Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow

TL;DR

This paper develops and tests monolithic multigrid preconditioners tailored for fully-implicit Runge-Kutta methods in incompressible fluid flow simulations, demonstrating robustness and scalability across complex models.

Contribution

It extends classical relaxation schemes to implicit RK discretizations, enabling efficient solution of large coupled linear systems in fluid dynamics.

Findings

Robust multigrid methods for incompressible flow models

Effective relaxation schemes for saddle point problems

Scalable performance in 2D and 3D simulations

Abstract

Most research on preconditioners for time-dependent PDEs has focused on implicit multi-step or diagonally-implicit multi-stage temporal discretizations. In this paper, we consider monolithic multigrid preconditioners for fully-implicit multi-stage Runge-Kutta (RK) time integration methods. These temporal discretizations have very attractive accuracy and stability properties, but they couple the spatial degrees of freedom across multiple time levels, requiring the solution of very large linear systems. We extend the classical Vanka relaxation scheme to implicit RK discretizations of saddle point problems. We present numerical results for the incompressible Stokes, Navier-Stokes, and resistive magnetohydrodynamics equations, in two and three dimensions, confirming that these relaxation schemes lead to robust and scalable monolithic multigrid methods for a challenging range of incompressible fluid-flow models.