Stability of Overintegration Methods for Nodal Discontinuous Galerkin Spectral Element Methods

摘要

We perform stability analyses for discontinuous Galerkin spectral elementapproximations of linear variable coefficient hyperbolic systems in threedimensional domains with curved elements. Although high order, the precision ofthe quadratures used are typically too low with respect to polynomial orderassociated with their arguments, which introduces aliasing errors that candestabilize an approximation, especially when the solution is underresolved. Weshow that using a larger number of points in the volume quadrature, oftencalled "overintegration", can eliminate the aliasing term associated with thevolume, but introduces new aliasing errors at the surfaces that can destabilizethe solution. Increased quadrature precision on both the volume and surfaceterms, on the other hand, leads to a stable approximation. The results supportthe findings of Mengaldo et al. [Dealiasing techniques for high-order spectralelement methods on regular and irregular grids. Journal of ComputationalPhysics, 299:56 -- 81, 2015] who found that fully consistent integration wasmore robust for the solution of compressible flows than the volume onlyversion.