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.
Highlights
In physics, shock waves are small transition layers of abrupt change of physical states such as density, pressure or temperature
This study presented the implementation of some typical finite difference schemes solving the compressible Euler equations
2.3.2 First-order Upwind Flux Vector Splitting. Another approach of the upwind method is flux vector splitting (FVS), which is based on the special property of the Euler equations, namely the homogeneity property
Summary
Shock waves are small transition layers of abrupt change of physical states such as density, pressure or temperature. 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 Flux-limited method, Flux-corrected method. While the basic methods have the linear numerical dissipation and distributed evenly for all grid points, which makes it not possible to capture strong shock, whereas the modern. SCIENCE & TECHNOLOGY DEVELOPMENT, Vol., No.K2 - 2015 methods can adaptively distribute the non-linear numerical dissipation to each grid point, the shock waves can be moderately captured. Drikakis [17] have studied the propagation of plane blast wave over a cylinder by solving the Euler equations and the Navier-Stokes equations with an adaptive-grid method and a second-order Godunov scheme. As categorized in [1], the numerical methods will be studied are the Flux Difference scheme, whose Roe’s Approximate Riemann Solver [9] is the representative; the Flux Vector Splitting scheme, whose Steger-Warming method [10] is the representative; the Flux Limited Method, whose TVD Sweby’s method [11] is the representative and the Flux-Corrected Method whose Essentially Non-Oscillatory Scheme [12], [13] is the representative
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