Exploiting symmetries for preconditioning Poisson's equation in CFD simulations

Abstract

Divergence constraints are present in the governing equations of many physical phenomena, and they usually lead to a Poisson equation whose solution is one of the most challenging parts of scientific simulation codes. Indeed, it is the main bottleneck of incompressible Computational Fluid Dynamics (CFD) simulations, and developing efficient and scalable Poisson solvers is a critical task. This work presents an enhanced variant of the Factored Sparse Approximate Inverse (FSAI) preconditioner. It arises from exploiting s spatial reflection symmetries, which are often present in academic and industrial configurations and allow transforming Poisson's equation into a set of 2s fully-decoupled subsystems. Then, we introduce another level of approximation by taking advantage of the subsystems' close similarity and applying the same FSAI to all of them. This leads to substantial memory savings and notable increases in the arithmetic intensity resulting from employing the more compute-intensive sparse matrix-matrix product. Of course, recycling the same preconditioner on all the subsystems worsens its convergence. However, this effect was much smaller than expected and made us introduce relatively cheap but very effective low-rank corrections. A key feature of these corrections is that thanks to being applied to each subsystem independently, the more symmetries being exploited, the more effective they become, leading to up to 5.7x faster convergences than the standard FSAI. Numerical experiments on up to 1.07 billion grids confirm the quality of our low-rank corrected FSAI, which, despite being 2.6x lighter, outperforms the standard FSAI by a factor of up to 4.4x.