A simple multigrid scheme for solving the Poisson equation with arbitrary domain boundaries

TL;DR

This paper introduces a new multigrid method tailored for solving the Poisson equation with irregular boundaries on Cartesian grids, optimized for adaptive mesh refinement in astrophysical simulations.

Contribution

It presents a memory-efficient multigrid scheme with a boundary condition reconstruction method suitable for complex geometries in AMR frameworks.

Findings

Achieves good linear convergence on complex grids.

Successfully implemented in the RAMSES astrophysics code.

Demonstrates efficient timing performance and accuracy.

Abstract

We present a new multigrid scheme for solving the Poisson equation with Dirichlet boundary conditions on a Cartesian grid with irregular domain boundaries. This scheme was developed in the context of the Adaptive Mesh Refinement (AMR) schemes based on a graded-octree data structure. The Poisson equation is solved on a level-by-level basis, using a "one-way interface" scheme in which boundary conditions are interpolated from the previous coarser level solution. Such a scheme is particularly well suited for self-gravitating astrophysical flows requiring an adaptive time stepping strategy. By constructing a multigrid hierarchy covering the active cells of each AMR level, we have designed a memory-efficient algorithm that can benefit fully from the multigrid acceleration. We present a simple method for capturing the boundary conditions across the multigrid hierarchy, based on a second-order accurate reconstruction of the boundaries of the multigrid levels. In case of very complex boundaries, small scale features become smaller than the discretization cell size of coarse multigrid levels and convergence problems arise. We propose a simple solution to address these issues. Using our scheme, the convergence rate usually depends on the grid size for complex grids, but good linear convergence is maintained. The proposed method was successfully implemented on distributed memory architectures in the RAMSES code, for which we present and discuss convergence and accuracy properties as well as timing performances.