[Paper Review] Entropy stable discontinuous Galerkin approximation for the Relativistic Hydrodynamic Equations.

Biswarup Biswas, Harish Kumar|arXiv (Cornell University)|Nov 18, 2019

Computational Fluid Dynamics and Aerodynamics4 references4 citations

TL;DR

This paper proposes a fourth-order entropy stable discontinuous Galerkin scheme for the special relativistic hydrodynamic equations using an entropy conservative numerical flux. The method ensures stability and reduces oscillations compared to lower-order schemes, with bound-preserving limiting maintaining physical solution validity, validated through extensive numerical experiments.

ABSTRACT

This paper presents the higher-order discontinuous Galerkin entropy stable schemes for special relativistic hydrodynamic equations. A suitable entropy conservative flux is used to construct the scheme. It is studied that the presented fourth-order scheme provides less oscillatory approximation than the third-order scheme. Bound preserving limiter is used to keep the computed solution in the physical domain. Extensive numerical results are presented to validate the accuracy and robustness of the schemes.

Motivation & Objective

• To develop high-order entropy stable discontinuous Galerkin schemes for special relativistic hydrodynamics.
• To improve numerical stability and reduce oscillations in the solution compared to lower-order schemes.
• To ensure the computed solution remains within the physical domain using a bound-preserving limiter.
• To validate the accuracy and robustness of the scheme through extensive numerical results.

Proposed method

• A suitable entropy conservative numerical flux is constructed to ensure entropy stability in the discontinuous Galerkin framework.
• A fourth-order accurate spatial discretization is implemented using high-order polynomial approximations in each element.
• The scheme employs a bound-preserving limiter to enforce physical bounds on the solution variables.
• The method is formulated in a weak form using numerical integration and local L2 projections.
• The entropy stability is enforced by ensuring the discrete entropy inequality holds element-wise.
• Numerical fluxes are constructed to preserve entropy conservation in the continuous limit.

Experimental results

Research questions

• RQ1 Can a high-order discontinuous Galerkin scheme be constructed to be entropy stable for the special relativistic hydrodynamic equations?
• RQ2 How does the fourth-order scheme compare to the third-order scheme in terms of oscillation suppression and solution accuracy?
• RQ3 Can the bound-preserving limiter effectively maintain the solution within the physical domain without compromising accuracy?
• RQ4 What is the robustness and accuracy of the scheme in resolving complex relativistic flow features?

Key findings

• The fourth-order entropy stable discontinuous Galerkin scheme produces less oscillatory solutions than the third-order scheme.
• The entropy conservative flux ensures that the discrete scheme satisfies the entropy inequality, enhancing numerical stability.
• The bound-preserving limiter successfully maintains the solution within the physical domain for all test cases.
• Extensive numerical results confirm the high accuracy and robustness of the proposed scheme across a range of relativistic hydrodynamic problems.