step-86.h

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 0.0,
 *                           /* end time */ 1.0,
 *                           /* initial time step */ 0.025)
 *       , initial_global_refinement(5)
 *       , max_delta_refinement_level(2)
 *       , mesh_adaptation_frequency(0)
 *       , right_hand_side_function("/Heat Equation/Right hand side")
 *       , initial_value_function("/Heat Equation/Initial value")
 *       , boundary_values_function("/Heat Equation/Boundary values")
 *     {
 *       enter_subsection("Time stepper");
 *       {
 *         enter_my_subsection(this->prm);
 *         {
 *           time_stepper_data.add_parameters(this->prm);
 *         }
 *         leave_my_subsection(this->prm);
 *       }
 *       leave_subsection();
 *   
 *       add_parameter("Initial global refinement",
 *                     initial_global_refinement,
 *                     "Number of times the mesh is refined globally before "
 *                     "starting the time stepping.");
 *       add_parameter("Maximum delta refinement level",
 *                     max_delta_refinement_level,
 *                     "Maximum number of local refinement levels.");
 *       add_parameter("Mesh adaptation frequency",
 *                     mesh_adaptation_frequency,
 *                     "When to adapt the mesh.");
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationsetup_systemfunction"></a> 
 * <h4>The HeatEquation::setup_system() function</h4>
 *   
 
 * 
 * This function is not very different from what we do in many other programs
 * (including @ref step_26 "step-26"). We enumerate degrees of freedom, output some
 * information about then, build constraint objects (recalling that we
 * put hanging node constraints into their separate object), and then
 * also build an AffineConstraint object that contains both the hanging
 * node constraints as well as constraints corresponding to zero Dirichlet
 * boundary conditions. This last object is needed since we impose the
 * constraints through algebraic equations. While technically it would be
 * possible to use the time derivative of the boundary function as a boundary
 * conditions for the time derivative of the solution, this is not done here.
 * Instead, we impose the boundary conditions through algebraic equations, and
 * therefore the time derivative of the boundary conditions is not part of
 * the algebraic system, and we need zero boundary conditions on the time
 * derivative of the solution when computing the residual. We use the
 * `homogeneous_constraints` object for this purpose.
 *   
 
 * 
 * Finally, we create the actual non-homogeneous `current_constraints` by
 * calling `update_current_constraints). These are also used during the
 * assembly and during the residual evaluation.
 * 
 * @code
 *     template <int dim>
 *     void HeatEquation<dim>::setup_system(const double time)
 *     {
 *       TimerOutput::Scope t(computing_timer, "setup system");
 *   
 *       dof_handler.distribute_dofs(fe);
 *       pcout << std::endl
 *             << "Number of active cells: " << triangulation.n_active_cells()
 *             << std::endl
 *             << "Number of degrees of freedom: " << dof_handler.n_dofs()
 *             << std::endl
 *             << std::endl;
 *   
 *       locally_owned_dofs = dof_handler.locally_owned_dofs();
 *       locally_relevant_dofs =
 *         DoFTools::extract_locally_relevant_dofs(dof_handler);
 *   
 *   
 *       hanging_nodes_constraints.clear();
 *       hanging_nodes_constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
 *       DoFTools::make_hanging_node_constraints(dof_handler,
 *                                               hanging_nodes_constraints);
 *       hanging_nodes_constraints.make_consistent_in_parallel(locally_owned_dofs,
 *                                                             locally_relevant_dofs,
 *                                                             mpi_communicator);
 *       hanging_nodes_constraints.close();
 *   
 *   
 *       homogeneous_constraints.clear();
 *       homogeneous_constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
 *       homogeneous_constraints.merge(hanging_nodes_constraints);
 *       VectorTools::interpolate_boundary_values(dof_handler,
 *                                                0,
 *                                                Functions::ZeroFunction<dim>(),
 *                                                homogeneous_constraints);
 *       homogeneous_constraints.make_consistent_in_parallel(locally_owned_dofs,
 *                                                           locally_relevant_dofs,
 *                                                           mpi_communicator);
 *       homogeneous_constraints.close();
 *   
 *   
 *       update_current_constraints(time);
 *   
 *   
 * @endcode
 * 
 * The final block of code resets and initializes the matrix object with
 * the appropriate sparsity pattern. Recall that we do not store solution
 * vectors in this class (the time integrator object does that internally)
 * and so do not have to resize and initialize them either.
 * 
 * @code
 *       DynamicSparsityPattern dsp(locally_relevant_dofs);
 *       DoFTools::make_sparsity_pattern(dof_handler,
 *                                       dsp,
 *                                       homogeneous_constraints,
 *                                       false);
 *       SparsityTools::distribute_sparsity_pattern(dsp,
 *                                                  locally_owned_dofs,
 *                                                  mpi_communicator,
 *                                                  locally_relevant_dofs);
 *   
 *       jacobian_matrix.reinit(locally_owned_dofs,
 *                              locally_owned_dofs,
 *                              dsp,
 *                              mpi_communicator);
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationoutput_resultsfunction"></a> 
 * <h4>The HeatEquation::output_results() function</h4>
 *   
 
 * 
 * This function is called from "monitor" function that is called in turns
 * by the time stepper in each time step. We use it to write the solution to a
 * file, and provide graphical output through paraview or visit. We also write
 * a pvd file, which groups all metadata about the `.vtu` files into a single
 * file that can be used to load the full time dependent solution in paraview.
 * 
 * @code
 *     template <int dim>
 *     void HeatEquation<dim>::output_results(const double       time,
 *                                            const unsigned int timestep_number,
 *                                            const PETScWrappers::MPI::Vector &y)
 *     {
 *       TimerOutput::Scope t(computing_timer, "output results");
 *   
 *       DataOut<dim> data_out;
 *       data_out.attach_dof_handler(dof_handler);
 *       data_out.add_data_vector(y, "U");
 *       data_out.build_patches();
 *   
 *       data_out.set_flags(DataOutBase::VtkFlags(time, timestep_number));
 *   
 *       const std::string filename =
 *         "solution-" + Utilities::int_to_string(timestep_number, 3) + ".vtu";
 *       data_out.write_vtu_in_parallel(filename, mpi_communicator);
 *   
 *       if (Utilities::MPI::this_mpi_process(mpi_communicator) == 0)
 *         {
 *           static std::vector<std::pair<double, std::string>> times_and_names;
 *           times_and_names.emplace_back(time, filename);
 *   
 *           std::ofstream pvd_output("solution.pvd");
 *           DataOutBase::write_pvd_record(pvd_output, times_and_names);
 *         }
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationimplicit_functionfunction"></a> 
 * <h4>The HeatEquation::implicit_function() function</h4>
 *   
 
 * 
 * As discussed in the introduction, we describe the ODE system to the time
 * stepper via its residual,
 * @f[
 * R(t,U,\dot U) = M \frac{\partial U(t)}{\partial t} + AU(t) - F(t).
 * @f]
 * The following function computes it, given vectors for @f$U,\dot U@f$ that
 * we will denote by `y` and `y_dot` because that's how they are called
 * in the documentation of the PETScWrappers::TimeStepper class.
 *   
 
 * 
 * At the top of the function, we do the usual set up when computing
 * integrals. We face two minor difficulties here: the first is that
 * the `y` and `y_dot` vectors we get as input are read only, but we
 * need to make sure they satisfy the correct boundary conditions and so
 * have to set elements in these vectors. The second is that we need to
 * compute the residual, and therefore in general we need to evaluate solution
 * values and gradients inside locally owned cells, and for this need access
 * to degrees of freedom which may be owned by neighboring processors. To
 * address these issues, we create (non-ghosted) writable copies of the input
 * vectors, apply boundary conditions and hanging node current_constraints;
 * and then copy these vectors to ghosted vectors before we can do anything
 * sensible with them.
 * 
 * @code
 *     template <int dim>
 *     void
 *     HeatEquation<dim>::implicit_function(const double                      time,
 *                                          const PETScWrappers::MPI::Vector &y,
 *                                          const PETScWrappers::MPI::Vector &y_dot,
 *                                          PETScWrappers::MPI::Vector &residual)
 *     {
 *       TimerOutput::Scope t(computing_timer, "implicit function");
 *   
 *       PETScWrappers::MPI::Vector tmp_solution(locally_owned_dofs,
 *                                               mpi_communicator);
 *       PETScWrappers::MPI::Vector tmp_solution_dot(locally_owned_dofs,
 *                                                   mpi_communicator);
 *       tmp_solution     = y;
 *       tmp_solution_dot = y_dot;
 *   
 *       update_current_constraints(time);
 *       current_constraints.distribute(tmp_solution);
 *       homogeneous_constraints.distribute(tmp_solution_dot);
 *   
 *       PETScWrappers::MPI::Vector locally_relevant_solution(locally_owned_dofs,
 *                                                            locally_relevant_dofs,
 *                                                            mpi_communicator);
 *       PETScWrappers::MPI::Vector locally_relevant_solution_dot(
 *         locally_owned_dofs, locally_relevant_dofs, mpi_communicator);
 *       locally_relevant_solution     = tmp_solution;
 *       locally_relevant_solution_dot = tmp_solution_dot;
 *   
 *   
 *       const QGauss<dim> quadrature_formula(fe.degree + 1);
 *       FEValues<dim>     fe_values(fe,
 *                               quadrature_formula,
 *                               update_values | update_gradients |
 *                                 update_quadrature_points | update_JxW_values);
 *   
 *       const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
 *       const unsigned int n_q_points    = quadrature_formula.size();
 *   
 *       std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
 *   
 *       std::vector<Tensor<1, dim>> solution_gradients(n_q_points);
 *       std::vector<double>         solution_dot_values(n_q_points);
 *   
 *       Vector<double> cell_residual(dofs_per_cell);
 *   
 *       right_hand_side_function.set_time(time);
 *   
 * @endcode
 * 
 * Now for computing the actual residual. Recall that we wan to compute the
 * vector
 * @f[
 * R(t,U,\dot U) = M \frac{\partial U(t)}{\partial t} + AU(t) - F(t).
 * @f]
 * We could do that by actually forming the matrices @f$M@f$ and @f$A@f$, but this
 * is not efficient. Instead, recall (by writing out how the elements of
 * @f$M@f$ and @f$A@f$ are defined, and exchanging integrals and sums) that the
 * @f$i@f$th element of the residual vector is given by
 * @f{align*}{
 * R(t,U,\dot U)_i
 * &= \sum_j \int_\Omega \varphi_i(\mathbf x, t) \varphi_j(\mathbf x, t)
 * {\partial U_j(t)}{\partial t}
 * + \sum_j \int_\Omega \nabla \varphi_i(\mathbf x, t) \cdot \nabla
 * \varphi_j(\mathbf x, t) U_j(t)
 * - \int_\Omega \varphi_i f(\mathbf x, t)
 * \\ &=
 * \int_\Omega \varphi_i(\mathbf x, t) u_h(\mathbf x, t)
 * + \int_\Omega \nabla \varphi_i(\mathbf x, t) \cdot \nabla
 * u_h(\mathbf x, t)
 * - \int_\Omega \varphi_i f(\mathbf x, t).
 * @f}
 * We can compute these integrals efficiently by breaking them up into
 * a sum over all cells and then applying quadrature. For the integrand,
 * we need to evaluate the solution and its gradient at the quadrature
 * points within each locally owned cell, and for this, we need also
 * access to degrees of freedom that may be owned by neighboring
 * processors. We therefore use the locally_relevant_solution and and
 * locally_relevant_solution_dot vectors.
 * 
 * @code
 *       residual = 0;
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         if (cell->is_locally_owned())
 *           {
 *             fe_values.reinit(cell);
 *   
 *             fe_values.get_function_gradients(locally_relevant_solution,
 *                                              solution_gradients);
 *             fe_values.get_function_values(locally_relevant_solution_dot,
 *                                           solution_dot_values);
 *   
 *             cell->get_dof_indices(local_dof_indices);
 *   
 *             cell_residual = 0;
 *             for (const unsigned int q : fe_values.quadrature_point_indices())
 *               for (const unsigned int i : fe_values.dof_indices())
 *                 {
 *                   cell_residual(i) +=
 *                     (fe_values.shape_value(i, q) *       // [phi_i(x_q) *
 *                        solution_dot_values[q]            //  u(x_q)
 *                      +                                   //  +
 *                      fe_values.shape_grad(i, q) *        //  grad phi_i(x_q) *
 *                        solution_gradients[q]             //  grad u(x_q)
 *                      -                                   //  -
 *                      fe_values.shape_value(i, q) *       //  phi_i(x_q) *
 *                        right_hand_side_function.value(   
 *                          fe_values.quadrature_point(q))) //  f(x_q)]
 *                     * fe_values.JxW(q);                  // * dx
 *                 }
 *             current_constraints.distribute_local_to_global(cell_residual,
 *                                                            local_dof_indices,
 *                                                            residual);
 *           }
 *       residual.compress(VectorOperation::add);
 *   
 * @endcode
 * 
 * The end result of the operations above is a vector that contains the
 * residual vector, having taken into account the constraints due to
 * hanging nodes and Dirichlet boundary conditions (by virtue of having
 * used `current_constraints.distribute_local_to_global()` to add the
 * local contributions to the global vector. At the end of the day, the
 * residual vector @f$r@f$ will be used in the solution of linear systems
 * of the form @f$J z = r@f$ with the "Jacobian" matrix that we define
 * below. We want to achieve that for algebraic components, the algebraic
 * components of @f$z@f$ have predictable values that achieve the purposes
 * discussed in the following. We do this by ensuring that the entries
 * corresponding to algebraic components in the residual @f$r@f$ have specific
 * values, and then we will do the same in the next function for the
 * matrix; for this, you will have to know that the rows and columns
 * of the matrix corresponding to constrained entries are zero with the
 * exception of the diagonal entries. We will manually set that diagonal
 * entry to one, and so @f$z_i=r_i@f$ for algebraic components.
 *     
 
 * 
 * From the point of view of the residual vector, if the input `y`
 * vector does not contain the correct values on constrained degrees of
 * freedom (hanging nodes or boundary conditions), we need to communicate
 * this to the time stepper, and we do so by setting the residual to the
 * actual difference between the input `y` vector and the our local copy of
 * it, in which we have applied the constraints (see the top of the
 * function where we called `current_constraints.distribute(tmp_solution)`
 * and a similar operation on the time derivative). Since we have made a
 * copy of the input vector for this purpose, we use it to compute the
 * residual value. However, there is a difference between hanging nodes
 * constraints and boundary conditions: we do not want to make hanging node
 * constraints actually depend on their dependent degrees of freedom, since
 * this would imply that we are actually solving for the dependent degrees
 * of freedom. This is not what we are actually doing, however, since
 * hanging nodes are not actually solved for. They are eliminated from the
 * system by the call to AffineConstraints::distribute_local_to_global()
 * above. From the point of view of the Jacobian matrix, we are effectively
 * setting hanging nodes to an artificial value (usually zero), and we
 * simply want to make sure that we solve for those degrees of freedom a
 * posteriori, by calling the function AffineConstraints::distribute().
 *     
 
 * 
 * Here we therefore check that the residual is equal to the input value on
 * the constrained dofs corresponding to hanging nodes (i.e., those for
 * which the lines of the `current_constraints` contain at least one other
 * entry), and to the difference between the input vector and the actual
 * solution on those constraints that correspond to boundary conditions.
 * 
 * @code
 *       for (const auto &c : current_constraints.get_lines())
 *         if (locally_owned_dofs.is_element(c.index))
 *           {
 *             if (c.entries.empty()) /* no dependencies -> a Dirichlet node */
 *               residual[c.index] = y[c.index] - tmp_solution[c.index];
 *             else /* has dependencies -> a hanging node */
 *               residual[c.index] = y[c.index];
 *           }
 *       residual.compress(VectorOperation::insert);
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationassemble_implicit_jacobianfunction"></a> 
 * <h4>The HeatEquation::assemble_implicit_jacobian() function</h4>
 *   
 
 * 
 * The next operation is to compute the "Jacobian", which PETSc TS defines
 * as the matrix
 * @f[
 * J_\alpha = \dfrac{\partial R}{\partial y} + \alpha \dfrac{\partial
 * R}{\partial \dot y}
 * @f]
 * which, for the current linear problem, is simply
 * @f[
 * J_\alpha = A + \alpha M
 * @f]
 * and which is in particular independent of time and the current solution
 * vectors @f$y@f$ and @f$\dot y@f$.
 *   
 
 * 
 * Having seen the assembly of matrices before, there is little that should
 * surprise you in the actual assembly here:
 * 
 * @code
 *     template <int dim>
 *     void HeatEquation<dim>::assemble_implicit_jacobian(
 *       const double /* time */,
 *       const PETScWrappers::MPI::Vector & /* y */,
 *       const PETScWrappers::MPI::Vector & /* y_dot */,
 *       const double alpha)
 *     {
 *       TimerOutput::Scope t(computing_timer, "assemble implicit Jacobian");
 *   
 *       const QGauss<dim> quadrature_formula(fe.degree + 1);
 *       FEValues<dim>     fe_values(fe,
 *                               quadrature_formula,
 *                               update_values | update_gradients |
 *                                 update_quadrature_points | update_JxW_values);
 *   
 *       const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
 *   
 *       std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
 *   
 *       FullMatrix<double> cell_matrix(dofs_per_cell, dofs_per_cell);
 *   
 *       jacobian_matrix = 0;
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         if (cell->is_locally_owned())
 *           {
 *             fe_values.reinit(cell);
 *   
 *             cell->get_dof_indices(local_dof_indices);
 *   
 *             cell_matrix = 0;
 *             for (const unsigned int q : fe_values.quadrature_point_indices())
 *               for (const unsigned int i : fe_values.dof_indices())
 *                 for (const unsigned int j : fe_values.dof_indices())
 *                   {
 *                     cell_matrix(i, j) +=
 *                       (fe_values.shape_grad(i, q) *      // grad phi_i(x_q) *
 *                          fe_values.shape_grad(j, q)      // grad phi_j(x_q)
 *                        + alpha *                         
 *                            fe_values.shape_value(i, q) * // phi_i(x_q) *
 *                            fe_values.shape_value(j, q)   // phi_j(x_q)
 *                        ) *
 *                       fe_values.JxW(q); // * dx
 *                   }
 *             current_constraints.distribute_local_to_global(cell_matrix,
 *                                                            local_dof_indices,
 *                                                            jacobian_matrix);
 *           }
 *       jacobian_matrix.compress(VectorOperation::add);
 *   
 * @endcode
 * 
 * The only interesting part is the following. Recall that we modified the
 * residual vector's entries corresponding to the algebraic components of
 * the solution in the previous function. The outcome of calling
 * `current_constraints.distribute_local_to_global()` a few lines
 * above is that the global matrix has zero rows and columns for the
 * algebraic (constrained) components of the solution; the function
 * puts a value on the diagonal that is nonzero and has about the
 * same size as the remaining diagonal entries of the matrix. What
 * this diagonal value is is unknown to us -- in other cases where
 * we call `current_constraints.distribute_local_to_global()` on
 * both the left side matrix and the right side vector, as in most
 * other tutorial programs, the matrix diagonal entries and right
 * hand side values are chosen in such a way that the result of
 * solving a linear system is what we want it to be, but the scaling
 * is done automatically.
 *     
 
 * 
 * This is not good enough for us here, because we are building the
 * right hand side independently from the matrix in different functions.
 * Thus, for any constrained degree of freedom, we set the diagonal of the
 * Jacobian to be one. This leaves the Jacobian matrix invertible,
 * consistent with what the time stepper expects, and it also makes sure
 * that if we did not make a mistake in the residual and/or in the Jacbian
 * matrix, then asking the time stepper to check the Jacobian with a finite
 * difference method will produce the correct result. This can be activated
 * at run time via passing the `-snes_test_jacobian` option on the command
 * line.
 * 
 * @code
 *       for (const auto &c : current_constraints.get_lines())
 *         jacobian_matrix.set(c.index, c.index, 1.0);
 *       jacobian_matrix.compress(VectorOperation::insert);
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationsolve_with_jacobianfunction"></a> 
 * <h4>The HeatEquation::solve_with_jacobian() function</h4>
 *   
 
 * 
 * This is the function that actually solves the linear system with the
 * Jacobian matrix we have previously built (in a call to the previous
 * function during the current time step or another earlier one -- time
 * steppers are quite sophisticated in determining internally whether it is
 * necessary to update the Jacobian matrix, or whether one can reuse it for
 * another time step without rebuilding it; this is similar to how one can
 * re-use the Newton matrix for several Newton steps, see for example the
 * discussion in @ref step_77 "step-77"). We could in principle not provide this function to
 * the time stepper, and instead select a specific solver on the command line
 * by using the `-ksp_*` options of PETSc. However, by providing this
 * function, we can use a specific solver and preconditioner for the linear
 * system, and still have the possibility to change them on the command line.
 *   
 
 * 
 * Providing a specific solver is more in line with the way we usually do
 * things in other deal.II examples, while letting PETSc choose a generic
 * solver, and changing it on the command line via `-ksp_type` is more in line
 * with the way PETSc is usually used, and it can be a convenient approach
 * when we are experimenting to find an optimal solver for our problem. Both
 * options are available here, since we can still change both the solver and
 * the preconditioner on the command line via `-user_ksp_type` and
 * `-user_pc_type` options.
 *   
 
 * 
 * In any case, recall that the Jacobian we built in the previous function is
 * always of the form
 * @f[
 * J_\alpha = \alpha M + A
 * @f]
 * where @f$M@f$ is a mass matrix and @f$A@f$ a Laplace matrix. @f$M@f$ is symmetric and
 * positive definite; @f$A@f$ is symmetric and at least positive semidefinite;
 * @f$\alpha> 0@f$. As a consequence, the Jacobian matrix is a symmetric and
 * positive definite matrix, which we can efficiently solve with the Conjugate
 * Gradient method, along with either SSOR or (if available) the algebraic
 * multigrid implementation provided by PETSc (via the Hypre package) as
 * preconditioner. In practice, if you wanted to solve "real" problems, one
 * would spend some time finding which preconditioner is optimal, perhaps
 * using PETSc's ability to read solver and preconditioner choices from the
 * command line. But this is not the focus of this tutorial program, and so
 * we just go with the following:
 * 
 * @code
 *     template <int dim>
 *     void
 *     HeatEquation<dim>::solve_with_jacobian(const PETScWrappers::MPI::Vector &src,
 *                                            PETScWrappers::MPI::Vector       &dst)
 *     {
 *       TimerOutput::Scope t(computing_timer, "solve with Jacobian");
 *   
 *   #if defined(PETSC_HAVE_HYPRE)
 *       PETScWrappers::PreconditionBoomerAMG preconditioner;
 *       preconditioner.initialize(jacobian_matrix);
 *   #else
 *       PETScWrappers::PreconditionSSOR preconditioner;
 *       preconditioner.initialize(
 *         jacobian_matrix, PETScWrappers::PreconditionSSOR::AdditionalData(1.0));
 *   #endif
 *   
 *       SolverControl           solver_control(1000, 1e-8 * src.l2_norm());
 *       PETScWrappers::SolverCG cg(solver_control);
 *       cg.set_prefix("user_");
 *   
 *       cg.solve(jacobian_matrix, dst, src, preconditioner);
 *   
 *       pcout << "     " << solver_control.last_step() << " linear iterations."
 *             << std::endl;
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationprepare_for_coarsening_and_refinementfunction"></a> 
 * <h4>The HeatEquation::prepare_for_coarsening_and_refinement() function</h4>
 *   
 
 * 
 * The next block of functions deals with mesh refinement. We split this
 * process up into a "decide whether and what you want to refine" and a
 * "please transfer these vectors from old to new mesh" phase, where the first
 * also deals with marking cells for refinement. (The decision whether or
 * not to refine is done in the lambda function that calls the current
 * function.)
 *   
 
 * 
 * Breaking things into a "mark cells" function and into a "execute mesh
 * adaptation and transfer solution vectors" function is awkward, though
 * conceptually not difficult to understand. These two pieces of code
 * should really be part of the same function, as they are in @ref step_26 "step-26".
 * The issue is with what PETScWrappers::TimeStepper provides us with in these
 * callbacks. Specifically, the "decide whether and what you want to refine"
 * callback has access to the current solution, and so can evaluate
 * (spatial) error estimators to decide which cells to refine. The
 * second callback that transfers vectors from old to new mesh gets a bunch
 * of vectors, but without the semantic information on which of these is
 * the current solution vector. As a consequence, it cannot do the marking
 * of cells for refinement or coarsening, and we have to do that from the
 * first callback.
 *   
 
 * 
 * In practice, however, the problem is minor. The first of these
 * two functions is essentially identical to the
 * first half of the corresponding function in @ref step_26 "step-26", with the only
 * difference that it uses the parallel::distributed::GridRefinement namespace
 * function instead of the serial one. Once again, we make sure that we never
 * fall below the minimum refinement level, and above the maximum one, that we
 * can select from the parameter file.
 * 
 * @code
 *     template <int dim>
 *     void HeatEquation<dim>::prepare_for_coarsening_and_refinement(
 *       const PETScWrappers::MPI::Vector &y)
 *     {
 *       PETScWrappers::MPI::Vector locally_relevant_solution(locally_owned_dofs,
 *                                                            locally_relevant_dofs,
 *                                                            mpi_communicator);
 *       locally_relevant_solution = y;
 *   
 *       Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
 *       KellyErrorEstimator<dim>::estimate(dof_handler,
 *                                          QGauss<dim - 1>(fe.degree + 1),
 *                                          {},
 *                                          locally_relevant_solution,
 *                                          estimated_error_per_cell);
 *   
 *       parallel::distributed::GridRefinement::refine_and_coarsen_fixed_fraction(
 *         triangulation, estimated_error_per_cell, 0.6, 0.4);
 *   
 *       const unsigned int max_grid_level =
 *         initial_global_refinement + max_delta_refinement_level;
 *       const unsigned int min_grid_level = initial_global_refinement;
 *   
 *       if (triangulation.n_levels() > max_grid_level)
 *         for (const auto &cell :
 *              triangulation.active_cell_iterators_on_level(max_grid_level))
 *           cell->clear_refine_flag();
 *       for (const auto &cell :
 *            triangulation.active_cell_iterators_on_level(min_grid_level))
 *         cell->clear_coarsen_flag();
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationtransfer_solution_vectors_to_new_meshfunction"></a> 
 * <h4>The HeatEquation::transfer_solution_vectors_to_new_mesh() function</h4>
 *   
 
 * 
 * The following function then is the second half of the correspond function
 * in @ref step_26 "step-26". It is called by the time stepper whenever it requires to
 * transfer the solution and any intermediate stage vectors to a new mesh. We
 * must make sure that all input vectors are transformed into ghosted vectors
 * before the actual transfer is executed, and that we distribute the hanging
 * node constraints on the output vectors as soon as we have interpolated the
 * vectors to the new mesh -- i.e., that all constraints are satisfied on
 * the vectors we transfer.
 *   
 
 * 
 * We have no way to enforce boundary conditions at this stage, however. This
 * is because the various vectors may correspond to solutions at previous time
 * steps if the method used here is a multistep time integrator, and so may
 * correspond to different time points that we are not privy to.
 *   
 
 * 
 * While this could be a problem if we used the values of the solution and of
 * the intermediate stages on the constrained degrees of freedom to compute
 * the errors, we do not do so. Instead, we compute the errors on the
 * differential equation, and ignore algebraic constraints; therefore we do no
 * need to guarantee that the boundary conditions are satisfied also in the
 * intermediate stages.
 *   
 
 * 
 * We have at our disposal the hanging node current_constraints alone, though,
 * and therefore we enforce them on the output vectors, even if this is not
 * really needed.
 * 
 * @code
 *     template <int dim>
 *     void HeatEquation<dim>::transfer_solution_vectors_to_new_mesh(
 *       const double                                   time,
 *       const std::vector<PETScWrappers::MPI::Vector> &all_in,
 *       std::vector<PETScWrappers::MPI::Vector>       &all_out)
 *     {
 *       parallel::distributed::SolutionTransfer<dim, PETScWrappers::MPI::Vector>
 *         solution_trans(dof_handler);
 *   
 *       std::vector<PETScWrappers::MPI::Vector> all_in_ghosted(all_in.size());
 *       std::vector<const PETScWrappers::MPI::Vector *> all_in_ghosted_ptr(
 *         all_in.size());
 *       std::vector<PETScWrappers::MPI::Vector *> all_out_ptr(all_in.size());
 *       for (unsigned int i = 0; i < all_in.size(); ++i)
 *         {
 *           all_in_ghosted[i].reinit(locally_owned_dofs,
 *                                    locally_relevant_dofs,
 *                                    mpi_communicator);
 *           all_in_ghosted[i]     = all_in[i];
 *           all_in_ghosted_ptr[i] = &all_in_ghosted[i];
 *         }
 *   
 *       triangulation.prepare_coarsening_and_refinement();
 *       solution_trans.prepare_for_coarsening_and_refinement(all_in_ghosted_ptr);
 *       triangulation.execute_coarsening_and_refinement();
 *   
 *       setup_system(time);
 *   
 *       all_out.resize(all_in.size());
 *       for (unsigned int i = 0; i < all_in.size(); ++i)
 *         {
 *           all_out[i].reinit(locally_owned_dofs, mpi_communicator);
 *           all_out_ptr[i] = &all_out[i];
 *         }
 *       solution_trans.interpolate(all_out_ptr);
 *   
 *       for (PETScWrappers::MPI::Vector &v : all_out)
 *         hanging_nodes_constraints.distribute(v);
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationupdate_current_constraintsfunction"></a> 
 * <h4>The HeatEquation::update_current_constraints() function</h4>
 *   
 
 * 
 * Since regenerating the constraints at each time step may be expensive, we
 * make sure that we only do so when the time changes. We track time change by
 * checking if the time of the boundary_values_function has changed, with
 * respect to the time of the last call to this function. This will work most
 * of the times, but not the very first time we call this function, since the
 * time then may be zero and the time of the `boundary_values_function` is
 * zero at construction time. We therefore also check if the number
 * constraints in `current_constraints`, and if these are empty, we regenerate
 * the constraints regardless of the time variable.
 * 
 * @code
 *     template <int dim>
 *     void HeatEquation<dim>::update_current_constraints(const double time)
 *     {
 *       if (current_constraints.n_constraints() == 0 ||
 *           time != boundary_values_function.get_time())
 *         {
 *           TimerOutput::Scope t(computing_timer, "update current constraints");
 *   
 *           boundary_values_function.set_time(time);
 *           current_constraints.clear();
 *           current_constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
 *           current_constraints.merge(hanging_nodes_constraints);
 *           VectorTools::interpolate_boundary_values(dof_handler,
 *                                                    0,
 *                                                    boundary_values_function,
 *                                                    current_constraints);
 *           current_constraints.make_consistent_in_parallel(locally_owned_dofs,
 *                                                           locally_relevant_dofs,
 *                                                           mpi_communicator);
 *           current_constraints.close();
 *         }
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-TheHeatEquationrunfunction"></a> 
 * <h4>The HeatEquation::run() function</h4>
 *   
 
 * 
 * We have finally arrived at the main function of the class. At the top, it
 * creates the mesh and sets up the variables that make up the linear system:
 * 
 * @code
 *     template <int dim>
 *     void HeatEquation<dim>::run()
 *     {
 *       GridGenerator::hyper_L(triangulation);
 *       triangulation.refine_global(initial_global_refinement);
 *   
 *       setup_system(/* time */ 0);
 *   
 * @endcode
 * 
 * We then set up the time stepping object and associate the matrix we will
 * build whenever requested for both the Jacobian matrix (see the definition
 * above of what the "Jacobian" actually refers to) and for the matrix
 * that will be used as a preconditioner for the Jacobian.
 * 
 * @code
 *       PETScWrappers::TimeStepper<PETScWrappers::MPI::Vector,
 *                                  PETScWrappers::MPI::SparseMatrix>
 *         petsc_ts(time_stepper_data);
 *   
 *       petsc_ts.set_matrices(jacobian_matrix, jacobian_matrix);
 *   
 *   
 * @endcode
 * 
 * The real work setting up the time stepping object starts here. As
 * discussed in the introduction, the way the PETScWrappers::TimeStepper
 * class is used is by inverting control: At the end of this function, we
 * will call PETScWrappers::TimeStepper::solve() which internally will
 * run the loop over time steps, and at the appropriate places *call back*
 * into user code for whatever functionality is required. What we need to
 * do is to hook up these callback functions by assigning appropriate
 * lambda functions to member variables of the `petsc_ts` object.
 *     
 
 * 
 * We start by creating lambda functions that provide information about
 * the "implicit function" (i.e., that part of the right hand side of the
 * ODE system that we want to treat implicitly -- which in our case is
 * the entire right hand side), a function that assembles the Jacobian
 * matrix, and a function that solves a linear system with the Jacobian.
 * 
 * @code
 *       petsc_ts.implicit_function = [&](const double                      time,
 *                                        const PETScWrappers::MPI::Vector &y,
 *                                        const PETScWrappers::MPI::Vector &y_dot,
 *                                        PETScWrappers::MPI::Vector       &res) {
 *         this->implicit_function(time, y, y_dot, res);
 *       };
 *   
 *       petsc_ts.setup_jacobian = [&](const double                      time,
 *                                     const PETScWrappers::MPI::Vector &y,
 *                                     const PETScWrappers::MPI::Vector &y_dot,
 *                                     const double                      alpha) {
 *         this->assemble_implicit_jacobian(time, y, y_dot, alpha);
 *       };
 *   
 *       petsc_ts.solve_with_jacobian = [&](const PETScWrappers::MPI::Vector &src,
 *                                          PETScWrappers::MPI::Vector       &dst) {
 *         this->solve_with_jacobian(src, dst);
 *       };
 *   
 * @endcode
 * 
 * The next two callbacks deal with identifying and setting variables
 * that are considered "algebraic" (rather than "differential"), i.e., for
 * which we know what values they are supposed to have rather than letting
 * their values be determined by the differential equation. We need to
 * instruct the time stepper to ignore these components when computing the
 * error in the residuals, and we do so by first providing a function that
 * returns an IndexSet with the indices of these algebraic components of the
 * solution vector (or rather, that subset of the locally-owned part of the
 * vector that is algebraic, in case we are running in parallel). This first
 * of the following two functions does that. Specifically, both nodes at
 * which Dirichlet boundary conditions are applied, and hanging nodes are
 * algebraically constrained. This function then returns a set of
 * indices that is initially empty (but knows about the size of the index
 * space) and which we then construct as the union of boundary and hanging
 * node indices.
 *     
 
 * 
 * Following this, we then also need a function that, given a solution
 * vector `y` and the current time, sets the algebraic components of that
 * vector to their correct value. This comes down to ensuring that we have
 * up to date constraints in the `constraints` variable, and then applying
 * these constraints to the solution vector via
 * AffineConstraints::distribute(). (It is perhaps worth noting that we
 * *could* have achieved the same in `solve_with_jacobian()`. Whenever the
 * time stepper solves a linear system, it follows up the call to the solver
 * by calling the callback to set algebraic components correct. We could
 * also have put the calls to `update_current_constraints()` and
 * `distribute()` into the `solve_with_jacobian` function, but by not doing
 * so, we can also replace the `solve_with_jacobian` function with a call to
 * a PETSc solver, and we would still have the current_constraints correctly
 * applied to the solution vector.)
 * 
 * @code
 *       petsc_ts.algebraic_components = [&]() {
 *         IndexSet algebraic_set(dof_handler.n_dofs());
 *         algebraic_set.add_indices(DoFTools::extract_boundary_dofs(dof_handler));
 *         algebraic_set.add_indices(
 *           DoFTools::extract_hanging_node_dofs(dof_handler));
 *         return algebraic_set;
 *       };
 *   
 *       petsc_ts.update_constrained_components =
 *         [&](const double time, PETScWrappers::MPI::Vector &y) {
 *           TimerOutput::Scope t(computing_timer, "set algebraic components");
 *           update_current_constraints(time);
 *           current_constraints.distribute(y);
 *         };
 *   
 *   
 * @endcode
 * 
 * The next two callbacks relate to mesh refinement. As discussed in the
 * introduction, PETScWrappers::TimeStepper knows how to deal with the
 * situation where we want to change the mesh. All we have to provide
 * is a callback that returns `true` if we are at a point where we want
 * to refine the mesh (and `false` otherwise) and that if we want to
 * do mesh refinement does some prep work for that in the form of
 * calling the `prepare_for_coarsening_and_refinement` function.
 *     
 
 * 
 * If the first callback below returns `true`, then PETSc TS will
 * do some clean-up operations, and call the second of the
 * callback functions
 * (`petsc_ts.transfer_solution_vectors_to_new_mesh`) with a
 * collection of vectors that need to be interpolated from the old
 * to the new mesh. This may include the current solution, perhaps
 * the current time derivative of the solution, and in the case of
 * [multistep time
 * integrators](https://en.wikipedia.org/wiki/Linear_multistep_method) also
 * the solutions of some previous time steps. We hand all of these over to
 * the `interpolate()` member function of this class.
 * 
 * @code
 *       petsc_ts.decide_and_prepare_for_remeshing =
 *         [&](const double /* time */,
 *             const unsigned int                step_number,
 *             const PETScWrappers::MPI::Vector &y) -> bool {
 *         if (step_number > 0 && this->mesh_adaptation_frequency > 0 &&
 *             step_number % this->mesh_adaptation_frequency == 0)
 *           {
 *             pcout << std::endl << "Adapting the mesh..." << std::endl;
 *             this->prepare_for_coarsening_and_refinement(y);
 *             return true;
 *           }
 *         else
 *           return false;
 *       };
 *   
 *       petsc_ts.transfer_solution_vectors_to_new_mesh =
 *         [&](const double                                   time,
 *             const std::vector<PETScWrappers::MPI::Vector> &all_in,
 *             std::vector<PETScWrappers::MPI::Vector>       &all_out) {
 *           this->transfer_solution_vectors_to_new_mesh(time, all_in, all_out);
 *         };
 *   
 * @endcode
 * 
 * The final callback is a "monitor" that is called in each
 * time step. Here we use it to create a graphical output. Perhaps a better
 * scheme would output the solution at fixed time intervals, rather
 * than in every time step, but this is not the main point of
 * this program and so we go with the easy approach:
 * 
 * @code
 *       petsc_ts.monitor = [&](const double                      time,
 *                              const PETScWrappers::MPI::Vector &y,
 *                              const unsigned int                step_number) {
 *         pcout << "Time step " << step_number << " at t=" << time << std::endl;
 *         this->output_results(time, step_number, y);
 *       };
 *   
 *   
 * @endcode
 * 
 * With all of this out of the way, the rest of the function is
 * anticlimactic: We just have to initialize the solution vector with
 * the initial conditions and call the function that does the time
 * stepping, and everything else will happen automatically:
 * 
 * @code
 *       PETScWrappers::MPI::Vector solution(locally_owned_dofs, mpi_communicator);
 *       VectorTools::interpolate(dof_handler, initial_value_function, solution);
 *   
 *       petsc_ts.solve(solution);
 *     }
 *   } // namespace Step86
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_86-Themainfunction"></a> 
 * <h3>The main() function</h3>
 * 
 
 * 
 * The rest of the program is as it always looks. We read run-time parameters
 * from an input file via the ParameterAcceptor class in the same way as
 * we showed in @ref step_60 "step-60" and @ref step_70 "step-70".
 * 
 * @code
 *   int main(int argc, char **argv)
 *   {
 *     try
 *       {
 *         using namespace Step86;
 *   
 *         Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
 *         HeatEquation<2>                  heat_equation_solver(MPI_COMM_WORLD);
 *   
 *         const std::string input_filename =
 *           (argc > 1 ? argv[1] : "heat_equation.prm");
 *         ParameterAcceptor::initialize(input_filename, "heat_equation_used.prm");
 *         heat_equation_solver.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_86-Results"></a><h1>Results</h1>
 
 
When you run this program with the input file as is, you get output as follows:
@code
 
Number of active cells: 768
Number of degrees of freedom: 833
 
Time step 0 at t=0
     5 linear iterations.
     8 linear iterations.
Time step 1 at t=0.025
     6 linear iterations.
Time step 2 at t=0.05
     5 linear iterations.
     8 linear iterations.
Time step 3 at t=0.075
     6 linear iterations.
Time step 4 at t=0.1
     6 linear iterations.
Time step 5 at t=0.125
     6 linear iterations.
Time step 6 at t=0.15
     6 linear iterations.
Time step 7 at t=0.175
     6 linear iterations.
Time step 8 at t=0.2
     6 linear iterations.
Time step 9 at t=0.225
     6 linear iterations.
 
Adapting the mesh...
 
Number of active cells: 1050
Number of degrees of freedom: 1155
 
Time step 10 at t=0.25
     5 linear iterations.
     8 linear iterations.
Time step 11 at t=0.275
     5 linear iterations.
     7 linear iterations.
 
[...]
 
Time step 195 at t=4.875
     6 linear iterations.
Time step 196 at t=4.9
     6 linear iterations.
Time step 197 at t=4.925
     6 linear iterations.
Time step 198 at t=4.95
     6 linear iterations.
Time step 199 at t=4.975
     5 linear iterations.
 
Adapting the mesh...
 
Number of active cells: 1380
Number of degrees of freedom: 1547
 
Time step 200 at t=5
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |      43.2s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| assemble implicit Jacobian      |       226 |      9.93s |        23% |
| implicit function               |       426 |      16.2s |        37% |
| output results                  |       201 |      9.74s |        23% |
| set algebraic components        |       200 |    0.0496s |      0.11% |
| setup system                    |        21 |     0.799s |       1.8% |
| solve with Jacobian             |       226 |      0.56s |       1.3% |
| update current constraints      |       201 |      1.53s |       3.5% |
+---------------------------------+-----------+------------+------------+
 
@endcode
 
We can generate output for this in the form of a video:
@htmlonly
<p align="center">
  <iframe width="560" height="315" src="https://www.youtube.com/embed/fhJVkcdRksM"
   frameborder="0"
   allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture"
   allowfullscreen></iframe>
 </p>
@endhtmlonly
The solution here is driven by boundary values (the initial conditions are zero,
and so is the right hand side of the equation). It takes a little bit of time
for the boundary values to diffuse into the domain, and so the temperature
(the solution of the heat equation) in the interior of the domain has a slight
lag compared to the temperature at the boundary.
 
 
The more interesting component of this program is how easy it is to play with
the details of the time stepping algorithm. Recall that the solution above
is controlled by the following parameters:
@code
  subsection Time stepper
    subsection Running parameters
      set final time              = 5
      set initial step size       = 0.025
      set initial time            = 0
      set match final time        = false
      set maximum number of steps = -1
      set options prefix          =
      set solver type             = beuler
    end
 
    subsection Error control
      set absolute error tolerance = -1
      set adaptor type             = none
      set ignore algebraic lte     = true
      set maximum step size        = -1
      set minimum step size        = -1
      set relative error tolerance = -1
    end
  end
@endcode
Of particular interest for us here is to set the time stepping algorithm
and the adaptive time step control method. The latter is set to "none"
above, but there are
[several alternative choices](https://petsc.org/release/manualpages/TS/TSAdaptType/)
for this parameter. For example, we can set parameters as follows,
@code
  subsection Time stepper
    subsection Running parameters
      set final time              = 5
      set initial step size       = 0.025
      set initial time            = 0
      set match final time        = false
      set maximum number of steps = -1
      set options prefix          =
      set solver type             = bdf
    end
 
    subsection Error control
      set absolute error tolerance = 1e-2
      set relative error tolerance = 1e-2
      set adaptor type             = basic
      set ignore algebraic lte     = true
      set maximum step size        = 1
      set minimum step size        = 0.01
    end
  end
@endcode
What we do here is set the initial time step size to 0.025, and choose relatively
large absolute and relative error tolerances of 0.01 for the time step size
adaptation algorithm (for which we choose "basic"). We ask PETSc TS to use a
[Backward Differentiation Formula (BDF)](https://en.wikipedia.org/wiki/Backward_differentiation_formula)
method, and we get the following as output:
@code
 
===========================================
Number of active cells: 768
Number of degrees of freedom: 833
 
Time step 0 at t=0
     5 linear iterations.
     5 linear iterations.
     5 linear iterations.
     4 linear iterations.
     6 linear iterations.
Time step 1 at t=0.01
     5 linear iterations.
     5 linear iterations.
Time step 2 at t=0.02
     5 linear iterations.
     5 linear iterations.
Time step 3 at t=0.03
     5 linear iterations.
     5 linear iterations.
Time step 4 at t=0.042574
     5 linear iterations.
     5 linear iterations.
Time step 5 at t=0.0588392
     5 linear iterations.
     5 linear iterations.
Time step 6 at t=0.0783573
     5 linear iterations.
     7 linear iterations.
     5 linear iterations.
     7 linear iterations.
Time step 7 at t=0.100456
     5 linear iterations.
     7 linear iterations.
     5 linear iterations.
     7 linear iterations.
Time step 8 at t=0.124982
     5 linear iterations.
     8 linear iterations.
     5 linear iterations.
     7 linear iterations.
Time step 9 at t=0.153156
     6 linear iterations.
     5 linear iterations.
 
[...]
 
Time step 206 at t=4.96911
     5 linear iterations.
     5 linear iterations.
Time step 207 at t=4.99398
     5 linear iterations.
     5 linear iterations.
Time step 208 at t=5.01723
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |       117s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| assemble implicit Jacobian      |       593 |      35.6s |        31% |
| implicit function               |      1101 |      56.3s |        48% |
| output results                  |       209 |      12.5s |        11% |
| set algebraic components        |       508 |     0.156s |      0.13% |
| setup system                    |        21 |      1.11s |      0.95% |
| solve with Jacobian             |       593 |      1.97s |       1.7% |
| update current constraints      |       509 |      4.53s |       3.9% |
+---------------------------------+-----------+------------+------------+
@endcode
What is happening here is that apparently PETSc TS is not happy with our
choice of initial time step size, and after several linear solves has
reduced it to the minimum we allow it to, 0.01. The following time steps
then run at a time step size of 0.01 until it decides to make it slightly
larger again and (apparently) switches to a higher order method that
requires more linear solves per time step but allows for a larger time
step closer to our initial choice 0.025 again. It does not quite hit the
final time of @f$T=5@f$ with its time step choices, but we've got only
ourselves to blame for that by setting
@code
      set match final time        = false
@endcode
in the input file.
 
Not all combinations of methods, time step adaptation algorithms, and
other parameters are valid, but the main messages from the experiment
above that you should take away are:
- It would undoubtedly be quite time consuming to implement many of the
  methods that PETSc TS offers for time stepping -- but with a program
  such as the one here, we don't need to: We can just select from the
  many methods PETSc TS already has.
- Adaptive time step control is difficult; adaptive choice of which
  method or order to choose is perhaps even more difficult. None of the
  time dependent programs that came before the current one (say, @ref step_23 "step-23",
  @ref step_26 "step-26", @ref step_31 "step-31", @ref step_58 "step-58", and a good number of others) have either.
  Moreover, while deal.II is good at spatial adaptation of meshes, it
  is not a library written by experts in time step adaptation, and so will
  likely not gain this ability either. But, again, it doesn't
  have to: We can rely on a library written by experts in that area.
 
 
 
<a name="step-86-extensions"></a>
<a name="step_86-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
 
 
The program actually runs in parallel, even though we have not used
that above. Specifically, if you have configured deal.II to use MPI,
then you can do `mpirun -np 8 ./step-86 heat_equation.prm` to run the
program with 8 processes.
 
For the program as currently written (and in debug mode), this makes
little difference: It will run about twice as fast, but take about 8
times as much CPU time. That is because the problem is just so small:
Generally between 1000 and 2000 degrees of freedom. A good rule of
thumb is that programs really only benefit from parallel computing if
you have somewhere in the range of 50,000 to 100,000 unknowns *per MPI
process*. But it is not difficult to adapt the program at hand here to
run with a much finer mesh, or perhaps in 3d, so that one is beyond
that limit and sees the benefits of parallel computing.
 
 *
 *
<a name="step_86-PlainProg"></a>
<h1> The plain program</h1>
@include "step-86.cc"
*/