Linear Solvers

Evolving the anelastic equation set requires solution of a Poisson equation in which we solve for the update to the perturbational pressure at cell centers. ERF uses several solver options available through AMReX: geometric multigrid, Fast Fourier Transforms (FFTs) and preconditioned GMRES. For simulations without terrain-fitted coordinates or grid stretching, one of the FFT options is generally the fastest solver, followed by multigrid. We note that the multigrid solver has the option to “ignore” a coordinate direction if the domain is only one cell wide in that direction; this allows for efficient solution of effectively 2D problems. Multigrid can also be used when the union of grids at a level is not in itself rectangular; the FFT solvers do not work in that general case.

For simulations using grid stretching in the vertical but a flat lower boundary, we must use the hybrid FFT solver in which we perform 2D transforms only in the lateral directions and couple the solution in the vertical direction with a tridiagonal solve. In both these cases we use a 7-point stencil.

To solve the Poisson equation using terrain-fitted coordinates with general terrain, we rely on the FFT-preconditioned GMRES solver since the stencil effectively has variable coefficients and requires 19 points.

Note

The FFT solver / preconditioner can only be used when the union of grids at a level is itself rectangular.