Jacobian-free Multigrid Preconditioner for Discontinuous Galerkin Methods applied to Numerical Weather Prediction

TL;DR

This paper introduces a Jacobian-free multigrid preconditioner tailored for high order Discontinuous Galerkin methods in Numerical Weather Prediction, enhancing solver efficiency for stiff, low Mach number flows.

Contribution

It develops a novel Jacobian-free multigrid preconditioner for high order DG discretizations, enabling efficient, parallelizable implicit solutions in weather modeling.

Findings

Improved solver performance on atmospheric flow problems.

Effective mass conservative grid mapping.

Parallelizable explicit Runge-Kutta smoothers.

Abstract

Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in $z$-direction and low Mach numbers, making the system stiff. Thus, implicit time integration is required and for this a fast, highly parallel, low-memory iterative solver for the resulting algebraic systems. As a basic framework, we use inexact Jacobian-Free Newton-GMRES with a preconditioner. For low order finite volume discretizations, multigrid methods have been successfully applied to steady and unsteady fluid flows. However, for high order DG methods, such solvers are currently lacking. %The lack of efficient solvers suitable for contemporary computer architectures inhibits wider adoption of DG methods. This motivates our research to construct a Jacobian-free precondtioner for high order DG discretizations. The preconditioner is based on a multigrid method constructed for a low order finite volume discretization defined on a subgrid of the DG mesh. We design a computationally efficient and mass conservative mapping between the grids. As smoothers, explicit Runge-Kutta pseudo time iterations are used, which can be implemented in parallel in a Jacobian-free low-memory manner. We consider DG Methods for the Euler equations and for viscous flow equations in 2D, both with gravity, in a well balanced formulation. Numerical experiments in the software framework DUNE-FEM on atmospheric flow problems show the benefit of this approach.