Overview
The study focuses on the numerical investigation of dissipative weak (DW) solutions pertinent to the Euler equations governing gas dynamics. This research employs a range of numerical methods, specifically the local characteristic decomposition-based central-upwind (LCDCU), low-dissipation central-upwind (LDCU), and viscous finite volume (VFV) methods. These methods' higher-order extensions are developed using the alternative weighted essentially non-oscillatory (A-WENO) schemes framework.
Approach
The methodology involved applying first-, second-, third-, fifth-, seventh-, and ninth-order LCDCU, LDCU, and VFV methods. These schemes were utilized to address several benchmark problems. Among the chosen benchmark problems were multiple two-dimensional Riemann problems and a Kelvin-Helmholtz instability test. The numerical results obtained from these applications formed the basis for the study's conclusions.
Specifically, the computational process included calculating the associated Young measures for each of the methods investigated. Subsequently, the DW solutions derived from these methods were compared. This comparison was facilitated through the application of entropy production and energy defect criteria.
Findings
The numerical results indicated that for methods exhibiting only weak convergence in both space and time, the resultant limiting solutions correspond to generalized DW solutions. These solutions were observed to be approximated in the sense of $\mathcal{K}$-convergence.
A further finding from this study is that these generalized DW solutions exhibit a dependency on the specific numerical scheme employed. This suggests variability in the characteristics of the limiting solutions based on the method used for approximation.
The comparison of DW solutions using entropy production and energy defect criteria was conducted for all studied methods.