A Simple Cellwise High-order Implicit Discontinuous Galerkin Scheme for Unsteady Turbulent Flows

Abstract

A simple cellwise implicit time integration scheme for high-order discontinuous Galerkin (DG) methods is presented for solving unsteady turbulent flows. One motivation of this study is to utilize the advantages of the cellwise feature of DG methods in implicit time integration to accurately predict the time evolution of unsteady turbulent flows. Our approach is to extend a block Jacobi (BJ) scheme to high-order implicit DG schemes. With the BJ scheme, the cellwise feature is ensured because sweeps referring to up-to-date solutions in nearby cells are not required, and accurate time evolution can be simulated by iteratively solving a linear system formulated in implicit time integration. Although the computational cost required for iterative solutions is of some concern, we found that few sub-iterations are needed to simulate unsteady turbulent flows accurately when the flowfield is well-resolved by high-order DG methods. These advantages are demonstrated through problems such as canonical vortex advection and the inviscid Taylor–Green vortex. The developed BJ-based implicit DG methods were applied to a large eddy simulation of turbulent channel flow, and the results show that, in addition to being a cellwise scheme, the proposed scheme outperforms explicit time integration in terms of the computation time.