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

In (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 charac-teristic-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.