Maximum principle preserving time implicit DGSEM for nonlinear scalar conservation laws

TL;DR

This paper develops maximum principle preserving and entropy stable implicit DGSEM schemes for nonlinear scalar conservation laws, ensuring well-posedness and demonstrating their effectiveness through numerical experiments.

Contribution

It introduces graph viscosities in space and time for implicit DGSEM to achieve maximum principle preservation and entropy stability in multiple dimensions.

Findings

Schemes are maximum principle preserving and entropy stable.

Existence and uniqueness of solutions are established.

Numerical experiments confirm theoretical properties.

Abstract

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. We consider either the DGSEM with a backward Euler time stepping, or a space-time DGSEM discretization to remove the restriction on the time step. We design 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. Numerical experiments in one space dimension are presented to illustrate the properties of these schemes.