Non-oscillatory entropy stable DG schemes for hyperbolic conservation law

TL;DR

This paper introduces a novel class of non-oscillatory, entropy-stable discontinuous Galerkin schemes for hyperbolic conservation laws, effectively controlling entropy and oscillations while maintaining high accuracy.

Contribution

The paper develops a new entropy-stable DG scheme with artificial viscosity and advanced time integration, improving stability and accuracy near discontinuities.

Findings

Effective suppression of spurious oscillations.

Maintains high-order accuracy in smooth regions.

Compatible with positivity-preserving limiters.

Abstract

In this paper, we propose a class of non-oscillatory, entropy-stable discontinuous Galerkin (NOES-DG) schemes for solving hyperbolic conservation laws. By incorporating a specific form of artificial viscosity, our new scheme directly controls entropy production and suppresses spurious oscillations. To address the stiffness introduced by the artificial terms, which can restrict severely time step sizes, we employ the integration factor strong stability-preserving Runge-Kutta method for time discretization. Furthermore, our method remains compatible with positivity-preserving limiters under suitable CFL conditions in extreme cases. Various numerical examples demonstrate the efficiency of the proposed scheme, showing that it maintains high-order accuracy in smooth regions and avoids spurious oscillations near discontinuities.