Reference documentation for deal.II version 9.6.0
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step-25.h
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1
511 = 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:
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="step_25-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="step_25-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="step_25-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="step_25-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(Utilities::fixed_power<2>(time_step * theta),
730 *   laplace_matrix);
731 *  
732 *   SparseMatrix<double> tmp_matrix(sparsity_pattern);
733 *   compute_nl_matrix(old_solution, solution, tmp_matrix);
734 *   system_matrix.add(Utilities::fixed_power<2>(time_step * theta), tmp_matrix);
735 *  
736 * @endcode
737 *
738 * Next we compute the right-hand side vector. This is just the
739 * combination of matrix-vector products implied by the description of
740 * @f$-F_h(U^{n,l})@f$ in the introduction.
741 *
742 * @code
743 *   system_rhs = 0.;
744 *  
745 *   Vector<double> tmp_vector(solution.size());
746 *  
747 *   mass_matrix.vmult(system_rhs, solution);
748 *   laplace_matrix.vmult(tmp_vector, solution);
749 *   system_rhs.add(Utilities::fixed_power<2>(time_step * theta), tmp_vector);
750 *  
751 *   mass_matrix.vmult(tmp_vector, old_solution);
752 *   system_rhs.add(-1.0, tmp_vector);
753 *   laplace_matrix.vmult(tmp_vector, old_solution);
754 *   system_rhs.add(Utilities::fixed_power<2>(time_step) * theta * (1 - theta),
755 *   tmp_vector);
756 *  
757 *   system_rhs.add(-time_step, M_x_velocity);
758 *  
759 *   compute_nl_term(old_solution, solution, tmp_vector);
760 *   system_rhs.add(Utilities::fixed_power<2>(time_step) * theta, tmp_vector);
761 *  
762 *   system_rhs *= -1.;
763 *   }
764 *  
765 * @endcode
766 *
767 *
768 * <a name="step_25-SineGordonProblemcompute_nl_term"></a>
769 * <h4>SineGordonProblem::compute_nl_term</h4>
770 *
771
772 *
773 * This function computes the vector @f$S(\cdot,\cdot)@f$, which appears in the
774 * nonlinear term in both equations of the split formulation. This
775 * function not only simplifies the repeated computation of this term, but
776 * it is also a fundamental part of the nonlinear iterative solver that we
777 * use when the time stepping is implicit (i.e. @f$\theta\ne 0@f$). Moreover, we
778 * must allow the function to receive as input an "old" and a "new"
779 * solution. These may not be the actual solutions of the problem stored in
780 * <code>old_solution</code> and <code>solution</code>, but are simply the
781 * two functions we linearize about. For the purposes of this function, let
782 * us call the first two arguments @f$w_{\mathrm{old}}@f$ and @f$w_{\mathrm{new}}@f$
783 * in the documentation of this class below, respectively.
784 *
785
786 *
787 * As a side-note, it is perhaps worth investigating what order quadrature
788 * formula is best suited for this type of integration. Since @f$\sin(\cdot)@f$
789 * is not a polynomial, there are probably no quadrature formulas that can
790 * integrate these terms exactly. It is usually sufficient to just make sure
791 * that the right hand side is integrated up to the same order of accuracy
792 * as the discretization scheme is, but it may be possible to improve on the
793 * constant in the asymptotic statement of convergence by choosing a more
794 * accurate quadrature formula.
795 *
796 * @code
797 *   template <int dim>
798 *   void SineGordonProblem<dim>::compute_nl_term(const Vector<double> &old_data,
799 *   const Vector<double> &new_data,
800 *   Vector<double> &nl_term) const
801 *   {
802 *   nl_term = 0;
803 *   const QGauss<dim> quadrature_formula(fe.degree + 1);
804 *   FEValues<dim> fe_values(fe,
805 *   quadrature_formula,
808 *  
809 *   const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
810 *   const unsigned int n_q_points = quadrature_formula.size();
811 *  
812 *   Vector<double> local_nl_term(dofs_per_cell);
813 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
814 *   std::vector<double> old_data_values(n_q_points);
815 *   std::vector<double> new_data_values(n_q_points);
816 *  
817 *   for (const auto &cell : dof_handler.active_cell_iterators())
818 *   {
819 *   local_nl_term = 0;
820 * @endcode
821 *
822 * Once we re-initialize our <code>FEValues</code> instantiation to
823 * the current cell, we make use of the
824 * <code>get_function_values</code> routine to get the values of the
825 * "old" data (presumably at @f$t=t_{n-1}@f$) and the "new" data
826 * (presumably at @f$t=t_n@f$) at the nodes of the chosen quadrature
827 * formula.
828 *
829 * @code
830 *   fe_values.reinit(cell);
831 *   fe_values.get_function_values(old_data, old_data_values);
832 *   fe_values.get_function_values(new_data, new_data_values);
833 *  
834 * @endcode
835 *
836 * Now, we can evaluate @f$\int_K \sin\left[\theta w_{\mathrm{new}} +
837 * (1-\theta) w_{\mathrm{old}}\right] \,\varphi_j\,\mathrm{d}x@f$ using
838 * the desired quadrature formula.
839 *
840 * @code
841 *   for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
842 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
843 *   local_nl_term(i) +=
844 *   (std::sin(theta * new_data_values[q_point] +
845 *   (1 - theta) * old_data_values[q_point]) *
846 *   fe_values.shape_value(i, q_point) * fe_values.JxW(q_point));
847 *  
848 * @endcode
849 *
850 * We conclude by adding up the contributions of the integrals over
851 * the cells to the global integral.
852 *
853 * @code
854 *   cell->get_dof_indices(local_dof_indices);
855 *  
856 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
857 *   nl_term(local_dof_indices[i]) += local_nl_term(i);
858 *   }
859 *   }
860 *  
861 * @endcode
862 *
863 *
864 * <a name="step_25-SineGordonProblemcompute_nl_matrix"></a>
865 * <h4>SineGordonProblem::compute_nl_matrix</h4>
866 *
867
868 *
869 * This is the second function dealing with the nonlinear scheme. It
870 * computes the matrix @f$N(\cdot,\cdot)@f$, which appears in the nonlinear
871 * term in the Jacobian of @f$F(\cdot)@f$. Just as <code>compute_nl_term</code>,
872 * we must allow this function to receive as input an "old" and a "new"
873 * solution, which we again call @f$w_{\mathrm{old}}@f$ and @f$w_{\mathrm{new}}@f$
874 * below, respectively.
875 *
876 * @code
877 *   template <int dim>
878 *   void SineGordonProblem<dim>::compute_nl_matrix(
879 *   const Vector<double> &old_data,
880 *   const Vector<double> &new_data,
881 *   SparseMatrix<double> &nl_matrix) const
882 *   {
883 *   const QGauss<dim> quadrature_formula(fe.degree + 1);
884 *   FEValues<dim> fe_values(fe,
885 *   quadrature_formula,
888 *  
889 *   const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
890 *   const unsigned int n_q_points = quadrature_formula.size();
891 *  
892 *   FullMatrix<double> local_nl_matrix(dofs_per_cell, dofs_per_cell);
893 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
894 *   std::vector<double> old_data_values(n_q_points);
895 *   std::vector<double> new_data_values(n_q_points);
896 *  
897 *   for (const auto &cell : dof_handler.active_cell_iterators())
898 *   {
899 *   local_nl_matrix = 0;
900 * @endcode
901 *
902 * Again, first we re-initialize our <code>FEValues</code>
903 * instantiation to the current cell.
904 *
905 * @code
906 *   fe_values.reinit(cell);
907 *   fe_values.get_function_values(old_data, old_data_values);
908 *   fe_values.get_function_values(new_data, new_data_values);
909 *  
910 * @endcode
911 *
912 * Then, we evaluate @f$\int_K \cos\left[\theta w_{\mathrm{new}} +
913 * (1-\theta) w_{\mathrm{old}}\right]\, \varphi_i\,
914 * \varphi_j\,\mathrm{d}x@f$ using the desired quadrature formula.
915 *
916 * @code
917 *   for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
918 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
919 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
920 *   local_nl_matrix(i, j) +=
921 *   (std::cos(theta * new_data_values[q_point] +
922 *   (1 - theta) * old_data_values[q_point]) *
923 *   fe_values.shape_value(i, q_point) *
924 *   fe_values.shape_value(j, q_point) * fe_values.JxW(q_point));
925 *  
926 * @endcode
927 *
928 * Finally, we add up the contributions of the integrals over the
929 * cells to the global integral.
930 *
931 * @code
932 *   cell->get_dof_indices(local_dof_indices);
933 *  
934 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
935 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
936 *   nl_matrix.add(local_dof_indices[i],
937 *   local_dof_indices[j],
938 *   local_nl_matrix(i, j));
939 *   }
940 *   }
941 *  
942 *  
943 *  
944 * @endcode
945 *
946 *
947 * <a name="step_25-SineGordonProblemsolve"></a>
948 * <h4>SineGordonProblem::solve</h4>
949 *
950
951 *
952 * As discussed in the Introduction, this function uses the CG iterative
953 * solver on the linear system of equations resulting from the finite
954 * element spatial discretization of each iteration of Newton's method for
955 * the (nonlinear) first equation of the split formulation. The solution to
956 * the system is, in fact, @f$\delta U^{n,l}@f$ so it is stored in
957 * <code>solution_update</code> and used to update <code>solution</code> in
958 * the <code>run</code> function.
959 *
960
961 *
962 * Note that we re-set the solution update to zero before solving for
963 * it. This is not necessary: iterative solvers can start from any point and
964 * converge to the correct solution. If one has a good estimate about the
965 * solution of a linear system, it may be worthwhile to start from that
966 * vector, but as a general observation it is a fact that the starting point
967 * doesn't matter very much: it has to be a very, very good guess to reduce
968 * the number of iterations by more than a few. It turns out that for this
969 * problem, using the previous nonlinear update as a starting point actually
970 * hurts convergence and increases the number of iterations needed, so we
971 * simply set it to zero.
972 *
973
974 *
975 * The function returns the number of iterations it took to converge to a
976 * solution. This number will later be used to generate output on the screen
977 * showing how many iterations were needed in each nonlinear iteration.
978 *
979 * @code
980 *   template <int dim>
981 *   unsigned int SineGordonProblem<dim>::solve()
982 *   {
983 *   SolverControl solver_control(1000, 1e-12 * system_rhs.l2_norm());
984 *   SolverCG<Vector<double>> cg(solver_control);
985 *  
986 *   PreconditionSSOR<SparseMatrix<double>> preconditioner;
987 *   preconditioner.initialize(system_matrix, 1.2);
988 *  
989 *   cg.solve(system_matrix, solution_update, system_rhs, preconditioner);
990 *  
991 *   return solver_control.last_step();
992 *   }
993 *  
994 * @endcode
995 *
996 *
997 * <a name="step_25-SineGordonProblemoutput_results"></a>
998 * <h4>SineGordonProblem::output_results</h4>
999 *
1000
1001 *
1002 * This function outputs the results to a file. It is pretty much identical
1003 * to the respective functions in @ref step_23 "step-23" and @ref step_24 "step-24":
1004 *
1005 * @code
1006 *   template <int dim>
1007 *   void SineGordonProblem<dim>::output_results(
1008 *   const unsigned int timestep_number) const
1009 *   {
1010 *   DataOut<dim> data_out;
1011 *  
1012 *   data_out.attach_dof_handler(dof_handler);
1013 *   data_out.add_data_vector(solution, "u");
1014 *   data_out.build_patches();
1015 *  
1016 *   const std::string filename =
1017 *   "solution-" + Utilities::int_to_string(timestep_number, 3) + ".vtu";
1018 *   DataOutBase::VtkFlags vtk_flags;
1020 *   data_out.set_flags(vtk_flags);
1021 *   std::ofstream output(filename);
1022 *   data_out.write_vtu(output);
1023 *   }
1024 *  
1025 * @endcode
1026 *
1027 *
1028 * <a name="step_25-SineGordonProblemrun"></a>
1029 * <h4>SineGordonProblem::run</h4>
1030 *
1031
1032 *
1033 * This function has the top-level control over everything: it runs the
1034 * (outer) time-stepping loop, the (inner) nonlinear-solver loop, and
1035 * outputs the solution after each time step.
1036 *
1037 * @code
1038 *   template <int dim>
1039 *   void SineGordonProblem<dim>::run()
1040 *   {
1041 *   make_grid_and_dofs();
1042 *  
1043 * @endcode
1044 *
1045 * To acknowledge the initial condition, we must use the function @f$u_0(x)@f$
1046 * to compute @f$U^0@f$. To this end, below we will create an object of type
1047 * <code>InitialValues</code>; note that when we create this object (which
1048 * is derived from the <code>Function</code> class), we set its internal
1049 * time variable to @f$t_0@f$, to indicate that the initial condition is a
1050 * function of space and time evaluated at @f$t=t_0@f$.
1051 *
1052
1053 *
1054 * Then we produce @f$U^0@f$ by projecting @f$u_0(x)@f$ onto the grid using
1055 * <code>VectorTools::project</code>. We have to use the same construct
1056 * using hanging node constraints as in @ref step_21 "step-21": the VectorTools::project
1057 * function requires a hanging node constraints object, but to be used we
1058 * first need to close it:
1059 *
1060 * @code
1061 *   {
1062 *   AffineConstraints<double> constraints;
1063 *   constraints.close();
1064 *   VectorTools::project(dof_handler,
1065 *   constraints,
1066 *   QGauss<dim>(fe.degree + 1),
1067 *   InitialValues<dim>(1, time),
1068 *   solution);
1069 *   }
1070 *  
1071 * @endcode
1072 *
1073 * For completeness, we output the zeroth time step to a file just like
1074 * any other time step.
1075 *
1076 * @code
1077 *   output_results(0);
1078 *  
1079 * @endcode
1080 *
1081 * Now we perform the time stepping: at every time step we solve the
1082 * matrix equation(s) corresponding to the finite element discretization
1083 * of the problem, and then advance our solution according to the time
1084 * stepping formulas we discussed in the Introduction.
1085 *
1086 * @code
1087 *   unsigned int timestep_number = 1;
1088 *   for (time += time_step; time <= final_time;
1089 *   time += time_step, ++timestep_number)
1090 *   {
1091 *   old_solution = solution;
1092 *  
1093 *   std::cout << std::endl
1094 *   << "Time step #" << timestep_number << "; "
1095 *   << "advancing to t = " << time << '.' << std::endl;
1096 *  
1097 * @endcode
1098 *
1099 * At the beginning of each time step we must solve the nonlinear
1100 * equation in the split formulation via Newton's method ---
1101 * i.e. solve for @f$\delta U^{n,l}@f$ then compute @f$U^{n,l+1}@f$ and so
1102 * on. The stopping criterion for this nonlinear iteration is that
1103 * @f$\|F_h(U^{n,l})\|_2 \le 10^{-6} \|F_h(U^{n,0})\|_2@f$. Consequently,
1104 * we need to record the norm of the residual in the first iteration.
1105 *
1106
1107 *
1108 * At the end of each iteration, we output to the console how many
1109 * linear solver iterations it took us. When the loop below is done,
1110 * we have (an approximation of) @f$U^n@f$.
1111 *
1112 * @code
1113 *   double initial_rhs_norm = 0.;
1114 *   bool first_iteration = true;
1115 *   do
1116 *   {
1117 *   assemble_system();
1118 *  
1119 *   if (first_iteration == true)
1120 *   initial_rhs_norm = system_rhs.l2_norm();
1121 *  
1122 *   const unsigned int n_iterations = solve();
1123 *  
1124 *   solution += solution_update;
1125 *  
1126 *   if (first_iteration == true)
1127 *   std::cout << " " << n_iterations;
1128 *   else
1129 *   std::cout << '+' << n_iterations;
1130 *   first_iteration = false;
1131 *   }
1132 *   while (system_rhs.l2_norm() > 1e-6 * initial_rhs_norm);
1133 *  
1134 *   std::cout << " CG iterations per nonlinear step." << std::endl;
1135 *  
1136 * @endcode
1137 *
1138 * Upon obtaining the solution to the first equation of the problem at
1139 * @f$t=t_n@f$, we must update the auxiliary velocity variable
1140 * @f$V^n@f$. However, we do not compute and store @f$V^n@f$ since it is not a
1141 * quantity we use directly in the problem. Hence, for simplicity, we
1142 * update @f$MV^n@f$ directly:
1143 *
1144 * @code
1145 *   Vector<double> tmp_vector(solution.size());
1146 *   laplace_matrix.vmult(tmp_vector, solution);
1147 *   M_x_velocity.add(-time_step * theta, tmp_vector);
1148 *  
1149 *   laplace_matrix.vmult(tmp_vector, old_solution);
1150 *   M_x_velocity.add(-time_step * (1 - theta), tmp_vector);
1151 *  
1152 *   compute_nl_term(old_solution, solution, tmp_vector);
1153 *   M_x_velocity.add(-time_step, tmp_vector);
1154 *  
1155 * @endcode
1156 *
1157 * Oftentimes, in particular for fine meshes, we must pick the time
1158 * step to be quite small in order for the scheme to be
1159 * stable. Therefore, there are a lot of time steps during which
1160 * "nothing interesting happens" in the solution. To improve overall
1161 * efficiency -- in particular, speed up the program and save disk
1162 * space -- we only output the solution every
1163 * <code>output_timestep_skip</code> time steps:
1164 *
1165 * @code
1166 *   if (timestep_number % output_timestep_skip == 0)
1167 *   output_results(timestep_number);
1168 *   }
1169 *   }
1170 *   } // namespace Step25
1171 *  
1172 * @endcode
1173 *
1174 *
1175 * <a name="step_25-Thecodemaincodefunction"></a>
1176 * <h3>The <code>main</code> function</h3>
1177 *
1178
1179 *
1180 * This is the main function of the program. It creates an object of top-level
1181 * class and calls its principal function. If exceptions are thrown during the
1182 * execution of the run method of the <code>SineGordonProblem</code> class, we
1183 * catch and report them here. For more information about exceptions the
1184 * reader should consult @ref step_6 "step-6".
1185 *
1186 * @code
1187 *   int main()
1188 *   {
1189 *   try
1190 *   {
1191 *   using namespace Step25;
1192 *  
1193 *   SineGordonProblem<1> sg_problem;
1194 *   sg_problem.run();
1195 *   }
1196 *   catch (std::exception &exc)
1197 *   {
1198 *   std::cerr << std::endl
1199 *   << std::endl
1200 *   << "----------------------------------------------------"
1201 *   << std::endl;
1202 *   std::cerr << "Exception on processing: " << std::endl
1203 *   << exc.what() << std::endl
1204 *   << "Aborting!" << std::endl
1205 *   << "----------------------------------------------------"
1206 *   << std::endl;
1207 *  
1208 *   return 1;
1209 *   }
1210 *   catch (...)
1211 *   {
1212 *   std::cerr << std::endl
1213 *   << std::endl
1214 *   << "----------------------------------------------------"
1215 *   << std::endl;
1216 *   std::cerr << "Unknown exception!" << std::endl
1217 *   << "Aborting!" << std::endl
1218 *   << "----------------------------------------------------"
1219 *   << std::endl;
1220 *   return 1;
1221 *   }
1222 *  
1223 *   return 0;
1224 *   }
1225 * @endcode
1226<a name="step_25-Results"></a><h1>Results</h1>
1227
1228The 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.
1229
1230In the simulations below, we solve the sine-Gordon equation on the interval @f$\Omega =
1231[-10,10]@f$ in 1D and on the square @f$\Omega = [-10,10]\times [-10,10]@f$ in 2D. In
1232each case, the respective grid is refined uniformly 6 times, i.e. @f$h\sim
12332^{-6}@f$.
1234
1235<a name="step_25-An11dSolution"></a><h3>An (1+1)-d Solution</h3>
1236
1237The first example we discuss is the so-called 1D (stationary) breather
1238solution of the sine-Gordon equation. The breather has the following
1239closed-form expression, as mentioned in the Introduction:
1240\f[
1241u_{\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),
1242\f]
1243where @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.
1244
1245<img src="https://www.dealii.org/images/steps/developer/step-25.1d-breather.gif" alt="Animation of the 1D stationary breather.">
1246
1247Though not shown how to do this in the program, another way to visualize the
1248(1+1)-d solution is to use output generated by the DataOutStack class; it
1249allows to "stack" the solutions of individual time steps, so that we get
12502D space-time graphs from 1D time-dependent
1251solutions. This produces the space-time plot below instead of the animation
1252above.
1253
1254<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.">
1255
1256Furthermore, since the breather is an analytical solution of the sine-Gordon
1257equation, we can use it to validate our code, although we have to assume that
1258the error introduced by our choice of Neumann boundary conditions is small
1259compared to the numerical error. Under this assumption, one could use the
1260VectorTools::integrate_difference function to compute the difference between
1261the numerical solution and the function described by the
1262<code>ExactSolution</code> class of this program. For the
1263simulation shown in the two images above, the @f$L^2@f$ norm of the error in the
1264finite element solution at each time step remained on the order of
1265@f$10^{-2}@f$. Hence, we can conclude that the numerical method has been
1266implemented correctly in the program.
1267
1268
1269<a name="step_25-Afew21DSolutions"></a><h3>A few (2+1)D Solutions</h3>
1270
1271
1272The 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:
1273 @f[
1274 u(x,y,t) = 4 \arctan \left[a_0 e^{s\xi}\right]
1275 @f]
1276with
1277 @f[
1278 \xi = x \cos\vartheta + \sin(\vartheta) (y\cosh\lambda + t\sinh \lambda)
1279 @f]
1280where @f$a_0@f$, @f$\vartheta@f$ and @f$\lambda@f$ are constants. In the simulation below
1281we have chosen @f$a_0=\lambda=1@f$. Notice that if @f$\vartheta=\pi@f$ the kink is
1282stationary, hence it would make a good solution against which we can
1283validate the program in 2D because no reflections off the boundary of the
1284domain occur.
1285
1286The 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.
1287
1288<img src="https://www.dealii.org/images/steps/developer/step-25.2d-kink.png" alt="Stationary 2D kink.">
1289
1290Now that we have validated the code in 1D and 2D, we move to a problem where the analytical solution is unknown.
1291
1292To this end, we rotate the kink solution discussed above about the @f$z@f$
1293axis: we let @f$\vartheta=\frac{\pi}{4}@f$. The latter results in a
1294solitary wave that is not aligned with the grid, so reflections occur
1295at the boundaries of the domain immediately. For the simulation shown
1296below, we have taken @f$u_0(x)=u_{\mathrm{kink}}(x,t_0)@f$,
1297@f$\theta=\frac{2}{3}@f$, @f$k=20h@f$, @f$t_0=0@f$ and @f$t_f=20@f$. Moreover, we had
1298to pick @f$\theta=\frac{2}{3}@f$ because for any @f$\theta\le\frac{1}{2}@f$
1299oscillations arose at the boundary, which are likely due to the scheme
1300and not the equation, thus picking a value of @f$\theta@f$ a good bit into
1301the "exponentially damped" spectrum of the time stepping schemes
1302assures these oscillations are not created.
1303
1304<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.">
1305
1306Another interesting solution to the sine-Gordon equation (which cannot be
1307obtained analytically) can be produced by using two 1D breathers to construct
1308the following separable 2D initial condition:
1309\f[
1310 u_0(x) =
1311 u_{\mathrm{pseudobreather}}(x,t_0) =
1312 16\arctan \left(
1313 \frac{m}{\sqrt{1-m^2}}
1314 \frac{\sin\left(\sqrt{1-m^2}t_0\right)}{\cosh(mx_1)} \right)
1315 \arctan \left(
1316 \frac{m}{\sqrt{1-m^2}}
1317 \frac{\sin\left(\sqrt{1-m^2}t_0\right)}{\cosh(mx_2)} \right),
1318\f]
1319where @f$x=(x_1,x_2)\in{R}^2@f$, @f$m=0.5<1@f$ as in the 1D case we discussed
1320above. For the simulation shown below, we have chosen @f$\theta=\frac{1}{2}@f$,
1321@f$k=10h@f$, @f$t_0=-5.4414@f$ and @f$t_f=2.7207@f$. The solution is pretty interesting
1322--- it acts like a breather (as far as the pictures are concerned); however,
1323it appears to break up and reassemble, rather than just oscillate.
1324
1325<img src="https://www.dealii.org/images/steps/developer/step-25.2d-pseudobreather.gif" alt="Animation of a 2D pseudobreather.">
1326
1327
1328<a name="step-25-extensions"></a>
1329<a name="step_25-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
1330
1331
1332It is instructive to change the initial conditions. Most choices will not lead
1333to solutions that stay localized (in the soliton community, such
1334solutions are called "stationary", though the solution does change
1335with time), but lead to solutions where the wave-like
1336character of the equation dominates and a wave travels away from the location
1337of a localized initial condition. For example, it is worth playing around with
1338the <code>InitialValues</code> class, by replacing the call to the
1339<code>ExactSolution</code> class by something like this function:
1340@f[
1341 u_0(x,y) = \cos\left(\frac x2\right)\cos\left(\frac y2\right)
1342@f]
1343if @f$|x|,|y|\le \frac\pi 2@f$, and @f$u_0(x,y)=0@f$ outside this region.
1344
1345A second area would be to investigate whether the scheme is
1346energy-preserving. For the pure wave equation, discussed in @ref
1347step_23 "step-23", this is the case if we choose the time stepping
1348parameter such that we get the Crank-Nicolson scheme. One could do a
1349similar thing here, noting that the energy in the sine-Gordon solution
1350is defined as
1351@f[
1352 E(t) = \frac 12 \int_\Omega \left(\frac{\partial u}{\partial
1353 t}\right)^2
1354 + \left(\nabla u\right)^2 + 2 (1-\cos u) \; dx.
1355@f]
1356(We use @f$1-\cos u@f$ instead of @f$-\cos u@f$ in the formula to ensure that all
1357contributions to the energy are positive, and so that decaying solutions have
1358finite energy on unbounded domains.)
1359
1360Beyond this, there are two obvious areas:
1361
1362- Clearly, adaptivity (i.e. time-adaptive grids) would be of interest
1363 to problems like these. Their complexity leads us to leave this out
1364 of this program again, though the general comments in the
1365 introduction of @ref step_23 "step-23" remain true.
1366
1367- Faster schemes to solve this problem. While computers today are
1368 plenty fast enough to solve 2d and, frequently, even 3d stationary
1369 problems within not too much time, time dependent problems present
1370 an entirely different class of problems. We address this topic in
1371 @ref step_48 "step-48" where we show how to solve this problem in parallel and
1372 without assembling or inverting any matrix at all.
1373 *
1374 *
1375<a name="step_25-PlainProg"></a>
1376<h1> The plain program</h1>
1377@include "step-25.cc"
1378*/
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())
unsigned int n_active_cells() const
void refine_global(const unsigned int times=1)
unsigned int n_cells() const
Point< 2 > second
Definition grid_out.cc:4624
Point< 2 > first
Definition grid_out.cc:4623
unsigned int level
Definition grid_out.cc:4626
__global__ void set(Number *val, const Number s, const size_type N)
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
Definition loop.h:564
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints={}, 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.
#define DEAL_II_NOT_IMPLEMENTED()
std::vector< value_type > split(const typename ::Triangulation< dim, spacedim >::cell_iterator &parent, const value_type parent_value)
const Event initial
Definition event.cc:64
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:57
void create_mass_matrix(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim > &q, MatrixType &matrix, const Function< spacedim, typename MatrixType::value_type > *const a=nullptr, const AffineConstraints< typename MatrixType::value_type > &constraints=AffineConstraints< typename MatrixType::value_type >())
void create_laplace_matrix(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim > &q, MatrixType &matrix, const Function< spacedim, typename MatrixType::value_type > *const a=nullptr, const AffineConstraints< typename MatrixType::value_type > &constraints=AffineConstraints< typename MatrixType::value_type >())
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:191
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)
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()
constexpr T fixed_power(const T t)
Definition utilities.h:942
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
Definition utilities.cc:470
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 >()), 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 > &)
inline ::VectorizedArray< Number, width > sinh(const ::VectorizedArray< Number, width > &x)
::VectorizedArray< Number, width > cos(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
inline ::VectorizedArray< Number, width > cosh(const ::VectorizedArray< Number, width > &x)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
inline ::VectorizedArray< Number, width > atan(const ::VectorizedArray< Number, width > &x)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DataOutBase::CompressionLevel compression_level
void advance(std::tuple< I1, I2 > &t, const unsigned int n)