# Efficient discontinuous Galerkin implementations and preconditioners for implicit unsteady compressible flow simulations

## TL;DR

This paper compares high-order discontinuous Galerkin methods and introduces a p-multigrid preconditioner for efficient implicit unsteady compressible flow simulations, demonstrating improved scalability and reduced memory usage.

## Contribution

It presents a matrix-free DG implementation, a hybridizable DG approach, and a primal formulation, along with a p-multigrid preconditioner that enhances efficiency for stiff flow problems.

## Key findings

- p-multigrid preconditioner improves convergence for stiff systems
- Matrix-free implementation reduces memory footprint
- Preconditioner shows excellent parallel scalability

## Abstract

This work presents and compares efficient implementations of high-order discontinuous Galerkin methods: a modal matrix-free discontinuous Galerkin (DG) method, a hybridizable discontinuous Galerkin (HDG) method, and a primal formulation of HDG, applied to the implicit solution of unsteady compressible flows. The matrix-free implementation allows for a reduction of the memory footprint of the solver when dealing with implicit time-accurate discretizations. HDG reduces the number of globally-coupled degrees of freedom relative to DG, at high order, by statically condensing element-interior degrees of freedom from the system in favor of face unknowns. The primal formulation further reduces the element-interior degrees of freedom by eliminating the gradient as a separate unknown. This paper introduces a $p$-multigrid preconditioner implementation for these discretizations and presents results for various flow problems. Benefits of the $p$-multigrid strategy relative to simpler, less expensive, preconditioners are observed for stiff systems, such as those arising from low-Mach number flows at high-order approximation. The $p$-multigrid preconditioner also shows excellent scalability for parallel computations. Additional savings in both speed and memory occur with a matrix-free/reduced version of the preconditioner.