On the Role of Split Formulations on Aliasing Errors and Entropy Stability of Discontinuous Galerkin Schemes

Abstract

In this work, we formally investigate the dealiasing properties of split form discontinuous Galerkin (DG) discretizations for the one-dimensional Burgers problem. By generalizing the proof of Blaisdell et al. for spectral discretizations, we show that split form DG schemes achieve dealiasing in the weakly underresolved range through integration error cancellation on the conservative and non-conservative forms. As a corollary, we identify quadrature- and order-dependent pairs of splitting coefficients that eliminate the dominant component of the aliasing error. While these ``optimized'' splitting coefficients minimize integration errors on the numerical scheme, we show that the skew-symmetric aliasing pattern resulting from the entropy stable split is required to maintain long-term stability of the numerical solution. The proposed framework provides an alternative proof for the entropy stability of the DG discretization of the (1/3, 2/3) split formulation introduced by Gassner which clarifies the connection between DG aliasing errors and entropy stability. Finally, we also show that the entropy stable split is associated with lower aliasing errors in the weakly underresolved range compared to the conservative form.