High fidelity fluid simulations have important applications in science and engineering, examples include numerical weather prediction and simulation aided design. Discontinuous Galerkin (DG) methods are promising high order discretizations for simulating unsteady compressible 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 successfully applied to steady and unsteady fluid flows. But for high order DG methods applied to flow 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 preconditioner 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.
Birken et al. (Mon,) studied this question.