Stability of Overintegration Methods for Nodal Discontinuous Galerkin Spectral Element Methods

TL;DR

This paper analyzes the stability of overintegration methods in nodal discontinuous Galerkin spectral element methods, showing that increased quadrature precision in both volume and surface integrals enhances stability, especially for high-order, underresolved solutions.

Contribution

The study demonstrates that fully consistent quadrature in both volume and surface integrals improves stability in high-order DG spectral element methods, extending previous findings on dealiasing techniques.

Findings

Overintegration reduces aliasing errors in volume integrals.

Full quadrature consistency stabilizes the solution for underresolved problems.

Surface aliasing errors can destabilize solutions if not properly addressed.

Abstract

We perform stability analyses for discontinuous Galerkin spectral element approximations of linear variable coefficient hyperbolic systems in three dimensional domains with curved elements. Although high order, the precision of the quadratures used are typically too low with respect to polynomial order associated with their arguments, which introduces aliasing errors that can destabilize an approximation, especially when the solution is underresolved. We show that using a larger number of points in the volume quadrature, often called "overintegration", can eliminate the aliasing term associated with the volume, but introduces new aliasing errors at the surfaces that can destabilize the solution. Increased quadrature precision on both the volume and surface terms, on the other hand, leads to a stable approximation. The results support the findings of Mengaldo et al. [Dealiasing techniques for high-order spectral element methods on regular and irregular grids. Journal of Computational Physics, 299:56 -- 81, 2015] who found that fully consistent integration was more robust for the solution of compressible flows than the volume only version.