On the Convergence of Monolithic Multigrid for Implicit Runge–Kutta Time Stepping of Finite Element Problems

Abstract

SIAM Journal on Scientific Computing, Ahead of Print.
Abstract. Finite element discretizations of time-dependent problems also require effective time-stepping schemes. While implicit Runge–Kutta methods provide favorable accuracy and stability properties, they give rise to large and complicated systems of equations to solve for each time step. These algebraic systems couple all Runge–Kutta stages together, giving a much larger system than for single-stage methods. We consider an approach to these systems based on monolithic smoothing. If stage-coupled smoothers possess a certain kind of structure, then the question of convergence of a two-grid or multigrid iteration reduces to convergence of a related strategy for a single-stage system with a complex-valued time step. In addition to providing a general theoretical approach to the convergence of monolithic multigrid methods, several numerical examples are given to illustrate the theory and show how higher-order Runge–Kutta methods can be made effective in practice. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and Data Available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/rckirby/CodeForMMGPaper as well as in the supplemental material.