Pure and Applied Mathematics Quarterly

Volume 14 (2018)

Number 1

Special Issue: In Honor of Chi-Wang Shu

Guest Editors: Jian-Guo Liu and Yong-Tao Zhang

Hybrid Fourier-Continuation method and WENO-Z finite difference scheme for multi-dimensional detonation structure simulations

Pages: 27 – 55

DOI: https://dx.doi.org/10.4310/PAMQ.2018.v14.n1.a2


Peng Li (Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, China)

Zhen Gao (School of Mathematical Sciences, Ocean University of China, Qingdao, China)

Wai Sun Don (School of Mathematical Sciences, Ocean University of China, Qingdao, China)


In [27] (Li et al., J. Sci. Comput. 2015, 64: 670–695), a Hybrid FC-WENO-Z scheme (Hybrid) conjugating the Fourier-Continuation (FC) method with the improved fifth order characteristic- wise weighted essentially non-oscillatory (WENO-Z) finite difference scheme for solving the system of hyperbolic conservation laws was developed. 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 a linear, essentially non-dissipative and non-dispersive FC method. A high order multi-resolution algorithm by Harten is used for measuring the smoothness of the solutions. In this study, the Hybrid scheme is employed in the long time simulations of multi-dimensional detonation structures which contain both discontinuous and complex smooth structures for the first time. The fine scale structures behind the detonation front and the quasi-steady state cellular structures of the peak pressure in the half reaction zone are well captured. A classical stable two-dimensional detonation waves shows that an improved resolution of the more fine scale structures of detonation waves as computed by the Hybrid scheme with less CPU times when compares with the pure WENO-Z scheme. The influence of initial and boundary conditions on the formation and evolution of the detonation structures are also illustrated with examples. Finally, the in-phase rectangular, out-of-phase rectangular and in-phase diagonal cellular structures in the three-dimensional detonation simulations are shown to demonstrate the ability of the Hybrid scheme in capturing the intrinsic evolution of the detonation fronts, which are in good agreement with the published results in the literature.


weighted essentially non-oscillatory, Fourier-Continuation, multi-resolution, Hybrid, detonation wave structures

The authors would like to acknowledge the funding support of this research by National Natural Science Foundation of China (11801383, 11871443), National Science and Technology Major Project (20101010), Shandong Provincial Natural Science Foundation (ZR2017MA016) and Fundamental Research Funds for the Central Universities (201562012). The author Wai Sun Don would like also to thank the Ocean University of China for providing the startup fund (201712011) that is used in supporting this work.

Received 15 June 2017

Published 2 April 2019