High Resolution, Entropy-Consistent Scheme Using Flux Limiter for Hyperbolic Systems of Conservation Laws

Abstract

Existing entropy-consistent Euler flux avoids spurious oscillations and exactly preserves the stationary contact discontinuity but still leaves much room for further improvement in resolution and other applications. In this spirit, we propose a new high resolution entropy-consistent scheme to track discontinuities in hyperbolic systems of conservation laws. The new high resolution scheme, termed ECL/EC2L scheme, is based on two main ingredients: (1) the entropy-consistent flux, and (2) suitable flux limiter. And the same as entropy-consistent flux function, we obtain a high resolution entropy-consistent flux function (ECL-M-M/EC2L-M-M) precisely satisfies the discrete second law of thermodynamics. Several numerical simulations of the ECL/EC2L scheme have been tested on one-dimensional test cases. For Burgers equations, the ECL-M-M scheme exactly model rarefaction with a stationary shock and compression wave, and the numerical results are comparable to second order entropy consistent scheme. The second kind of case is the Euler equations with different initial value problems. The numerical results such as height, density, velocity and pressure are analyzed and then compared with the second order entropy-consistent scheme. The third kind of case is the shallow water equations with different kinds of dam break. Those excellent numerical results show the desired resolution and robustness of our ECL-M-M/EC2L-M-M scheme. Moreover, the ECL-M-M/EC2L-M-M flux is completely shock stable which will use to avoid multi-dimensional shock instability, particularly the carbuncle phenomenon.