Stability of Overintegration Methods for Nodal Discontinuous Galerkin Spectral Element Methods
Provides theoretical stability analysis for high-order methods in computational fluid dynamics, confirming previous empirical findings.
The paper analyzes stability of overintegration methods for discontinuous Galerkin spectral element methods, showing that overintegration of volume terms alone introduces surface aliasing errors that can destabilize the solution, while consistent integration of both volume and surface terms yields stability.
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.