Solvers

Parthenon provides a number of linear solvers, including a geometric multigrid solver, a CG solver, a BiCGSTAB solver, and multigrid preconditioned versions of the latter two solvers.

Solvers are templated on a type defining the system of equations they are solving. The type defining the system of equations must provide two methods and a TypeList of all of the fields that make up the vector space: .. code:: c++

class MySystemOfEquations {

using IndependentVars = parthenon::TypeList<var1_t, var2_t>;

TaskId Ax(TaskList &tl, TaskID depends_on,

std::shared_ptr<parthenon::MeshData<Real>> &md_mat, std::shared_ptr<parthenon::MeshData<Real>> &md_in, std::shared_ptr<parthenon::MeshData<Real>> &md_out);

TaskStatus SetDiagonal(std::shared_ptr<parthenon::MeshData<Real>> &md_mat,

std::shared_ptr<parthenon::MeshData<Real>> &md_diag)

};

The routine Ax must calculate the matrix vector product y <- A.x by taking a container md_mat which contains all of the fields required to reconstruct the matrix A associated with the system of linear equations, the container md_in which will store the vector x in the fields in the typelist IndependentVars, and md_out which will hold the vector y.

The routine SetDiagonal takes the same container md_mat as Ax and returns the (approximate) diagonal of A in the container md_diag. This only needs to be approximate since it is only used in preconditioners/smoothers.

With such a class defining a linear system of equations, one can then define and use a solver with code along the lines of: .. code:: c++

std::string base_cont_name = “base”; std::string u_cont_name = “u”; std::string rhs_cont_name = “rhs”;

MySystemOfEquations eqs(….); std::shared_ptr<SolverBase> psolver = std::make_shared<BiCGSTABSolverStages<MySystemOfEquations>>(

base_cont_name, u_cont_name, rhs_cont_name, pin, “location/of/solver_params”, eqs);

…

auto partitions = pmesh->GetDefaultBlockPartitions(); const int num_partitions = partitions.size(); TaskRegion &region = tc.AddRegion(num_partitions); for (int partition = 0; partition < num_partitions; ++partition) {

TaskList &tl = region[partition]; auto &md = pmesh->mesh_data.Add(base_cont_name, partitions[partition]); auto &md_u = pmesh->mesh_data.Add(u_cont_name, md); auto &md_rhs = pmesh->mesh_data.Add(rhs_cont_name, md);

// Do some stuff to fill the base container with information necessary to define A // if it wasn’t defined during initialization or something

// Do some stuff to fill the rhs container

auto setup = psolver->AddSetupTasks(tl, dependence, partition, pmesh); auto solve = psolver->AddTasks(tl, setup, partition, pmesh);

// Do some stuff with the solution stored in md_u

}

Some notes: - All solvers inherit from SolverBase, so the best practice is to stash a shared pointer to a

SolverBase object in params during initialization and pull this solver out while building a task list. This should make switching between solvers trivial.

• For any solver involving geometric multigrid, the input parameter parthenon/mesh/multigrid must be set to true. This tells the Mesh to build the coarsened blocks associated with the multi-grid hierarchy.

• For geometric multigrid based solvers, it is possible to define block interior prolongation operators that are separate from the standard prolongation machinery in Parthenon. This allows for defining boundary aware prolongation operators and having different prolongation operators in the ghost cells of blocks from the prolongation operators used in their interiors. Users can easily define their own prolongation operators. An example of using these interior prolongation operators is contained in the poisson_gmg example.

Some implementation notes about geometric multi-grid can be found in these notes.

Stencil

This class provides a very simple and efficient means of storing a sparse matrix with the special form that every row has identical entries relative to the matrix diagonal. A good example of this is in the straightforward finite difference discretization of the Poisson equation (see here for example usage). The Stencil object is extremely efficient at storing these sparse matrices because it only has to store the matrix values and offsets from the diagnonal for a single row. The Stencil class provides member functions to compute matrix vector products (MatVec) and Jacobi iterates (Jacobi). Both are designed to be called from within kernels and operate on a single matrix row at a time.

SparseMatrixAccessor

This is a helper class that allows one to store a more general sparse matrix than Stencil provides. Like Stencil, the SparseMatrixAccessor class assumes that the location of the nonzero matrix elements have fixed offsets from the diagonal in every row. Here, though, the values of the matrix elements can be different from row to row. The sparse matrix itself can be stored in a normal Variable with the number of components equal to the number of nonzero elements in a row of the matrix. The SparseMatrixAccessor class than associates each of these components with a particular matrix element. Like Stencil, the SparseMatrixAccessor class provides MatVec and Jacobi member functions. A simple demonstration of usage can be found in the Poisson example.