A scalable high-order multigrid-FFT Poisson solver for unbounded domains on adaptive multiresolution grids

TL;DR

This paper introduces a scalable multigrid-FFT Poisson solver capable of handling various boundary conditions on adaptive multiresolution grids, significantly improving efficiency and accuracy for large-scale computational physics problems.

Contribution

It presents a flexible, high-performance multigrid solver using Fourier-based direct methods and high-order stencils within an adaptive grid framework, supporting unbounded and semi-unbounded domains.

Findings

Validated against analytical solutions for different domains.

Achieved scalability up to 16,384 cores on HPC systems.

Enhanced accuracy with high-order compact stencils.

Abstract

Multigrid solvers are among the most efficient methods for solving the Poisson equation, which is ubiquitous in computational physics. For example, in the context of incompressible flows, it is typically the costliest operation. The present document expounds upon the implementation of a flexible multigrid solver that is capable of handling any type of boundary conditions within murphy, a multiresolution framework for solving partial differential equations (PDEs) on collocated adaptive grids. The utilization of a Fourier-based direct solver facilitates the attainment of flexibility and enhanced performance by accommodating any combination of unbounded and semi-unbounded boundary conditions. The employment of high-order compact stencils contributes to the reduction of communication demands while concurrently enhancing the accuracy of the system. The resulting solver is validated against analytical solutions for periodic and unbounded domains. In conclusion, the solver has been demonstrated to demonstrate scalability to 16,384 cores within the context of leading European high-performance computing infrastructures.