/*
* == Test 2: Solving a linear parabolic PDE ==
*
* Now we solve a parabolic PDE. We choose a simple problem so that the code changes
* needed from the elliptic case are clearer. We will solve
* du/dt = div(grad u) + u, in 3d, with boundary conditions u=1 on the boundary, and initial
* conditions u=1.
*
*/
void TestSolvingParabolicPde() throw(Exception)
{
/* Create a 10 by 10 by 10 mesh in 3D, this time using the {{{ConstructRegularSlabMesh}}} method
* on the mesh. The first parameter is the cartesian space-step and the other three parameters are the width, height and depth of the mesh.*/
TetrahedralMesh<3,3> mesh;
mesh.ConstructRegularSlabMesh(0.1, 1.0, 1.0, 1.0);
/* Our PDE object should be a class that is derived from the {{{AbstractLinearParabolicPde}}}.
* We could write it ourselves as in the previous test, but since the PDE we want to solve is
* so simple, it has already been defined (look it up! - it is located in pde/test/pdes).
*/
HeatEquationWithSourceTerm<3> pde;
/* Create a new boundary conditions container and specify u=1.0 on the boundary. */
BoundaryConditionsContainer<3,3,1> bcc;
bcc.DefineConstantDirichletOnMeshBoundary(&mesh, 1.0);
/* Create an instance of the solver, passing in the mesh, pde and boundary conditions.
*/
SimpleLinearParabolicSolver<3,3> solver(&mesh,&pde,&bcc);
/* For parabolic problems, initial conditions are also needed. The solver will expect
* a PETSc vector, where the i-th entry is the initial solution at node i, to be passed
* in. To create this PETSc {{{Vec}}}, we will use a helper function in the {{{PetscTools}}}
* class to create a {{{Vec}}} of size num_nodes, with each entry set to 1.0. Then we
* set the initial condition on the solver. */
Vec initial_condition = PetscTools::CreateAndSetVec(mesh.GetNumNodes(), 1.0);
solver.SetInitialCondition(initial_condition);
/* Next define the start time, end time, and timestep, and set them. */
double t_start = 0;
double t_end = 1;
double dt = 0.01;
solver.SetTimes(t_start, t_end);
solver.SetTimeStep(dt);
/* HOW_TO_TAG PDE
* Output results to file for time-dependent PDE solvers
*/
/* When we call Solve() below we will just get the solution at the final time. If we want
* to have intermediate solutions written to file, we do the following. We start by
* specifying an output directory and filename prefix for our results file:
*/
solver.SetOutputDirectoryAndPrefix("ParabolicSolverTutorial","results");
/* When an output directory has been specified, the solver writes output in HDF5 format. To
* convert this to another output format, we call the relevant method. Here, we convert
* the output to plain text files (VTK or cmgui formats are also possible). We also say how
* often to write the data, telling the solver to output results to file every tenth timestep.
* The solver will create one file for each variable (in this case there is only one variable)
* and for each time, so for example, the file
* `results_Variable_0_10` is the results for u, over all nodes, at the 11th printed time.
* Have a look in the output directory after running the test. (For comments on visualising the data in
* matlab or octave, see the end of the tutorial UserTutorials/WritingPdeSolvers.)
*/
solver.SetOutputToTxt(true);
solver.SetPrintingTimestepMultiple(10);
/* Now we can solve the problem. The {{{Vec}}} that is returned can be passed into a
* {{{ReplicatableVector}}} as before.
*/
Vec solution = solver.Solve();
ReplicatableVector solution_repl(solution);
/* Let's also solve the equivalent static PDE, i.e. set du/dt=0, so 0=div(gradu) + u. This
* is easy, as the PDE class has already been defined. */
SimplePoissonEquation<3,3> static_pde;
SimpleLinearEllipticSolver<3,3> static_solver(&mesh, &static_pde, &bcc);
Vec static_solution = static_solver.Solve();
ReplicatableVector static_solution_repl(static_solution);
/* We can now compare the solution of the parabolic PDE at t=1 with the static solution,
* to see if the static equilibrium solution was reached in the former. (Ideally we should
* compute some relative error, but we just compute an absolute error for simplicity.) */
for (unsigned i=0; i<static_solution_repl.GetSize(); i++)
{
TS_ASSERT_DELTA( solution_repl[i], static_solution_repl[i], 1e-3);
}
/* All PETSc vectors should be destroyed when they are no longer needed. */
PetscTools::Destroy(initial_condition);
PetscTools::Destroy(solution);
PetscTools::Destroy(static_solution);
}
