An Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Conservation Laws: Entropy Stability

In Klingenberg, Schnucke and Xia (Math. Comp. Available via https://doi.org/ 10.1090/mcom/3126) an arbitrary Lagrangian-EulerianDiscontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be briefly presented. Furthermore, the semi-discrete method will be discretized by the so-called upsilon-method. The upsilon-method is a generalization of the forward or backward Euler step. In particular, the method degenerates to the forward Euler step for upsilon = 0 and to the backward Euler step for upsilon = 1. The corresponding fully discrete upsilon-P-k-ALE-DG method for scalar conservation laws will be analyzed with respect to entropy stability, where P-k denotes the space of polynomials of degree k which is used on a reference cell. The main results are a cell entropy inequality for the fully discrete upsilon-P-k-ALE-DG method with respect to the square entropy function, when. has a lower bound given by a mesh parameter depending constant, and a cell entropy inequality for the fully discrete upsilon-P-0-ALE-DG method with respect to the Kruzkov entropy functions.