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Mapped Hybrid Central-WENO Finite Difference Scheme for Detonation Waves Simulations

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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.

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Acknowledgements

The authors would like to acknowledge the funding support of this research by the RGC grant HKBU-200910 from Hong Kong University Grant Council. The first author also acknowledges the funding support of this research by the Fundamental Research Funds for the Central Universities (201213009) and Natural Science Foundation of Shandong Province (ZR2012AQ003). The second author would like to thank the School of Mathematical Sciences of Ocean University of China for hosting his visit while part of the research was conducted during the visit.

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Authors and Affiliations

  1. Key Laboratory of Marine Environment & Ecology, Ministry of Education, and School of Mathematical Sciences, Ocean University of China, Qingdao, China

    Zhen Gao

  2. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China

    Wai Sun Don

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  1. Zhen Gao
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  2. Wai Sun Don
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Correspondence to Wai Sun Don.

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Gao, Z., Don, W.S. Mapped Hybrid Central-WENO Finite Difference Scheme for Detonation Waves Simulations. J Sci Comput 55, 351–371 (2013). https://doi.org/10.1007/s10915-012-9635-2

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  • DOI: https://doi.org/10.1007/s10915-012-9635-2

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