Entropy stable finite difference schemes for One-Fluid Two-Temperature Euler Non-equilibrium Hydrodynamics

TL;DR

This paper develops entropy-stable finite difference schemes for the complex One-Fluid Two-Temperature Euler equations modeling non-equilibrium hydrodynamics, ensuring numerical stability and accuracy.

Contribution

The work introduces a reformulation of the OFTT-Euler equations and constructs higher-order entropy-stable schemes using Tadmor's relation and entropy-scaled eigenvectors.

Findings

Schemes are demonstrated to be accurate and stable in 1D and 2D test cases.

Reformulation ensures the non-conservative part does not affect entropy stability.

Entropy-dissipation terms are effectively designed using entropy-scaled eigenvectors.

Abstract

In this work, we consider the One-Fluid Two-Temperature Euler (OFTT-Euler) equations used for modeling non-equilibrium hydrodynamics. The model comprises a system of nonlinear hyperbolic partial differential equations with non-conservative products. The model decomposed the total pressure into two scalar components: one for electrons and one for ions. Our aim in this work is to design entropy-stable finite difference numerical schemes for the model. This is achieved by reformulating the equations such that the reformulated non-conservative part does not contribute to the entropy. Then, we design higher-order entropy-conservative numerical schemes by using Tadmor's relation for the conservative part and higher-order central differences for the non-conservative parts. Finally, we design the entropy-dissipation terms using the entropy-scaled right eigenvectors of the conservative part, thereby deriving the entropy inequality for the entire system. We present several test cases in one and two dimensions to demonstrate the accuracy and stability of the proposed schemes.