tailieunhanh - Comparative study of numerical schemes for strong shock simulation using the Euler equations

This study presented the implementation of some typical finite difference schemes solving the compressible Euler equations. As categorized in, the numerical methods will be studied are the Flux Difference scheme, whose Roe’s Approximate Riemann Solver is the representative; the Flux Vector Splitting scheme, whose Steger-Warming method is the representative; the Flux Limited Method. | TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K2- 2015 Comparative study of numerical schemes for strong shock simulation using the Euler equations Nguyen Huy Binh1,2 Le Song Giang1 1 Ho Chi Minh City University of Technology, VNU-HCM 2 Vietnamese-German University (Manuscript Received on January 7th, 2015; Manuscript Revised May 08th, 2015) ABSTRACT A numerical study of extremely strong shocks was presented. Various types of numerical schemes with first-order accuracy and higherorder accuracy with adaptive stencils were implemented to solve the one and twodimensional Euler equations based on the explicit finite difference method, including Roe’s first-order upwind, Steger-Warming Flux Vector splitting (FVS), Sweby’s flux-limited and Essentially Non-oscillatory (ENO) scheme. The result comparisons were carried out to discuss which scheme is the most suitable for strong shock problem. The dissipative nature of the firstorder scheme can be easily seen from the numerical solutions. High order ENO scheme had the best resolution for the case having weak discontinuity, but it over- predicted the shock wave location for the case of strong discontinuity. Keywords: numerical schemes, strong shock, Euler equations 1. INTRODUCTION In physics, shock waves are small transition layers of abrupt change of physical states such as density, pressure or temperature. For engineering problems, shock waves are often observed in dam break problem or from an explosion. In order to predict and evaluate the effects of strong shocks, computational approach is a robust and low cost method to investigate the nature of the shock waves. This study concerned strong shock, . shock from an explosion, which has large discontinuities. Many robust, stable and accurate numerical methods have been developed for shock-capturing problem including some basic methods such as Lax-Friedrichs’s method, LaxWendroff’s method, MacCormark’s method Godunov’s method and some modern methods such as .