Stability of Overintegration Methods for Nodal Discontinuous Galerkin Spectral Element Methods

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 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 destabilizing errors associated with the volume, but introduces new errors at the surfaces that can also 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. (J Comput Phys 299:56–81, 2015) who found that fully consistent integration was more robust for the solution of compressible flows than the volume only version.