A Multigrid preconditioner for Discontinuous Galerkin Methods Applied to Numerical Weather Prediction

Abstract

High fidelity fluid simulations have important applications in science and engineer ing, examples include numerical weather prediction and simulation aided design. Discontinuous Galerkin (DG) methods are promising high order discretizations for simulating unsteady com pressible fluid flow in three dimensions. Systems arising from such discretizations are often stiff and require implicit time integration. This motivates the study of fast, parallel, low-memory solvers for the resulting algebraic equation systems. For (low order) finite volume (FV) discretizations, multigrid (MG) methods have been suc cessfully applied to steady and unsteady fluid flows. But for high order DG methods applied to f low problems, 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 precondi tioner 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. Numerical experiments on atmospheric flow problems show the benefit of this approach.

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