.quadrature_point_indices())
 *             {
 *               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)
 *             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
 *   
 *       const FESystem<dim> fe;
 *       const 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 p = 0; p < n_evaluation_points; ++p)
 *         {
 *           Tensor<1, dim + 2> solution;
 *           for (unsigned int d = 0; d < dim + 2; ++d)
 *             solution[d] = inputs.solution_values[p](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[p](d) = velocity[d];
 *           computed_quantities[p](dim)     = pressure;
 *           computed_quantities[p](dim + 1) = std::sqrt(gamma * pressure / density);
 *   
 *           if (do_schlieren_plot == true)
 *             computed_quantities[p](dim + 2) =
 *               inputs.solution_gradients[p][0] * inputs.solution_gradients[p][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:
 *             DEAL_II_NOT_IMPLEMENTED();
 *         }
 *   
 *       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_v<Number, double> ? "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
 *       {
 *         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;
 *   }
 * @endcode
<a name="step_76-Results"></a><h1>Results</h1>
 
 
Running the program with the default settings on a machine with 24 processes
in release mode and AVX2 vectorization produces the following output:
 
@code
Running with 24 MPI processes
Vectorization over 4 doubles = 256 bits (AVX)
Number of degrees of freedom: 230,400 ( = 4 [vars] x 1,600 [cells] x 36 [dofs/cell/var] )
Time step size: 0.000295952, minimal h: 0.0075, initial transport scaling: 0.00441179
 
Time:       0, dt:   0.0003, norm rho:   4.17e-16, rho * u:  1.629e-16, energy: 1.381e-15
Time:  0.0501, dt:  0.00025, norm rho:    0.02076, rho * u:      0.038, energy:   0.08774
 
...
 
+-------------------------------------+------------------+------------+------------------+
| Total wallclock time elapsed        |     8.231s     8 |     8.235s |     8.236s     5 |
|                                     |                  |                               |
| Section                 | no. calls |   min time  rank |   avg time |   max time  rank |
+-------------------------------------+------------------+------------+------------------+
| compute errors          |        41 |  0.002513s    17 |  0.002564s |  0.002631s    13 |
| compute transport speed |      1731 |    0.1581s    17 |    0.1608s |    0.1636s    11 |
| output                  |        41 |    0.7818s     0 |    0.7832s |    0.7847s    17 |
| rk time stepping total  |      8648 |      7.23s    23 |     7.233s |     7.237s    13 |
+-------------------------------------+------------------+------------+------------------+
@endcode
 
and the following visual output, which was taken from @ref step_67 "step-67" (for a slightly different test case):
 
<table align="center" class="doxtable" style="width:85%">
  <tr>
    <td>
        <img src="https://www.dealii.org/images/steps/developer/step-67.pressure_010.png" alt="" width="100%">
    </td>
    <td>
        <img src="https://www.dealii.org/images/steps/developer/step-67.pressure_025.png" alt="" width="100%">
    </td>
  </tr>
  <tr>
    <td>
        <img src="https://www.dealii.org/images/steps/developer/step-67.pressure_050.png" alt="" width="100%">
    </td>
    <td>
        <img src="https://www.dealii.org/images/steps/developer/step-67.pressure_100.png" alt="" width="100%">
    </td>
  </tr>
</table>
 
As a reference, the results of @ref step_67 "step-67" for test case 1 and the same resolution
(note this is not the default test case nor the default resolution in @ref step_67 "step-67")
using FCL are:
 
@code
Running with 24 MPI processes
Vectorization over 4 doubles = 256 bits (AVX)
Number of degrees of freedom: 230,400 ( = 4 [vars] x 1,600 [cells] x 36 [dofs/cell/var] )
Time step size: 0.000295952, minimal h: 0.0075, initial transport scaling: 0.00441179
 
Time:       0, dt:   0.0003, norm rho:   4.17e-16, rho * u:  1.629e-16, energy: 1.381e-15
Time:  0.0501, dt:  0.00025, norm rho:    0.02076, rho * u:    0.03801, energy:   0.08774
 
...
 
+-------------------------------------------+------------------+------------+------------------+
| Total wallclock time elapsed              |     8.166s     4 |     8.168s |     8.169s    14 |
|                                           |                  |                               |
| Section                       | no. calls |   min time  rank |   avg time |   max time  rank |
+-------------------------------------------+------------------+------------+------------------+
| compute errors                |        41 |  0.002265s    18 |  0.004323s |  0.006903s     0 |
| compute transport speed       |      1731 |    0.1471s    18 |    0.2393s |    0.3772s     0 |
| output                        |        41 |     0.791s     0 |    0.7925s |    0.7934s     1 |
| rk time stepping total        |      8648 |     6.955s     0 |     7.096s |     7.189s    18 |
| rk_stage - integrals L_h      |     43240 |     5.537s     5 |     5.833s |     6.171s    13 |
| rk_stage - inv mass + vec upd |     43240 |    0.7758s     4 |      1.15s |     1.373s     5 |
+-------------------------------------------+------------------+------------+------------------+
@endcode
 
While the performance in this small test case is almost identical, the
difference depends on the hardware and the dimension and size of the setup.
In larger computations we have seen that the modifications shown in this
tutorial were able to achieve a speedup of 27% for the Runge-Kutta stages.
 
<a name="step_76-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
 
 
The algorithms are easily extendable to higher dimensions: a high-dimensional
<a href="https://github.com/hyperdeal/hyperdeal/blob/a9e67b4e625ff1dde2fed93ad91cdfacfaa3acdf/include/hyper.deal/operators/advection/advection_operation.h#L219-L569">advection operator based on cell-centric loops</a>
is part of the hyper.deal library. An extension of cell-centric loops
to locally-refined meshes is more involved.
 
<a name="step_76-ExtensiontothecompressibleNavierStokesequations"></a><h4>Extension to the compressible Navier-Stokes equations</h4>
 
 
The solver presented in this tutorial program can also be extended to the
compressible Navier–Stokes equations by adding viscous terms, as also
suggested in @ref step_67 "step-67". To keep as much of the performance obtained here despite
the additional cost of elliptic terms, e.g. via an interior penalty method, that
tutorial has proposed to switch the basis from FE_DGQ to FE_DGQHermite like in
the @ref step_59 "step-59" tutorial program. The reasoning behind this switch is that in the
case of FE_DGQ all values of neighboring cells (i.e., @f$k+1@f$ layers) are needed,
whilst in the case of FE_DGQHermite only 2 layers, making the latter
significantly more suitable for higher degrees. The additional layers have to be,
on the one hand, loaded from main memory during flux computation and, one the
other hand, have to be communicated. Using the shared-memory capabilities
introduced in this tutorial, the second point can be eliminated on a single
compute node or its influence can be reduced in a hybrid context.
 
<a name="step_76-BlockGaussSeidellikepreconditioners"></a><h4>Block Gauss-Seidel-like preconditioners</h4>
 
 
Cell-centric loops could be used to create block Gauss-Seidel preconditioners
that are multiplicative within one process and additive over processes. These
type of preconditioners use during flux computation, in contrast to Jacobi-type
preconditioners, already updated values from neighboring cells. The following
pseudo-code visualizes how this could in principal be achieved:
 
@code
// vector monitor if cells have been updated or not
Vector<Number> visit_flags(data.n_cell_batches () + data.n_ghost_cell_batches ());
 
// element centric loop with a modified kernel
data.template loop_cell_centric<VectorType, VectorType>(
  [&](const auto &data, auto &dst, const auto &src, const auto cell_range) {
 
    for (unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
      {
        // cell integral as usual (not shown)
 
        for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
          {
            const auto boundary_id = data.get_faces_by_cells_boundary_id(cell, face)[0];
 
            if (boundary_id == numbers::internal_face_boundary_id)
              {
                phi_p.reinit(cell, face);
 
                const auto flags = phi_p.read_cell_data(visit_flags);
                const auto all_neighbors_have_been_updated =
                  std::min(flags.begin(),
                           flags().begin() + data.n_active_entries_per_cell_batch(cell) == 1;
 
                if(all_neighbors_have_been_updated)
                  phi_p.gather_evaluate(dst, EvaluationFlags::values);
                else
                  phi_p.gather_evaluate(src, EvaluationFlags::values);
 
                // continue as usual (not shown)
              }
            else
              {
                // boundary integral as usual (not shown)
              }
          }
 
        // continue as above and apply your favorite algorithm to invert
        // the cell-local operator (not shown)
 
        // make cells as updated
        phi.set_cell_data(visit_flags, VectorizedArrayType(1.0));
      }
  },
  dst,
  src,
  true,
  MatrixFree<dim, Number, VectorizedArrayType>::DataAccessOnFaces::values);
@endcode
 
For this purpose, one can exploit the cell-data vector capabilities of
MatrixFree and the range-based iteration capabilities of VectorizedArray.
 
Please note that in the given example we process <code>VectorizedArrayType::size()</code>
number of blocks, since each lane corresponds to one block. We consider blocks
as updated if all blocks processed by a vector register have been updated. In
the case of Cartesian meshes this is a reasonable approach, however, for
general unstructured meshes this conservative approach might lead to a decrease in the
efficiency of the preconditioner. A reduction of cells processed in parallel
by explicitly reducing the number of lanes used by <code>VectorizedArrayType</code>
might increase the quality of the preconditioner, but with the cost that each
iteration might be more expensive. This dilemma leads us to a further
"possibility for extension": vectorization within an element.
 *
 *
<a name="step_76-PlainProg"></a>
<h1> The plain program</h1>
@include "step-76.cc"
*/