Multigrid Solvers

pyro solves elliptic problems (like Laplace’s equation or Poisson’s equation) through multigrid. This accelerates the convergence of qsimple relaxation by moving the solution down and up through a series of grids. Chapter 9 of the pdf notes gives an introduction to solving elliptic equations, including multigrid.

There are three solvers:

• The core solver, provided in the class MG.CellCenterMG2d solves constant-coefficient Helmholtz problems of the form

  \[(\alpha - \beta \nabla^2) \phi = f\]

• The class variable_coeff_MG.VarCoeffCCMG2d solves variable coefficient Poisson problems of the form

  \[\nabla \cdot (\eta \nabla \phi ) = f\]

  This class inherits the core functionality from MG.CellCenterMG2d.

• The class general_MG.GeneralMG2d solves a general elliptic equation of the form

  \[\alpha \phi + \nabla \cdot ( \beta \nabla \phi) + \gamma \cdot \nabla \phi = f\]

  This class inherits the core functionality from MG.CellCenterMG2d.

  This solver is the only one to support inhomogeneous boundary conditions.

We simply use V-cycles in our implementation, and restrict ourselves to square grids with zoning a power of 2.

Note

The multigrid solver is not controlled through pyro_sim.py since there is no time-dependence in pure elliptic problems. Instead, there are a few scripts in the multigrid/ subdirectory (see: python-hydro/pyro2) that demonstrate its use.