pyro solves elliptic problems (like Laplace's equation or Poisson's equation) through multigrid. This accelerates the convergence of simple relaxation by moving the solution down and up through a series of grids. Chapter 8 of the notes describe the basics of multigrid:
There are three solvers:
The core solver, provided in the class MG.CellCenterMG2d
solves constant-coefficient Helmholtz problems of the form:
(α - β ∇2) φ = f
The class variable_coeff_MG.VarCoeffCCMG2d solves variable
coefficient Poisson problems of the form:
∇· (η ∇ φ) = f
This class inherits the core functionality from MG.CellCenterMG2d.
The class general_MG.GeneralMG2d solves a general elliptic
equation of the form:
αφ + ∇· (β ∇ φ) + γ · ∇ φ = 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.
The multigrid solver is not controlled through pyro.py since there is no time-dependence in pure elliptic problems. Instead, there are a few scripts in the multigrid/ subdirectory that demonstrate its use.
A basic multigrid test is run as:
The mg_test_simple.py script solves a Poisson equation with a known analytic solution. This particular example comes from the text "A Multigrid Tutorial, 2nd Ed.". The example is:
uxx + uyy =
-2 [(1-6x2)y2(1-y2) +
on [0,1] × [0,1] with u = 0 on the boundary.
The solution to this is shown below.
Since this has a known analytic solution:
u(x,y) = (x2 - x4)(y4 - y2)
We can assess the convergence of our solver by running at a variety of resolutions and computing the norm of the error with respect to the analytic solution. This is shown below:
The dotted line is 2nd order convergence, which we match perfectly.
The movie below shows the smoothing at each level to realize this solution:
Another example that uses multigrid to extract the divergence free part of a velocity field is run as:
Given a vector field, U, we can decompose it into a divergence free part, Ud, and the gradient of a scalar, φ:
U = Ud + ∇ φ
We can project out the divergence free part by taking the divergence, leading to an elliptic equation:
∇2 φ = ∇ · U
The project-periodic.py script starts with a divergence free velocity field, adds to it the gradient of a scalar, and then projects it to recover the divergence free part. The error can found by comparing the original velocity field to the recovered field. The results are shown below:
Left is the original u velocity, middle is the modified field after adding the gradient of the scalar, and right is the recovered field.
A jupyter notebook showing how to use the basic solver can be found here: multigrid-examples.ipynb
- Try doing just smoothing, no multigrid. Show that it still converges second order if you use enough iterations, but that the amount of time needed to get a solution is much greater.
- Implement inhomogeneous dirichlet boundary conditions
- Add a different bottom solver to the multigrid algorithm
- Make the multigrid solver work for non-square domains