Numerical Study of Dissipative Weak Solutions for the Euler Equations of Gas Dynamics

TL;DR

This paper investigates dissipative weak solutions of the Euler equations in gas dynamics using high-order numerical schemes, analyzing their convergence, dependence on schemes, and properties via Young measures and entropy criteria.

Contribution

It introduces a comprehensive numerical analysis framework for dissipative weak solutions using advanced high-order schemes and compares their properties through Young measures and entropy measures.

Findings

Higher-order schemes converge to generalized dissipative weak solutions.

Solutions depend on the numerical scheme used, affecting their properties.

Young measures and entropy criteria effectively characterize solution differences.

Abstract

We study dissipative weak (DW) solutions of the Euler equations of gas dynamics using the first-, second-, third-, fifth-, seventh-, and ninth-order local characteristic decomposition-based central-upwind (LCDCU), low-dissipation central-upwind (LDCU), and viscous finite volume (VFV) methods, whose higher-order extensions are obtained via the framework of the alternative weighted essentially non-oscillatory (A-WENO) schemes. These methods are applied to several benchmark problems, including several two-dimensional Riemann problems and a Kelvin-Helmholtz instability test. The numerical results demonstrate that for methods converging only weakly in space and time, the limiting solutions are generalized DW solutions, approximated in the sense of ${\cal K}$-convergence and dependent on the numerical scheme. For all of the studied methods, we compute the associated Young measures and compare the DW solutions using entropy production and energy defect criteria.