Decoupling of hyperbolic conservation laws using an improved eigenvector matrix

Abstract

The resolution of Harten’s non-oscillatory explicit second-order accurate total variation diminishing scheme was relatively high for computations of nonlinear wave equations. However, this scheme shows false dissipation for sharp discontinuities, such as shocks and contacts, when applied to 1D Euler equation. The selection of the entropy fixing parameter, flux limiter, and appropriate eigenvector matrix is crucial to control the amount of such dissipation. This study focuses on the formulation of eigenvector matrix, which is used for the decoupling of a highly coupled nonlinear 1D Euler equation into a nonlinear wave equation. The selection of the eigenvector matrix is crucial to limit the unwanted false dissipation, which is inherent to the numerical schemes derived for nonlinear wave equations. A generalized form of the scaling factor for the eigenvector matrix is then derived, which is satisfied by the scaling factors used by Harten, Hofmann, and Yee. LAX, SOD, and inverse shock test cases of the shock tube problem are solved to examine and analyze the corresponding physical aspects. The findings show that multiplying the scaling factor by itself or by a constant does not affect the results. However, the variables inside the scaling factor, as well as powers of these variables play a vital role in controlling false dissipation. These scaling factors can effectively capture physics in one case, but might not exhibit satisfactory performance for other problems. In addition, Yee’s scaling factor can effectively capture shock and contact discontinuities, especially for the inverse shock test case.