Comparison of different entropy stabilization techniques for discontinuous Galerkin spectral element methods

Abstract

We review and compare two techniques to get entropy stability for nodal Discontinuous Galerkin Spectral Element Methods (DG) in compressible flows. One technique is based on entropy split-forms, e.g., [8, 15, 5] and one is based on a direct algebraic correction [1]. We have implemented the Flux Differencing methodology for both, Legendre-Gauss-Lobatto (Lobatto) and Legendre-Gauss (Gauss) based spectral element basis functions. While the Lobatto operators belong to the class of diagonal norm summation-by-parts (SBP) operators, the Gauss operators belong to the generalized class of SBP operators, where it is not necessary that the boundary nodes are included. To reach entropy stability, respectively guaranteed entropy dissipation, a key ingredient is an entropy conserving numerical flux function. With this ingredient, only the volume integral term of the DG method has to be modified accordingly. We have also implemented an alternative technique, which is in general applicable for a wide range of discretizations. Abgrall [R. Abgrall, A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes. Journal of Computational Physics, vol 372, 2018] introduced an algebraic correction term that retains conservation of the primary quantities and is furthermore constructed such, that an entropy (in-) equality can be shown. The second technique is at first sight a simpler alternative to the split-form based approach. Hence, questions regarding its advantages and disadvantages naturally come up.