FFT-based Poisson Solver for large scale numerical simulations of incompressible flows

Abstract

In the context of time-accurate numerical simulation of incompressible flows, a Poisson equation needs to be solved at least once per time-step to project the velocity field onto a divergence-free space. Due to the non-local nature of its solution, this elliptic system is one of the most time consuming and difficult to parallelise parts of the code. In this paper, a parallel direct Poisson solver restricted to problems with one uniform periodic direction is presented. It is a combination of a Direct Schur-complement based Decomposition (DSD) and a Fourier diagonalisation. The latter decomposes the original system into a set of mutually independent 2D systems which are solved by means of the DSD algorithm. Since no restrictions are imposed in the non-periodic directions, the overall algorithm is well-suited for solving problems discretised on extruded 2D unstructured meshes. A new overall parallelisation strategy with respect to our earlier works is presented. This has allowed us to solve discrete Poisson equations with up to 109 grid points in less than half a second, using up to 8192 CPU cores of the MareNostrum Supercomputer.