Energy Bounds for Discontinuous Galerkin Spectral Element Approximations of Well-Posed Overset Grid Problems for Hyperbolic Systems

TL;DR

This paper investigates energy bounds for Discontinuous Galerkin Spectral Element Methods applied to hyperbolic systems with overset grids, introducing a novel penalty method to ensure stability and spectral convergence in one dimension.

Contribution

It introduces a new penalty method that adds dissipation within overlaps to stabilize overset grid coupling for hyperbolic problems.

Findings

Energy bounds hold only for fixed polynomial order, mesh, and time.

Coupling without dissipation can be destabilizing due to positive eigenvalues.

The proposed penalty method achieves spectral convergence with sufficient dissipation.

Abstract

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied.