Reference documentation for deal.II version 9.5.0
\(\newcommand{\dealvcentcolon}{\mathrel{\mathop{:}}}\) \(\newcommand{\dealcoloneq}{\dealvcentcolon\mathrel{\mkern-1.2mu}=}\) \(\newcommand{\jump}[1]{\left[\!\left[ #1 \right]\!\right]}\) \(\newcommand{\average}[1]{\left\{\!\left\{ #1 \right\}\!\right\}}\)
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step-25.h
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1 = 0) const override
512 *   {
513 *   const double t = this->get_time();
514 *  
515 *   switch (dim)
516 *   {
517 *   case 1:
518 *   {
519 *   const double m = 0.5;
520 *   const double c1 = 0.;
521 *   const double c2 = 0.;
522 *   return -4. * std::atan(m / std::sqrt(1. - m * m) *
523 *   std::sin(std::sqrt(1. - m * m) * t + c2) /
524 *   std::cosh(m * p[0] + c1));
525 *   }
526 *  
527 *   case 2:
528 *   {
529 *   const double theta = numbers::PI / 4.;
530 *   const double lambda = 1.;
531 *   const double a0 = 1.;
532 *   const double s = 1.;
533 *   const double arg = p[0] * std::cos(theta) +
534 *   std::sin(theta) * (p[1] * std::cosh(lambda) +
535 *   t * std::sinh(lambda));
536 *   return 4. * std::atan(a0 * std::exp(s * arg));
537 *   }
538 *  
539 *   case 3:
540 *   {
541 *   const double theta = numbers::PI / 4;
542 *   const double phi = numbers::PI / 4;
543 *   const double tau = 1.;
544 *   const double c0 = 1.;
545 *   const double s = 1.;
546 *   const double arg = p[0] * std::cos(theta) +
547 *   p[1] * std::sin(theta) * std::cos(phi) +
548 *   std::sin(theta) * std::sin(phi) *
549 *   (p[2] * std::cosh(tau) + t * std::sinh(tau));
550 *   return 4. * std::atan(c0 * std::exp(s * arg));
551 *   }
552 *  
553 *   default:
554 *   Assert(false, ExcNotImplemented());
555 *   return -1e8;
556 *   }
557 *   }
558 *   };
559 *  
560 * @endcode
561 *
562 * In the second part of this section, we provide the initial conditions. We
563 * are lazy (and cautious) and don't want to implement the same functions as
564 * above a second time. Rather, if we are queried for initial conditions, we
565 * create an object <code>ExactSolution</code>, set it to the correct time,
566 * and let it compute whatever values the exact solution has at that time:
567 *
568 * @code
569 *   template <int dim>
570 *   class InitialValues : public Function<dim>
571 *   {
572 *   public:
573 *   InitialValues(const unsigned int n_components = 1, const double time = 0.)
574 *   : Function<dim>(n_components, time)
575 *   {}
576 *  
577 *   virtual double value(const Point<dim> & p,
578 *   const unsigned int component = 0) const override
579 *   {
580 *   return ExactSolution<dim>(1, this->get_time()).value(p, component);
581 *   }
582 *   };
583 *  
584 *  
585 * @endcode
586 *
587 *
588 * <a name="ImplementationofthecodeSineGordonProblemcodeclass"></a>
589 * <h3>Implementation of the <code>SineGordonProblem</code> class</h3>
590 *
591
592 *
593 * Let's move on to the implementation of the main class, as it implements
594 * the algorithm outlined in the introduction.
595 *
596
597 *
598 *
599 * <a name="SineGordonProblemSineGordonProblem"></a>
600 * <h4>SineGordonProblem::SineGordonProblem</h4>
601 *
602
603 *
604 * This is the constructor of the <code>SineGordonProblem</code> class. It
605 * specifies the desired polynomial degree of the finite elements,
606 * associates a <code>DoFHandler</code> to the <code>triangulation</code>
607 * object (just as in the example programs @ref step_3 "step-3" and @ref step_4 "step-4"), initializes
608 * the current or initial time, the final time, the time step size, and the
609 * value of @f$\theta@f$ for the time stepping scheme. Since the solutions we
610 * compute here are time-periodic, the actual value of the start-time
611 * doesn't matter, and we choose it so that we start at an interesting time.
612 *
613
614 *
615 * Note that if we were to chose the explicit Euler time stepping scheme
616 * (@f$\theta = 0@f$), then we must pick a time step @f$k \le h@f$, otherwise the
617 * scheme is not stable and oscillations might arise in the solution. The
618 * Crank-Nicolson scheme (@f$\theta = \frac{1}{2}@f$) and the implicit Euler
619 * scheme (@f$\theta=1@f$) do not suffer from this deficiency, since they are
620 * unconditionally stable. However, even then the time step should be chosen
621 * to be on the order of @f$h@f$ in order to obtain a good solution. Since we
622 * know that our mesh results from the uniform subdivision of a rectangle,
623 * we can compute that time step easily; if we had a different domain, the
624 * technique in @ref step_24 "step-24" using GridTools::minimal_cell_diameter would work as
625 * well.
626 *
627 * @code
628 *   template <int dim>
629 *   SineGordonProblem<dim>::SineGordonProblem()
630 *   : fe(1)
631 *   , dof_handler(triangulation)
632 *   , n_global_refinements(6)
633 *   , time(-5.4414)
634 *   , final_time(2.7207)
635 *   , time_step(10 * 1. / std::pow(2., 1. * n_global_refinements))
636 *   , theta(0.5)
637 *   , output_timestep_skip(1)
638 *   {}
639 *  
640 * @endcode
641 *
642 *
643 * <a name="SineGordonProblemmake_grid_and_dofs"></a>
644 * <h4>SineGordonProblem::make_grid_and_dofs</h4>
645 *
646
647 *
648 * This function creates a rectangular grid in <code>dim</code> dimensions
649 * and refines it several times. Also, all matrix and vector members of the
650 * <code>SineGordonProblem</code> class are initialized to their appropriate
651 * sizes once the degrees of freedom have been assembled. Like @ref step_24 "step-24", we
652 * use <code>MatrixCreator</code> functions to generate a mass matrix @f$M@f$
653 * and a Laplace matrix @f$A@f$ and store them in the appropriate variables for
654 * the remainder of the program's life.
655 *
656 * @code
657 *   template <int dim>
658 *   void SineGordonProblem<dim>::make_grid_and_dofs()
659 *   {
661 *   triangulation.refine_global(n_global_refinements);
662 *  
663 *   std::cout << " Number of active cells: " << triangulation.n_active_cells()
664 *   << std::endl
665 *   << " Total number of cells: " << triangulation.n_cells()
666 *   << std::endl;
667 *  
668 *   dof_handler.distribute_dofs(fe);
669 *  
670 *   std::cout << " Number of degrees of freedom: " << dof_handler.n_dofs()
671 *   << std::endl;
672 *  
673 *   DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
674 *   DoFTools::make_sparsity_pattern(dof_handler, dsp);
675 *   sparsity_pattern.copy_from(dsp);
676 *  
677 *   system_matrix.reinit(sparsity_pattern);
678 *   mass_matrix.reinit(sparsity_pattern);
679 *   laplace_matrix.reinit(sparsity_pattern);
680 *  
681 *   MatrixCreator::create_mass_matrix(dof_handler,
682 *   QGauss<dim>(fe.degree + 1),
683 *   mass_matrix);
685 *   QGauss<dim>(fe.degree + 1),
686 *   laplace_matrix);
687 *  
688 *   solution.reinit(dof_handler.n_dofs());
689 *   solution_update.reinit(dof_handler.n_dofs());
690 *   old_solution.reinit(dof_handler.n_dofs());
691 *   M_x_velocity.reinit(dof_handler.n_dofs());
692 *   system_rhs.reinit(dof_handler.n_dofs());
693 *   }
694 *  
695 * @endcode
696 *
697 *
698 * <a name="SineGordonProblemassemble_system"></a>
699 * <h4>SineGordonProblem::assemble_system</h4>
700 *
701
702 *
703 * This function assembles the system matrix and right-hand side vector for
704 * each iteration of Newton's method. The reader should refer to the
705 * Introduction for the explicit formulas for the system matrix and
706 * right-hand side.
707 *
708
709 *
710 * Note that during each time step, we have to add up the various
711 * contributions to the matrix and right hand sides. In contrast to @ref step_23 "step-23"
712 * and @ref step_24 "step-24", this requires assembling a few more terms, since they depend
713 * on the solution of the previous time step or previous nonlinear step. We
714 * use the functions <code>compute_nl_matrix</code> and
715 * <code>compute_nl_term</code> to do this, while the present function
716 * provides the top-level logic.
717 *
718 * @code
719 *   template <int dim>
720 *   void SineGordonProblem<dim>::assemble_system()
721 *   {
722 * @endcode
723 *
724 * First we assemble the Jacobian matrix @f$F'_h(U^{n,l})@f$, where @f$U^{n,l}@f$
725 * is stored in the vector <code>solution</code> for convenience.
726 *
727 * @code
728 *   system_matrix.copy_from(mass_matrix);
729 *   system_matrix.add(std::pow(time_step * theta, 2), laplace_matrix);
730 *  
731 *   SparseMatrix<double> tmp_matrix(sparsity_pattern);
732 *   compute_nl_matrix(old_solution, solution, tmp_matrix);
733 *   system_matrix.add(std::pow(time_step * theta, 2), tmp_matrix);
734 *  
735 * @endcode
736 *
737 * Next we compute the right-hand side vector. This is just the
738 * combination of matrix-vector products implied by the description of
739 * @f$-F_h(U^{n,l})@f$ in the introduction.
740 *
741 * @code
742 *   system_rhs = 0.;
743 *  
744 *   Vector<double> tmp_vector(solution.size());
745 *  
746 *   mass_matrix.vmult(system_rhs, solution);
747 *   laplace_matrix.vmult(tmp_vector, solution);
748 *   system_rhs.add(std::pow(time_step * theta, 2), tmp_vector);
749 *  
750 *   mass_matrix.vmult(tmp_vector, old_solution);
751 *   system_rhs.add(-1.0, tmp_vector);
752 *   laplace_matrix.vmult(tmp_vector, old_solution);
753 *   system_rhs.add(std::pow(time_step, 2) * theta * (1 - theta), tmp_vector);
754 *  
755 *   system_rhs.add(-time_step, M_x_velocity);
756 *  
757 *   compute_nl_term(old_solution, solution, tmp_vector);
758 *   system_rhs.add(std::pow(time_step, 2) * theta, tmp_vector);
759 *  
760 *   system_rhs *= -1.;
761 *   }
762 *  
763 * @endcode
764 *
765 *
766 * <a name="SineGordonProblemcompute_nl_term"></a>
767 * <h4>SineGordonProblem::compute_nl_term</h4>
768 *
769
770 *
771 * This function computes the vector @f$S(\cdot,\cdot)@f$, which appears in the
772 * nonlinear term in both equations of the split formulation. This
773 * function not only simplifies the repeated computation of this term, but
774 * it is also a fundamental part of the nonlinear iterative solver that we
775 * use when the time stepping is implicit (i.e. @f$\theta\ne 0@f$). Moreover, we
776 * must allow the function to receive as input an "old" and a "new"
777 * solution. These may not be the actual solutions of the problem stored in
778 * <code>old_solution</code> and <code>solution</code>, but are simply the
779 * two functions we linearize about. For the purposes of this function, let
780 * us call the first two arguments @f$w_{\mathrm{old}}@f$ and @f$w_{\mathrm{new}}@f$
781 * in the documentation of this class below, respectively.
782 *
783
784 *
785 * As a side-note, it is perhaps worth investigating what order quadrature
786 * formula is best suited for this type of integration. Since @f$\sin(\cdot)@f$
787 * is not a polynomial, there are probably no quadrature formulas that can
788 * integrate these terms exactly. It is usually sufficient to just make sure
789 * that the right hand side is integrated up to the same order of accuracy
790 * as the discretization scheme is, but it may be possible to improve on the
791 * constant in the asymptotic statement of convergence by choosing a more
792 * accurate quadrature formula.
793 *
794 * @code
795 *   template <int dim>
796 *   void SineGordonProblem<dim>::compute_nl_term(const Vector<double> &old_data,
797 *   const Vector<double> &new_data,
798 *   Vector<double> &nl_term) const
799 *   {
800 *   nl_term = 0;
801 *   const QGauss<dim> quadrature_formula(fe.degree + 1);
802 *   FEValues<dim> fe_values(fe,
803 *   quadrature_formula,
806 *  
807 *   const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
808 *   const unsigned int n_q_points = quadrature_formula.size();
809 *  
810 *   Vector<double> local_nl_term(dofs_per_cell);
811 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
812 *   std::vector<double> old_data_values(n_q_points);
813 *   std::vector<double> new_data_values(n_q_points);
814 *  
815 *   for (const auto &cell : dof_handler.active_cell_iterators())
816 *   {
817 *   local_nl_term = 0;
818 * @endcode
819 *
820 * Once we re-initialize our <code>FEValues</code> instantiation to
821 * the current cell, we make use of the
822 * <code>get_function_values</code> routine to get the values of the
823 * "old" data (presumably at @f$t=t_{n-1}@f$) and the "new" data
824 * (presumably at @f$t=t_n@f$) at the nodes of the chosen quadrature
825 * formula.
826 *
827 * @code
828 *   fe_values.reinit(cell);
829 *   fe_values.get_function_values(old_data, old_data_values);
830 *   fe_values.get_function_values(new_data, new_data_values);
831 *  
832 * @endcode
833 *
834 * Now, we can evaluate @f$\int_K \sin\left[\theta w_{\mathrm{new}} +
835 * (1-\theta) w_{\mathrm{old}}\right] \,\varphi_j\,\mathrm{d}x@f$ using
836 * the desired quadrature formula.
837 *
838 * @code
839 *   for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
840 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
841 *   local_nl_term(i) +=
842 *   (std::sin(theta * new_data_values[q_point] +
843 *   (1 - theta) * old_data_values[q_point]) *
844 *   fe_values.shape_value(i, q_point) * fe_values.JxW(q_point));
845 *  
846 * @endcode
847 *
848 * We conclude by adding up the contributions of the integrals over
849 * the cells to the global integral.
850 *
851 * @code
852 *   cell->get_dof_indices(local_dof_indices);
853 *  
854 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
855 *   nl_term(local_dof_indices[i]) += local_nl_term(i);
856 *   }
857 *   }
858 *  
859 * @endcode
860 *
861 *
862 * <a name="SineGordonProblemcompute_nl_matrix"></a>
863 * <h4>SineGordonProblem::compute_nl_matrix</h4>
864 *
865
866 *
867 * This is the second function dealing with the nonlinear scheme. It
868 * computes the matrix @f$N(\cdot,\cdot)@f$, which appears in the nonlinear
869 * term in the Jacobian of @f$F(\cdot)@f$. Just as <code>compute_nl_term</code>,
870 * we must allow this function to receive as input an "old" and a "new"
871 * solution, which we again call @f$w_{\mathrm{old}}@f$ and @f$w_{\mathrm{new}}@f$
872 * below, respectively.
873 *
874 * @code
875 *   template <int dim>
876 *   void SineGordonProblem<dim>::compute_nl_matrix(
877 *   const Vector<double> &old_data,
878 *   const Vector<double> &new_data,
879 *   SparseMatrix<double> &nl_matrix) const
880 *   {
881 *   QGauss<dim> quadrature_formula(fe.degree + 1);
882 *   FEValues<dim> fe_values(fe,
883 *   quadrature_formula,
886 *  
887 *   const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
888 *   const unsigned int n_q_points = quadrature_formula.size();
889 *  
890 *   FullMatrix<double> local_nl_matrix(dofs_per_cell, dofs_per_cell);
891 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
892 *   std::vector<double> old_data_values(n_q_points);
893 *   std::vector<double> new_data_values(n_q_points);
894 *  
895 *   for (const auto &cell : dof_handler.active_cell_iterators())
896 *   {
897 *   local_nl_matrix = 0;
898 * @endcode
899 *
900 * Again, first we re-initialize our <code>FEValues</code>
901 * instantiation to the current cell.
902 *
903 * @code
904 *   fe_values.reinit(cell);
905 *   fe_values.get_function_values(old_data, old_data_values);
906 *   fe_values.get_function_values(new_data, new_data_values);
907 *  
908 * @endcode
909 *
910 * Then, we evaluate @f$\int_K \cos\left[\theta w_{\mathrm{new}} +
911 * (1-\theta) w_{\mathrm{old}}\right]\, \varphi_i\,
912 * \varphi_j\,\mathrm{d}x@f$ using the desired quadrature formula.
913 *
914 * @code
915 *   for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
916 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
917 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
918 *   local_nl_matrix(i, j) +=
919 *   (std::cos(theta * new_data_values[q_point] +
920 *   (1 - theta) * old_data_values[q_point]) *
921 *   fe_values.shape_value(i, q_point) *
922 *   fe_values.shape_value(j, q_point) * fe_values.JxW(q_point));
923 *  
924 * @endcode
925 *
926 * Finally, we add up the contributions of the integrals over the
927 * cells to the global integral.
928 *
929 * @code
930 *   cell->get_dof_indices(local_dof_indices);
931 *  
932 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
933 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
934 *   nl_matrix.add(local_dof_indices[i],
935 *   local_dof_indices[j],
936 *   local_nl_matrix(i, j));
937 *   }
938 *   }
939 *  
940 *  
941 *  
942 * @endcode
943 *
944 *
945 * <a name="SineGordonProblemsolve"></a>
946 * <h4>SineGordonProblem::solve</h4>
947 *
948
949 *
950 * As discussed in the Introduction, this function uses the CG iterative
951 * solver on the linear system of equations resulting from the finite
952 * element spatial discretization of each iteration of Newton's method for
953 * the (nonlinear) first equation of the split formulation. The solution to
954 * the system is, in fact, @f$\delta U^{n,l}@f$ so it is stored in
955 * <code>solution_update</code> and used to update <code>solution</code> in
956 * the <code>run</code> function.
957 *
958
959 *
960 * Note that we re-set the solution update to zero before solving for
961 * it. This is not necessary: iterative solvers can start from any point and
962 * converge to the correct solution. If one has a good estimate about the
963 * solution of a linear system, it may be worthwhile to start from that
964 * vector, but as a general observation it is a fact that the starting point
965 * doesn't matter very much: it has to be a very, very good guess to reduce
966 * the number of iterations by more than a few. It turns out that for this
967 * problem, using the previous nonlinear update as a starting point actually
968 * hurts convergence and increases the number of iterations needed, so we
969 * simply set it to zero.
970 *
971
972 *
973 * The function returns the number of iterations it took to converge to a
974 * solution. This number will later be used to generate output on the screen
975 * showing how many iterations were needed in each nonlinear iteration.
976 *
977 * @code
978 *   template <int dim>
979 *   unsigned int SineGordonProblem<dim>::solve()
980 *   {
981 *   SolverControl solver_control(1000, 1e-12 * system_rhs.l2_norm());
982 *   SolverCG<Vector<double>> cg(solver_control);
983 *  
984 *   PreconditionSSOR<SparseMatrix<double>> preconditioner;
985 *   preconditioner.initialize(system_matrix, 1.2);
986 *  
987 *   cg.solve(system_matrix, solution_update, system_rhs, preconditioner);
988 *  
989 *   return solver_control.last_step();
990 *   }
991 *  
992 * @endcode
993 *
994 *
995 * <a name="SineGordonProblemoutput_results"></a>
996 * <h4>SineGordonProblem::output_results</h4>
997 *
998
999 *
1000 * This function outputs the results to a file. It is pretty much identical
1001 * to the respective functions in @ref step_23 "step-23" and @ref step_24 "step-24":
1002 *
1003 * @code
1004 *   template <int dim>
1005 *   void SineGordonProblem<dim>::output_results(
1006 *   const unsigned int timestep_number) const
1007 *   {
1008 *   DataOut<dim> data_out;
1009 *  
1010 *   data_out.attach_dof_handler(dof_handler);
1011 *   data_out.add_data_vector(solution, "u");
1012 *   data_out.build_patches();
1013 *  
1014 *   const std::string filename =
1015 *   "solution-" + Utilities::int_to_string(timestep_number, 3) + ".vtu";
1016 *   DataOutBase::VtkFlags vtk_flags;
1018 *   data_out.set_flags(vtk_flags);
1019 *   std::ofstream output(filename);
1020 *   data_out.write_vtu(output);
1021 *   }
1022 *  
1023 * @endcode
1024 *
1025 *
1026 * <a name="SineGordonProblemrun"></a>
1027 * <h4>SineGordonProblem::run</h4>
1028 *
1029
1030 *
1031 * This function has the top-level control over everything: it runs the
1032 * (outer) time-stepping loop, the (inner) nonlinear-solver loop, and
1033 * outputs the solution after each time step.
1034 *
1035 * @code
1036 *   template <int dim>
1037 *   void SineGordonProblem<dim>::run()
1038 *   {
1039 *   make_grid_and_dofs();
1040 *  
1041 * @endcode
1042 *
1043 * To acknowledge the initial condition, we must use the function @f$u_0(x)@f$
1044 * to compute @f$U^0@f$. To this end, below we will create an object of type
1045 * <code>InitialValues</code>; note that when we create this object (which
1046 * is derived from the <code>Function</code> class), we set its internal
1047 * time variable to @f$t_0@f$, to indicate that the initial condition is a
1048 * function of space and time evaluated at @f$t=t_0@f$.
1049 *
1050
1051 *
1052 * Then we produce @f$U^0@f$ by projecting @f$u_0(x)@f$ onto the grid using
1053 * <code>VectorTools::project</code>. We have to use the same construct
1054 * using hanging node constraints as in @ref step_21 "step-21": the VectorTools::project
1055 * function requires a hanging node constraints object, but to be used we
1056 * first need to close it:
1057 *
1058 * @code
1059 *   {
1060 *   AffineConstraints<double> constraints;
1061 *   constraints.close();
1062 *   VectorTools::project(dof_handler,
1063 *   constraints,
1064 *   QGauss<dim>(fe.degree + 1),
1065 *   InitialValues<dim>(1, time),
1066 *   solution);
1067 *   }
1068 *  
1069 * @endcode
1070 *
1071 * For completeness, we output the zeroth time step to a file just like
1072 * any other time step.
1073 *
1074 * @code
1075 *   output_results(0);
1076 *  
1077 * @endcode
1078 *
1079 * Now we perform the time stepping: at every time step we solve the
1080 * matrix equation(s) corresponding to the finite element discretization
1081 * of the problem, and then advance our solution according to the time
1082 * stepping formulas we discussed in the Introduction.
1083 *
1084 * @code
1085 *   unsigned int timestep_number = 1;
1086 *   for (time += time_step; time <= final_time;
1087 *   time += time_step, ++timestep_number)
1088 *   {
1089 *   old_solution = solution;
1090 *  
1091 *   std::cout << std::endl
1092 *   << "Time step #" << timestep_number << "; "
1093 *   << "advancing to t = " << time << '.' << std::endl;
1094 *  
1095 * @endcode
1096 *
1097 * At the beginning of each time step we must solve the nonlinear
1098 * equation in the split formulation via Newton's method ---
1099 * i.e. solve for @f$\delta U^{n,l}@f$ then compute @f$U^{n,l+1}@f$ and so
1100 * on. The stopping criterion for this nonlinear iteration is that
1101 * @f$\|F_h(U^{n,l})\|_2 \le 10^{-6} \|F_h(U^{n,0})\|_2@f$. Consequently,
1102 * we need to record the norm of the residual in the first iteration.
1103 *
1104
1105 *
1106 * At the end of each iteration, we output to the console how many
1107 * linear solver iterations it took us. When the loop below is done,
1108 * we have (an approximation of) @f$U^n@f$.
1109 *
1110 * @code
1111 *   double initial_rhs_norm = 0.;
1112 *   bool first_iteration = true;
1113 *   do
1114 *   {
1115 *   assemble_system();
1116 *  
1117 *   if (first_iteration == true)
1118 *   initial_rhs_norm = system_rhs.l2_norm();
1119 *  
1120 *   const unsigned int n_iterations = solve();
1121 *  
1122 *   solution += solution_update;
1123 *  
1124 *   if (first_iteration == true)
1125 *   std::cout << " " << n_iterations;
1126 *   else
1127 *   std::cout << '+' << n_iterations;
1128 *   first_iteration = false;
1129 *   }
1130 *   while (system_rhs.l2_norm() > 1e-6 * initial_rhs_norm);
1131 *  
1132 *   std::cout << " CG iterations per nonlinear step." << std::endl;
1133 *  
1134 * @endcode
1135 *
1136 * Upon obtaining the solution to the first equation of the problem at
1137 * @f$t=t_n@f$, we must update the auxiliary velocity variable
1138 * @f$V^n@f$. However, we do not compute and store @f$V^n@f$ since it is not a
1139 * quantity we use directly in the problem. Hence, for simplicity, we
1140 * update @f$MV^n@f$ directly:
1141 *
1142 * @code
1143 *   Vector<double> tmp_vector(solution.size());
1144 *   laplace_matrix.vmult(tmp_vector, solution);
1145 *   M_x_velocity.add(-time_step * theta, tmp_vector);
1146 *  
1147 *   laplace_matrix.vmult(tmp_vector, old_solution);
1148 *   M_x_velocity.add(-time_step * (1 - theta), tmp_vector);
1149 *  
1150 *   compute_nl_term(old_solution, solution, tmp_vector);
1151 *   M_x_velocity.add(-time_step, tmp_vector);
1152 *  
1153 * @endcode
1154 *
1155 * Oftentimes, in particular for fine meshes, we must pick the time
1156 * step to be quite small in order for the scheme to be
1157 * stable. Therefore, there are a lot of time steps during which
1158 * "nothing interesting happens" in the solution. To improve overall
1159 * efficiency -- in particular, speed up the program and save disk
1160 * space -- we only output the solution every
1161 * <code>output_timestep_skip</code> time steps:
1162 *
1163 * @code
1164 *   if (timestep_number % output_timestep_skip == 0)
1165 *   output_results(timestep_number);
1166 *   }
1167 *   }
1168 *   } // namespace Step25
1169 *  
1170 * @endcode
1171 *
1172 *
1173 * <a name="Thecodemaincodefunction"></a>
1174 * <h3>The <code>main</code> function</h3>
1175 *
1176
1177 *
1178 * This is the main function of the program. It creates an object of top-level
1179 * class and calls its principal function. If exceptions are thrown during the
1180 * execution of the run method of the <code>SineGordonProblem</code> class, we
1181 * catch and report them here. For more information about exceptions the
1182 * reader should consult @ref step_6 "step-6".
1183 *
1184 * @code
1185 *   int main()
1186 *   {
1187 *   try
1188 *   {
1189 *   using namespace Step25;
1190 *  
1191 *   SineGordonProblem<1> sg_problem;
1192 *   sg_problem.run();
1193 *   }
1194 *   catch (std::exception &exc)
1195 *   {
1196 *   std::cerr << std::endl
1197 *   << std::endl
1198 *   << "----------------------------------------------------"
1199 *   << std::endl;
1200 *   std::cerr << "Exception on processing: " << std::endl
1201 *   << exc.what() << std::endl
1202 *   << "Aborting!" << std::endl
1203 *   << "----------------------------------------------------"
1204 *   << std::endl;
1205 *  
1206 *   return 1;
1207 *   }
1208 *   catch (...)
1209 *   {
1210 *   std::cerr << std::endl
1211 *   << std::endl
1212 *   << "----------------------------------------------------"
1213 *   << std::endl;
1214 *   std::cerr << "Unknown exception!" << std::endl
1215 *   << "Aborting!" << std::endl
1216 *   << "----------------------------------------------------"
1217 *   << std::endl;
1218 *   return 1;
1219 *   }
1220 *  
1221 *   return 0;
1222 *   }
1223 * @endcode
1224<a name="Results"></a><h1>Results</h1>
1225
1226The explicit Euler time stepping scheme (@f$\theta=0@f$) performs adequately for the problems we wish to solve. Unfortunately, a rather small time step has to be chosen due to stability issues --- @f$k\sim h/10@f$ appears to work for most the simulations we performed. On the other hand, the Crank-Nicolson scheme (@f$\theta=\frac{1}{2}@f$) is unconditionally stable, and (at least for the case of the 1D breather) we can pick the time step to be as large as @f$25h@f$ without any ill effects on the solution. The implicit Euler scheme (@f$\theta=1@f$) is "exponentially damped," so it is not a good choice for solving the sine-Gordon equation, which is conservative. However, some of the damped schemes in the continuum that is offered by the @f$\theta@f$-method were useful for eliminating spurious oscillations due to boundary effects.
1227
1228In the simulations below, we solve the sine-Gordon equation on the interval @f$\Omega =
1229[-10,10]@f$ in 1D and on the square @f$\Omega = [-10,10]\times [-10,10]@f$ in 2D. In
1230each case, the respective grid is refined uniformly 6 times, i.e. @f$h\sim
12312^{-6}@f$.
1232
1233<a name="An11dSolution"></a><h3>An (1+1)-d Solution</h3>
1234
1235The first example we discuss is the so-called 1D (stationary) breather
1236solution of the sine-Gordon equation. The breather has the following
1237closed-form expression, as mentioned in the Introduction:
1238\f[
1239u_{\mathrm{breather}}(x,t) = -4\arctan \left(\frac{m}{\sqrt{1-m^2}} \frac{\sin\left(\sqrt{1-m^2}t +c_2\right)}{\cosh(mx+c_1)} \right),
1240\f]
1241where @f$c_1@f$, @f$c_2@f$ and @f$m<1@f$ are constants. In the simulation below, we have chosen @f$c_1=0@f$, @f$c_2=0@f$, @f$m=0.5@f$. Moreover, it is know that the period of oscillation of the breather is @f$2\pi\sqrt{1-m^2}@f$, hence we have chosen @f$t_0=-5.4414@f$ and @f$t_f=2.7207@f$ so that we can observe three oscillations of the solution. Then, taking @f$u_0(x) = u_{\mathrm{breather}}(x,t_0)@f$, @f$\theta=0@f$ and @f$k=h/10@f$, the program computed the following solution.
1242
1243<img src="https://www.dealii.org/images/steps/developer/step-25.1d-breather.gif" alt="Animation of the 1D stationary breather.">
1244
1245Though not shown how to do this in the program, another way to visualize the
1246(1+1)-d solution is to use output generated by the DataOutStack class; it
1247allows to "stack" the solutions of individual time steps, so that we get
12482D space-time graphs from 1D time-dependent
1249solutions. This produces the space-time plot below instead of the animation
1250above.
1251
1252<img src="https://www.dealii.org/images/steps/developer/step-25.1d-breather_stp.png" alt="A space-time plot of the 1D stationary breather.">
1253
1254Furthermore, since the breather is an analytical solution of the sine-Gordon
1255equation, we can use it to validate our code, although we have to assume that
1256the error introduced by our choice of Neumann boundary conditions is small
1257compared to the numerical error. Under this assumption, one could use the
1258VectorTools::integrate_difference function to compute the difference between
1259the numerical solution and the function described by the
1260<code>ExactSolution</code> class of this program. For the
1261simulation shown in the two images above, the @f$L^2@f$ norm of the error in the
1262finite element solution at each time step remained on the order of
1263@f$10^{-2}@f$. Hence, we can conclude that the numerical method has been
1264implemented correctly in the program.
1265
1266
1267<a name="Afew21DSolutions"></a><h3>A few (2+1)D Solutions</h3>
1268
1269
1270The only analytical solution to the sine-Gordon equation in (2+1)D that can be found in the literature is the so-called kink solitary wave. It has the following closed-form expression:
1271 @f[
1272 u(x,y,t) = 4 \arctan \left[a_0 e^{s\xi}\right]
1273 @f]
1274with
1275 @f[
1276 \xi = x \cos\vartheta + \sin(\vartheta) (y\cosh\lambda + t\sinh \lambda)
1277 @f]
1278where @f$a_0@f$, @f$\vartheta@f$ and @f$\lambda@f$ are constants. In the simulation below
1279we have chosen @f$a_0=\lambda=1@f$. Notice that if @f$\vartheta=\pi@f$ the kink is
1280stationary, hence it would make a good solution against which we can
1281validate the program in 2D because no reflections off the boundary of the
1282domain occur.
1283
1284The simulation shown below was performed with @f$u_0(x) = u_{\mathrm{kink}}(x,t_0)@f$, @f$\theta=\frac{1}{2}@f$, @f$k=20h@f$, @f$t_0=1@f$ and @f$t_f=500@f$. The @f$L^2@f$ norm of the error of the finite element solution at each time step remained on the order of @f$10^{-2}@f$, showing that the program is working correctly in 2D, as well as 1D. Unfortunately, the solution is not very interesting, nonetheless we have included a snapshot of it below for completeness.
1285
1286<img src="https://www.dealii.org/images/steps/developer/step-25.2d-kink.png" alt="Stationary 2D kink.">
1287
1288Now that we have validated the code in 1D and 2D, we move to a problem where the analytical solution is unknown.
1289
1290To this end, we rotate the kink solution discussed above about the @f$z@f$
1291axis: we let @f$\vartheta=\frac{\pi}{4}@f$. The latter results in a
1292solitary wave that is not aligned with the grid, so reflections occur
1293at the boundaries of the domain immediately. For the simulation shown
1294below, we have taken @f$u_0(x)=u_{\mathrm{kink}}(x,t_0)@f$,
1295@f$\theta=\frac{2}{3}@f$, @f$k=20h@f$, @f$t_0=0@f$ and @f$t_f=20@f$. Moreover, we had
1296to pick @f$\theta=\frac{2}{3}@f$ because for any @f$\theta\le\frac{1}{2}@f$
1297oscillations arose at the boundary, which are likely due to the scheme
1298and not the equation, thus picking a value of @f$\theta@f$ a good bit into
1299the "exponentially damped" spectrum of the time stepping schemes
1300assures these oscillations are not created.
1301
1302<img src="https://www.dealii.org/images/steps/developer/step-25.2d-angled_kink.gif" alt="Animation of a moving 2D kink, at 45 degrees to the axes of the grid, showing boundary effects.">
1303
1304Another interesting solution to the sine-Gordon equation (which cannot be
1305obtained analytically) can be produced by using two 1D breathers to construct
1306the following separable 2D initial condition:
1307\f[
1308 u_0(x) =
1309 u_{\mathrm{pseudobreather}}(x,t_0) =
1310 16\arctan \left(
1311 \frac{m}{\sqrt{1-m^2}}
1312 \frac{\sin\left(\sqrt{1-m^2}t_0\right)}{\cosh(mx_1)} \right)
1313 \arctan \left(
1314 \frac{m}{\sqrt{1-m^2}}
1315 \frac{\sin\left(\sqrt{1-m^2}t_0\right)}{\cosh(mx_2)} \right),
1316\f]
1317where @f$x=(x_1,x_2)\in{R}^2@f$, @f$m=0.5<1@f$ as in the 1D case we discussed
1318above. For the simulation shown below, we have chosen @f$\theta=\frac{1}{2}@f$,
1319@f$k=10h@f$, @f$t_0=-5.4414@f$ and @f$t_f=2.7207@f$. The solution is pretty interesting
1320--- it acts like a breather (as far as the pictures are concerned); however,
1321it appears to break up and reassemble, rather than just oscillate.
1322
1323<img src="https://www.dealii.org/images/steps/developer/step-25.2d-pseudobreather.gif" alt="Animation of a 2D pseudobreather.">
1324
1325
1326<a name="extensions"></a>
1327<a name="Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
1328
1329
1330It is instructive to change the initial conditions. Most choices will not lead
1331to solutions that stay localized (in the soliton community, such
1332solutions are called "stationary", though the solution does change
1333with time), but lead to solutions where the wave-like
1334character of the equation dominates and a wave travels away from the location
1335of a localized initial condition. For example, it is worth playing around with
1336the <code>InitialValues</code> class, by replacing the call to the
1337<code>ExactSolution</code> class by something like this function:
1338@f[
1339 u_0(x,y) = \cos\left(\frac x2\right)\cos\left(\frac y2\right)
1340@f]
1341if @f$|x|,|y|\le \frac\pi 2@f$, and @f$u_0(x,y)=0@f$ outside this region.
1342
1343A second area would be to investigate whether the scheme is
1344energy-preserving. For the pure wave equation, discussed in @ref
1345step_23 "step-23", this is the case if we choose the time stepping
1346parameter such that we get the Crank-Nicolson scheme. One could do a
1347similar thing here, noting that the energy in the sine-Gordon solution
1348is defined as
1349@f[
1350 E(t) = \frac 12 \int_\Omega \left(\frac{\partial u}{\partial
1351 t}\right)^2
1352 + \left(\nabla u\right)^2 + 2 (1-\cos u) \; dx.
1353@f]
1354(We use @f$1-\cos u@f$ instead of @f$-\cos u@f$ in the formula to ensure that all
1355contributions to the energy are positive, and so that decaying solutions have
1356finite energy on unbounded domains.)
1357
1358Beyond this, there are two obvious areas:
1359
1360- Clearly, adaptivity (i.e. time-adaptive grids) would be of interest
1361 to problems like these. Their complexity leads us to leave this out
1362 of this program again, though the general comments in the
1363 introduction of @ref step_23 "step-23" remain true.
1364
1365- Faster schemes to solve this problem. While computers today are
1366 plenty fast enough to solve 2d and, frequently, even 3d stationary
1367 problems within not too much time, time dependent problems present
1368 an entirely different class of problems. We address this topic in
1369 @ref step_48 "step-48" where we show how to solve this problem in parallel and
1370 without assembling or inverting any matrix at all.
1371 *
1372 *
1373<a name="PlainProg"></a>
1374<h1> The plain program</h1>
1375@include "step-25.cc"
1376*/
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
void reinit(const TriaIterator< DoFCellAccessor< dim, spacedim, level_dof_access > > &cell)
void initialize(const MatrixType &A, const AdditionalData &parameters=AdditionalData())
Point< 2 > second
Definition grid_out.cc:4616
Point< 2 > first
Definition grid_out.cc:4615
unsigned int level
Definition grid_out.cc:4618
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
void loop(ITERATOR begin, std_cxx20::type_identity_t< ITERATOR > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(DOFINFO &, DOFINFO &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, ASSEMBLER &assembler, const LoopControl &lctrl=LoopControl())
Definition loop.h:439
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints=AffineConstraints< number >(), const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
@ update_values
Shape function values.
@ update_JxW_values
Transformed quadrature weights.
@ update_quadrature_points
Transformed quadrature points.
std::vector< value_type > split(const typename ::Triangulation< dim, spacedim >::cell_iterator &parent, const value_type parent_value)
const Event initial
Definition event.cc:65
void hyper_cube(Triangulation< dim, spacedim > &tria, const double left=0., const double right=1., const bool colorize=false)
@ matrix
Contents is actually a matrix.
void mass_matrix(FullMatrix< double > &M, const FEValuesBase< dim > &fe, const double factor=1.)
Definition l2.h:58
void create_mass_matrix(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim > &q, SparseMatrixType &matrix, const Function< spacedim, typename SparseMatrixType::value_type > *const a=nullptr, const AffineConstraints< typename SparseMatrixType::value_type > &constraints=AffineConstraints< typename SparseMatrixType::value_type >())
void create_laplace_matrix(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim > &q, SparseMatrixType &matrix, const Function< spacedim, typename SparseMatrixType::value_type > *const a=nullptr, const AffineConstraints< typename SparseMatrixType::value_type > &constraints=AffineConstraints< typename SparseMatrixType::value_type >())
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:189
Tensor< 2, dim, Number > l(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
VectorType::value_type * end(VectorType &V)
T reduce(const T &local_value, const MPI_Comm comm, const std::function< T(const T &, const T &)> &combiner, const unsigned int root_process=0)
std::string get_time()
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
Definition utilities.cc:471
void project(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const AffineConstraints< typename VectorType::value_type > &constraints, const Quadrature< dim > &quadrature, const Function< spacedim, typename VectorType::value_type > &function, VectorType &vec, const bool enforce_zero_boundary=false, const Quadrature< dim - 1 > &q_boundary=(dim > 1 ? QGauss< dim - 1 >(2) :Quadrature< dim - 1 >(0)), const bool project_to_boundary_first=false)
int(&) functions(const void *v1, const void *v2)
static constexpr double PI
Definition numbers.h:259
::VectorizedArray< Number, width > exp(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > cos(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DataOutBase::CompressionLevel compression_level
void advance(std::tuple< I1, I2 > &t, const unsigned int n)