Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow
arXiv:2202.07381 · doi:10.1016/j.jcp.2023.111961
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.
arXiv comment22 pages, 9 figures. Submitted to Journal of Computational Physics on Feb 14 2022, updated to match the final published version in Journal of Computational Physics fixing any typos and adding references
References in corpus (7)
- Firedrake: automating the finite element method by composing abstractions
- An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number
- Stage-parallel fully implicit Runge-Kutta solvers for discontinuous Galerkin fluid simulations
- Solver composition across the PDE/linear algebra barrier
- A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations
- Efficient mesh management in Firedrake using PETSc-DMPlex
- PCPATCH: software for the topological construction of multigrid relaxation methods