ex9.cpp

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//                                MFEM Example 9
//                             SUNDIALS Modification
//
// Compile with:
//    make ex9              (GNU make)
//    make sundials_ex9     (CMake)
//
// Sample runs:
//    ex9 -m ../../data/periodic-segment.mesh -p 0 -r 2 -s 7 -dt 0.005
//    ex9 -m ../../data/periodic-square.mesh  -p 1 -r 2 -s 8 -dt 0.005 -tf 9
//    ex9 -m ../../data/periodic-hexagon.mesh -p 0 -r 2 -s 7 -dt 0.0018 -vs 25
//    ex9 -m ../../data/periodic-hexagon.mesh -p 0 -r 2 -s 9 -dt 0.01 -vs 15
//    ex9 -m ../../data/amr-quad.mesh         -p 1 -r 2 -s 9 -dt 0.002 -tf 9
//    ex9 -m ../../data/star-q3.mesh          -p 1 -r 2 -s 9 -dt 0.005 -tf 9
//    ex9 -m ../../data/disc-nurbs.mesh       -p 1 -r 3 -s 7 -dt 0.005 -tf 9
//    ex9 -m ../../data/periodic-cube.mesh    -p 0 -r 2 -s 8 -dt 0.02 -tf 8 -o 2
//
// Device sample runs:
//    ex9 -pa
//    ex9 -ea
//    ex9 -fa
//    ex9 -pa -m ../../data/periodic-cube.mesh
//    ex9 -pa -m ../../data/periodic-cube.mesh -d cuda
//    ex9 -ea -m ../../data/periodic-cube.mesh -d cuda
//    ex9 -fa -m ../../data/periodic-cube.mesh -d cuda
//
// Description:  This example code solves the time-dependent advection equation
//               du/dt + v.grad(u) = 0, where v is a given fluid velocity, and
//               u0(x)=u(0,x) is a given initial condition.
//
//               The example demonstrates the use of Discontinuous Galerkin (DG)
//               bilinear forms in MFEM (face integrators), the use of implicit
//               and explicit ODE time integrators, the definition of periodic
//               boundary conditions through periodic meshes, as well as the use
//               of GLVis for persistent visualization of a time-evolving
//               solution. The saving of time-dependent data files for external
//               visualization with VisIt (visit.llnl.gov) and ParaView
//               (paraview.org) is also illustrated.
 
#include "mfem.hpp"
#include <fstream>
#include <iostream>
#include <algorithm>
 
#ifndef MFEM_USE_SUNDIALS
#error This example requires that MFEM is built with MFEM_USE_SUNDIALS=YES
#endif
 
using namespace std;
using namespace mfem;
 
// Choice for the problem setup. The fluid velocity, initial condition and
// inflow boundary condition are chosen based on this parameter.
int problem;
 
// Velocity coefficient
void velocity_function(const Vector &x, Vector &v);
 
// Initial condition
double u0_function(const Vector &x);
 
// Inflow boundary condition
double inflow_function(const Vector &x);
 
// Mesh bounding box
Vector bb_min, bb_max;
 
class DG_Solver : public Solver
{
private:
   SparseMatrix &M, &K, A;
   GMRESSolver linear_solver;
   BlockILU prec;
   double dt;
public:
   DG_Solver(SparseMatrix &M_, SparseMatrix &K_, const FiniteElementSpace &fes)
      : M(M_),
        K(K_),
        prec(fes.GetTypicalFE()->GetDof(),
             BlockILU::Reordering::MINIMUM_DISCARDED_FILL),
        dt(-1.0)
   {
      linear_solver.iterative_mode = false;
      linear_solver.SetRelTol(1e-9);
      linear_solver.SetAbsTol(0.0);
      linear_solver.SetMaxIter(100);
      linear_solver.SetPrintLevel(0);
      linear_solver.SetPreconditioner(prec);
   }
 
   void SetTimeStep(double dt_)
   {
      if (dt_ != dt)
      {
         dt = dt_;
         // Form operator A = M - dt*K
         A = K;
         A *= -dt;
         A += M;
 
         // this will also call SetOperator on the preconditioner
         linear_solver.SetOperator(A);
      }
   }
 
   void SetOperator(const Operator &op)
   {
      linear_solver.SetOperator(op);
   }
 
   virtual void Mult(const Vector &x, Vector &y) const
   {
      linear_solver.Mult(x, y);
   }
};
 
/** A time-dependent operator for the right-hand side of the ODE. The DG weak
    form of du/dt = -v.grad(u) is M du/dt = K u + b, where M and K are the mass
    and advection matrices, and b describes the flow on the boundary. This can
    be written as a general ODE, du/dt = M^{-1} (K u + b), and this class is
    used to evaluate the right-hand side. */
class FE_Evolution : public TimeDependentOperator
{
private:
   BilinearForm &M, &K;
   const Vector &b;
   Solver *M_prec;
   CGSolver M_solver;
   DG_Solver *dg_solver;
 
   mutable Vector z;
 
public:
   FE_Evolution(BilinearForm &M_, BilinearForm &K_, const Vector &b_);
 
   virtual void Mult(const Vector &x, Vector &y) const;
   virtual void ImplicitSolve(const double dt, const Vector &x, Vector &k);
 
   virtual ~FE_Evolution();
};
 
 
int main(int argc, char *argv[])
{
   // 0. Initialize SUNDIALS.
   Sundials::Init();
 
   // 1. Parse command-line options.
   problem = 0;
   const char *mesh_file = "../../data/periodic-hexagon.mesh";
   int ref_levels = 2;
   int order = 3;
   bool pa = false;
   bool ea = false;
   bool fa = false;
   const char *device_config = "cpu";
   int ode_solver_type = 7;
   double t_final = 10.0;
   double dt = 0.01;
   bool visualization = false;
   bool visit = false;
   bool paraview = false;
   bool binary = false;
   int vis_steps = 5;
 
   // Relative and absolute tolerances for CVODE and ARKODE.
   const double reltol = 1e-2, abstol = 1e-2;
 
   int precision = 8;
   cout.precision(precision);
 
   OptionsParser args(argc, argv);
   args.AddOption(&mesh_file, "-m", "--mesh",
                  "Mesh file to use.");
   args.AddOption(&problem, "-p", "--problem",
                  "Problem setup to use. See options in velocity_function().");
   args.AddOption(&ref_levels, "-r", "--refine",
                  "Number of times to refine the mesh uniformly.");
   args.AddOption(&order, "-o", "--order",
                  "Order (degree) of the finite elements.");
   args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
                  "--no-partial-assembly", "Enable Partial Assembly.");
   args.AddOption(&ea, "-ea", "--element-assembly", "-no-ea",
                  "--no-element-assembly", "Enable Element Assembly.");
   args.AddOption(&fa, "-fa", "--full-assembly", "-no-fa",
                  "--no-full-assembly", "Enable Full Assembly.");
   args.AddOption(&device_config, "-d", "--device",
                  "Device configuration string, see Device::Configure().");
   args.AddOption(&ode_solver_type, "-s", "--ode-solver",
                  "ODE solver:\n\t"
                  "1 - Forward Euler,\n\t"
                  "2 - RK2 SSP,\n\t"
                  "3 - RK3 SSP,\n\t"
                  "4 - RK4,\n\t"
                  "6 - RK6,\n\t"
                  "7 - CVODE (adaptive order implicit Adams),\n\t"
                  "8 - ARKODE default (4th order) explicit,\n\t"
                  "9 - ARKODE RK8.");
   args.AddOption(&t_final, "-tf", "--t-final",
                  "Final time; start time is 0.");
   args.AddOption(&dt, "-dt", "--time-step",
                  "Time step.");
   args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                  "--no-visualization",
                  "Enable or disable GLVis visualization.");
   args.AddOption(&visit, "-visit", "--visit-datafiles", "-no-visit",
                  "--no-visit-datafiles",
                  "Save data files for VisIt (visit.llnl.gov) visualization.");
   args.AddOption(&paraview, "-paraview", "--paraview-datafiles", "-no-paraview",
                  "--no-paraview-datafiles",
                  "Save data files for ParaView (paraview.org) visualization.");
   args.AddOption(&binary, "-binary", "--binary-datafiles", "-ascii",
                  "--ascii-datafiles",
                  "Use binary (Sidre) or ascii format for VisIt data files.");
   args.AddOption(&vis_steps, "-vs", "--visualization-steps",
                  "Visualize every n-th timestep.");
   args.Parse();
   if (!args.Good())
   {
      args.PrintUsage(cout);
      return 1;
   }
   // check for valid ODE solver option
   if (ode_solver_type < 1 || ode_solver_type > 9)
   {
      cout << "Unknown ODE solver type: " << ode_solver_type << '\n';
      return 3;
   }
   args.PrintOptions(cout);
 
   Device device(device_config);
   device.Print();
 
   // 2. Read the mesh from the given mesh file. We can handle geometrically
   //    periodic meshes in this code.
   Mesh mesh(mesh_file, 1, 1);
   int dim = mesh.Dimension();
 
   // 3. Refine the mesh to increase the resolution. In this example we do
   //    'ref_levels' of uniform refinement, where 'ref_levels' is a
   //    command-line parameter. If the mesh is of NURBS type, we convert it to
   //    a (piecewise-polynomial) high-order mesh.
   for (int lev = 0; lev < ref_levels; lev++)
   {
      mesh.UniformRefinement();
   }
   if (mesh.NURBSext)
   {
      mesh.SetCurvature(max(order, 1));
   }
   mesh.GetBoundingBox(bb_min, bb_max, max(order, 1));
 
   // 4. Define the discontinuous DG finite element space of the given
   //    polynomial order on the refined mesh.
   DG_FECollection fec(order, dim, BasisType::GaussLobatto);
   FiniteElementSpace fes(&mesh, &fec);
 
   cout << "Number of unknowns: " << fes.GetVSize() << endl;
 
   // 5. Set up and assemble the bilinear and linear forms corresponding to the
   //    DG discretization. The DGTraceIntegrator involves integrals over mesh
   //    interior faces.
   VectorFunctionCoefficient velocity(dim, velocity_function);
   FunctionCoefficient inflow(inflow_function);
   FunctionCoefficient u0(u0_function);
 
   BilinearForm m(&fes);
   BilinearForm k(&fes);
   if (pa)
   {
      m.SetAssemblyLevel(AssemblyLevel::PARTIAL);
      k.SetAssemblyLevel(AssemblyLevel::PARTIAL);
   }
   else if (ea)
   {
      m.SetAssemblyLevel(AssemblyLevel::ELEMENT);
      k.SetAssemblyLevel(AssemblyLevel::ELEMENT);
   }
   else if (fa)
   {
      m.SetAssemblyLevel(AssemblyLevel::FULL);
      k.SetAssemblyLevel(AssemblyLevel::FULL);
   }
   m.AddDomainIntegrator(new MassIntegrator);
   constexpr double alpha = -1.0;
   k.AddDomainIntegrator(new ConvectionIntegrator(velocity, alpha));
   k.AddInteriorFaceIntegrator(
      new NonconservativeDGTraceIntegrator(velocity, alpha));
   k.AddBdrFaceIntegrator(
      new NonconservativeDGTraceIntegrator(velocity, alpha));
 
   LinearForm b(&fes);
   b.AddBdrFaceIntegrator(
      new BoundaryFlowIntegrator(inflow, velocity, alpha));
 
   m.Assemble();
   int skip_zeros = 0;
   k.Assemble(skip_zeros);
   b.Assemble();
   m.Finalize();
   k.Finalize(skip_zeros);
 
   // 6. Define the initial conditions, save the corresponding grid function to
   //    a file and (optionally) save data in the VisIt format and initialize
   //    GLVis visualization.
   GridFunction u(&fes);
   u.ProjectCoefficient(u0);
 
   {
      ofstream omesh("ex9.mesh");
      omesh.precision(precision);
      mesh.Print(omesh);
      ofstream osol("ex9-init.gf");
      osol.precision(precision);
      u.Save(osol);
   }
 
   // Create data collection for solution output: either VisItDataCollection for
   // ascii data files, or SidreDataCollection for binary data files.
   DataCollection *dc = NULL;
   if (visit)
   {
      if (binary)
      {
#ifdef MFEM_USE_SIDRE
         dc = new SidreDataCollection("Example9", &mesh);
#else
         MFEM_ABORT("Must build with MFEM_USE_SIDRE=YES for binary output.");
#endif
      }
      else
      {
         dc = new VisItDataCollection("Example9", &mesh);
         dc->SetPrecision(precision);
      }
      dc->RegisterField("solution", &u);
      dc->SetCycle(0);
      dc->SetTime(0.0);
      dc->Save();
   }
 
   ParaViewDataCollection *pd = NULL;
   if (paraview)
   {
      pd = new ParaViewDataCollection("Example9", &mesh);
      pd->SetPrefixPath("ParaView");
      pd->RegisterField("solution", &u);
      pd->SetLevelsOfDetail(order);
      pd->SetDataFormat(VTKFormat::BINARY);
      pd->SetHighOrderOutput(true);
      pd->SetCycle(0);
      pd->SetTime(0.0);
      pd->Save();
   }
 
   socketstream sout;
   if (visualization)
   {
      char vishost[] = "localhost";
      int  visport   = 19916;
      sout.open(vishost, visport);
      if (!sout)
      {
         cout << "Unable to connect to GLVis server at "
              << vishost << ':' << visport << endl;
         visualization = false;
         cout << "GLVis visualization disabled.\n";
      }
      else
      {
         sout.precision(precision);
         sout << "solution\n" << mesh << u;
         sout << "pause\n";
         sout << flush;
         cout << "GLVis visualization paused."
              << " Press space (in the GLVis window) to resume it.\n";
      }
   }
 
   // 7. Define the time-dependent evolution operator describing the ODE
   //    right-hand side, and define the ODE solver used for time integration.
   FE_Evolution adv(m, k, b);
 
   double t = 0.0;
   adv.SetTime(t);
 
   // Create the time integrator
   ODESolver *ode_solver = NULL;
   CVODESolver *cvode = NULL;
   ARKStepSolver *arkode = NULL;
   switch (ode_solver_type)
   {
      case 1: ode_solver = new ForwardEulerSolver; break;
      case 2: ode_solver = new RK2Solver(1.0); break;
      case 3: ode_solver = new RK3SSPSolver; break;
      case 4: ode_solver = new RK4Solver; break;
      case 6: ode_solver = new RK6Solver; break;
      case 7:
         cvode = new CVODESolver(CV_ADAMS);
         cvode->Init(adv);
         cvode->SetSStolerances(reltol, abstol);
         cvode->SetMaxStep(dt);
         cvode->UseSundialsLinearSolver();
         ode_solver = cvode; break;
      case 8:
         arkode = new ARKStepSolver(ARKStepSolver::EXPLICIT);
         arkode->Init(adv);
         arkode->SetSStolerances(reltol, abstol);
         arkode->SetMaxStep(dt);
         arkode->SetOrder(4);
         ode_solver = arkode; break;
      case 9:
         arkode = new ARKStepSolver(ARKStepSolver::EXPLICIT);
         arkode->Init(adv);
         arkode->SetSStolerances(reltol, abstol);
         arkode->SetMaxStep(dt);
         arkode->SetERKTableNum(ARKODE_FEHLBERG_13_7_8);
         ode_solver = arkode; break;
   }
 
   // Initialize MFEM integrators, SUNDIALS integrators are initialized above
   if (ode_solver_type < 7) { ode_solver->Init(adv); }
 
   // 8. Perform time-integration (looping over the time iterations, ti,
   //    with a time-step dt).
   bool done = false;
   for (int ti = 0; !done; )
   {
      double dt_real = min(dt, t_final - t);
      ode_solver->Step(u, t, dt_real);
      ti++;
 
      done = (t >= t_final - 1e-8*dt);
 
      if (done || ti % vis_steps == 0)
      {
         cout << "time step: " << ti << ", time: " << t << endl;
         if (cvode) { cvode->PrintInfo(); }
         if (arkode) { arkode->PrintInfo(); }
 
         if (visualization)
         {
            sout << "solution\n" << mesh << u << flush;
         }
 
         if (visit)
         {
            dc->SetCycle(ti);
            dc->SetTime(t);
            dc->Save();
         }
 
         if (paraview)
         {
            pd->SetCycle(ti);
            pd->SetTime(t);
            pd->Save();
         }
      }
   }
 
   // 9. Save the final solution. This output can be viewed later using GLVis:
   //    "glvis -m ex9.mesh -g ex9-final.gf".
   {
      ofstream osol("ex9-final.gf");
      osol.precision(precision);
      u.Save(osol);
   }
 
   // 10. Free the used memory.
   delete ode_solver;
   delete pd;
   delete dc;
 
   return 0;
}
int main(int argc, char *argv[]) {…}
 
 
// Implementation of class FE_Evolution
FE_Evolution::FE_Evolution(BilinearForm &M_, BilinearForm &K_, const Vector &b_)
   : TimeDependentOperator(M_.FESpace()->GetTrueVSize()),
     M(M_), K(K_), b(b_), z(height)
{
   Array<int> ess_tdof_list;
   if (M.GetAssemblyLevel() == AssemblyLevel::LEGACY)
   {
      M_prec = new DSmoother(M.SpMat());
      M_solver.SetOperator(M.SpMat());
      dg_solver = new DG_Solver(M.SpMat(), K.SpMat(), *M.FESpace());
   }
   else
   {
      M_prec = new OperatorJacobiSmoother(M, ess_tdof_list);
      M_solver.SetOperator(M);
      dg_solver = NULL;
   }
   M_solver.SetPreconditioner(*M_prec);
   M_solver.iterative_mode = false;
   M_solver.SetRelTol(1e-9);
   M_solver.SetAbsTol(0.0);
   M_solver.SetMaxIter(100);
   M_solver.SetPrintLevel(0);
}
 
void FE_Evolution::Mult(const Vector &x, Vector &y) const
{
   // y = M^{-1} (K x + b)
   K.Mult(x, z);
   z += b;
   M_solver.Mult(z, y);
}
 
void FE_Evolution::ImplicitSolve(const double dt, const Vector &x, Vector &k)
{
   MFEM_VERIFY(dg_solver != NULL,
               "Implicit time integration is not supported with partial assembly");
   K.Mult(x, z);
   z += b;
   dg_solver->SetTimeStep(dt);
   dg_solver->Mult(z, k);
}
 
FE_Evolution::~FE_Evolution()
{
   delete M_prec;
   delete dg_solver;
}
 
// Velocity coefficient
void velocity_function(const Vector &x, Vector &v)
{
   int dim = x.Size();
 
   // map to the reference [-1,1] domain
   Vector X(dim);
   for (int i = 0; i < dim; i++)
   {
      double center = (bb_min[i] + bb_max[i]) * 0.5;
      X(i) = 2 * (x(i) - center) / (bb_max[i] - bb_min[i]);
   }
 
   switch (problem)
   {
      case 0:
      {
         // Translations in 1D, 2D, and 3D
         switch (dim)
         {
            case 1: v(0) = 1.0; break;
            case 2: v(0) = sqrt(2./3.); v(1) = sqrt(1./3.); break;
            case 3: v(0) = sqrt(3./6.); v(1) = sqrt(2./6.); v(2) = sqrt(1./6.);
               break;
         }
         break;
      }
      case 1:
      case 2:
      {
         // Clockwise rotation in 2D around the origin
         const double w = M_PI/2;
         switch (dim)
         {
            case 1: v(0) = 1.0; break;
            case 2: v(0) = w*X(1); v(1) = -w*X(0); break;
            case 3: v(0) = w*X(1); v(1) = -w*X(0); v(2) = 0.0; break;
         }
         break;
      }
      case 3:
      {
         // Clockwise twisting rotation in 2D around the origin
         const double w = M_PI/2;
         double d = max((X(0)+1.)*(1.-X(0)),0.) * max((X(1)+1.)*(1.-X(1)),0.);
         d = d*d;
         switch (dim)
         {
            case 1: v(0) = 1.0; break;
            case 2: v(0) = d*w*X(1); v(1) = -d*w*X(0); break;
            case 3: v(0) = d*w*X(1); v(1) = -d*w*X(0); v(2) = 0.0; break;
         }
         break;
      }
   }
}
void velocity_function(const Vector &x, Vector &v) {…}
 
// Initial condition
double u0_function(const Vector &x)
{
   int dim = x.Size();
 
   // map to the reference [-1,1] domain
   Vector X(dim);
   for (int i = 0; i < dim; i++)
   {
      double center = (bb_min[i] + bb_max[i]) * 0.5;
      X(i) = 2 * (x(i) - center) / (bb_max[i] - bb_min[i]);
   }
 
   switch (problem)
   {
      case 0:
      case 1:
      {
         switch (dim)
         {
            case 1:
               return exp(-40.*pow(X(0)-0.5,2));
            case 2:
            case 3:
            {
               double rx = 0.45, ry = 0.25, cx = 0., cy = -0.2, w = 10.;
               if (dim == 3)
               {
                  const double s = (1. + 0.25*cos(2*M_PI*X(2)));
                  rx *= s;
                  ry *= s;
               }
               return ( erfc(w*(X(0)-cx-rx))*erfc(-w*(X(0)-cx+rx)) *
                        erfc(w*(X(1)-cy-ry))*erfc(-w*(X(1)-cy+ry)) )/16;
            }
         }
      }
      case 2:
      {
         double x_ = X(0), y_ = X(1), rho, phi;
         rho = hypot(x_, y_);
         phi = atan2(y_, x_);
         return pow(sin(M_PI*rho),2)*sin(3*phi);
      }
      case 3:
      {
         const double f = M_PI;
         return sin(f*X(0))*sin(f*X(1));
      }
   }
   return 0.0;
}
double u0_function(const Vector &x) {…}
 
// Inflow boundary condition (zero for the problems considered in this example)
double inflow_function(const Vector &x)
{
   switch (problem)
   {
      case 0:
      case 1:
      case 2:
      case 3: return 0.0;
   }
   return 0.0;
}
double inflow_function(const Vector &x) {…}