Mapped Hybrid Central-WENO Finite Difference Scheme for Detonation Waves Simulations
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Mapped Hybrid Central-WENO Finite Difference Scheme for Detonation Waves Simulations Zhen Gao · Wai Sun Don
Received: 25 May 2012 / Revised: 28 July 2012 / Accepted: 9 August 2012 / Published online: 23 August 2012 © Springer Science+Business Media, LLC 2012
Abstract In this study, we employ the fifth order hybrid Central-WENO conservative finite difference scheme (Hybrid) in the simulation of detonation waves. The Hybrid scheme is used to keep the solutions parts displaying high gradients and discontinuities always captured by the WENO-Z scheme in an essentially non-oscillatory manner while the smooth parts are highly resolved by an efficient and accurate central finite difference scheme and to speedup the computation of the overall scheme. To detect the smooth and discontinuous parts of the solutions, a high order multi-resolution algorithm by Harten is used. A tangent domain mapping is used to cluster grid points near the detonation front in order to enhance the grid resolution within half reaction zone that drives the development of complex nonlinear wave structures behind the front. We conduct several numerical comparisons among the WENO-Z scheme with a uniformly spaced grid, the WENO-Z scheme and the Hybrid scheme with the domain mapping in simulations of classical stable and unstable detonation waves. One- and two-dimensional numerical examples show that the increased grid resolution in the half reaction zone by the Mapped WENO-Z scheme and the Mapped Hybrid scheme allows a significant increased efficiency and accuracy when compares with the solution obtained with a highly resolved one computed by the WENO-Z scheme with a uniformly spaced grid. Results of three-dimensional simulations of stable, slightly unstable and highly unstable detonation waves computed by the Mapped Hybrid scheme are also presented. Keywords Weighted essentially non-oscillatory · Central finite difference · Multi-resolution · Hybrid · Detonation wave
Z. Gao Key Laboratory of Marine Environment & Ecology, Ministry of Education, and School of Mathematical Sciences, Ocean University of China, Qingdao, China e-mail: [email protected] W.S. Don () Department of Mathematics, Hong Kong Baptist University, Hong Kong, China e-mail: [email protected]
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J Sci Comput (2013) 55:351–371
1 Introduction Detonation is a complex phenomenon that involves a shock front followed by a reaction zone. It usually happens inside a mine shaft or gas tubes in the kitchen and causes large property damages and loss of human lives. The detailed understanding and studies on the propagation of detonation waves has wide applications. However, direct experiments of these physical phenomenon, which exists in the most challenging and hostile environment, are difficult and safely conducted if and when possible. Accurate and efficient numerical simulations of a mathematical model of detonation waves provide a way to obtain insights in the physical problems and guide researchers to have a deeper understanding of the physics and to design better experiments. Direct nu
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