Entropy-Bounded Discontinuous Galerkin Scheme for Euler Equations

TL;DR

This paper introduces an entropy-bounded Discontinuous Galerkin scheme that stabilizes solutions near discontinuities while maintaining high accuracy for smooth flows, applicable to complex high-dimensional problems.

Contribution

The paper develops a novel entropy-bounded DG scheme with proven stability, optimal CFL conditions, and flexible entropy constraints for multiple discontinuities, suitable for high-order elements and complex meshes.

Findings

The scheme effectively stabilizes solutions near discontinuities.

It retains optimal accuracy for smooth solutions.

Numerical tests demonstrate robustness on arbitrary meshes.

Abstract

An entropy-bounded Discontinuous Galerkin (EBDG) scheme is proposed in which the solution is regularized by constraining the entropy. The resulting scheme is able to stabilize the solution in the vicinity of discontinuities and retains the optimal accuracy for smooth solutions. The properties of the limiting operator according to the entropy-minimum principle are proofed analytically, and an optimal CFL-criterion is derived. We provide a rigorous description for locally imposing entropy constraints to capture multiple discontinuities. Significant advantages of the EBDG-scheme are the general applicability to arbitrary high-order elements and its simple implementation for two- and three-dimensional configurations. Numerical tests confirm the properties of the scheme, and particular focus is attributed to the robustness in treating discontinuities on arbitrary meshes.