Maximum principle preserving and entropy stable time implicit DGSEM for nonlinear scalar conservation laws

This work concerns the analysis of the discontinuous Galerkin spectral element method (DGSEM) with implicit time stepping for the numerical approximation of nonlinear scalar conservation laws in multiple space dimensions. First we consider the DGSEM with a backward Euler time stepping, then a space-time DGSEM discretization to remove the restriction on the time step. We design first-order graph viscosities in space, and in time for the space-time DGSEM, to make the schemes maximum principle preserving and entropy stable for every admissible convex entropy. We also establish well-posedness of the discrete problems by showing existence and uniqueness of the solutions to the nonlinear implicit algebraic relations that need to be solved at each time step. We then use these low-order schemes as building blocks to design a two-step limiter that successfully captures the physical solution, imposes the maximum principle and entropy stability for any convex entropy imposed by the user on the high-order DGSEM scheme, while keeping its accuracy in smooth regions. These properties hold at any approximation order in space and time and without any constraint on the time step. Numerical experiments in one and two space dimensions are presented to illustrate the properties of these schemes.