Analysis and entropy stability of the line-based discontinuous Galerkin method

TL;DR

This paper introduces a computationally efficient line-based discontinuous Galerkin method that ensures entropy stability for hyperbolic conservation laws, improving robustness especially for discontinuous solutions across multiple dimensions.

Contribution

The paper develops a novel line-based DG method that guarantees discrete entropy stability using flux differencing, reducing computational cost compared to standard DG methods.

Findings

Method is entropy-stable and robust for discontinuous solutions.

Demonstrates effectiveness on Burgers' and Euler equations.

Applicable in 1D, 2D, and 3D cases.

Abstract

We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers' equation and the Euler equations, in one, two, and three spatial dimensions.