High Order Hybrid Schemes And Its Application In Numerical Simulation Of Gas Detonation

Weighted essentially non-oscillatory?WENO?finite difference scheme is a high order/resolution numerical method,which is extensively used in the hydrodynamics and simulations of detonation.However,the issues about the robustness of the WENO scheme to solve the extreme conditional problems and it's numerical symmetry property are not studied deeply.Another disadvantage of the WENO scheme is the expensive computation,and the relative large errors of numerical dispersion and dissipation for some flow fields containing complex structures.In order to overcome the above disadvantages of the WENO scheme,we study the robustness and the numerical symmetry of the WENO scheme in deep and propose a framework of the high order hybrid scheme for solving the hyperbolic conservation laws.It is verified in the simulation for the gas detonation.This framework reduces the errors of the numerical dispersion and dissipation,and improves the efficiency compared with the standard WENO scheme to solve the problem in the flow field with complex structures.The main works are listed as follows:?1?Two important properties of the high order WENO scheme are discussed in detail.On one hand,based on the global Lax-Friedrichs flux splitting and the local characteristic projection,the robustness of the WENO scheme is verified to solve some extreme problems.On the other hand,based on the reconstruction of the polynomials,the symmetry and stability of the high order WENO scheme is improved by optimizing the local smoothness indicators₆).By combining the fifth order WENO-Z scheme,grid mapping method with the PML boundary layer technique,we conduct three di-mensional detonation waves simulations and study the propagation of the detonation waves in a square channel in detail.?2?By combining the Fourier continuous method,WENO scheme with the MR shock detection method,the hybrid high order FC-WENO scheme in a single domain is designed to solve the hyperbolic conservation laws.The main idea,construction,acceleration and the symmetry of the Fourier continuous method is discussed in detail.The hybrid FC-WENO scheme is more efficient and lower errors in the numerical dissipation,dispersion than the standard WENO scheme in the numerical simulations of the high dimensional detonation waves.The effect of the initial perturbations,boundary conditions and the parameters of the detonation on the propagation of the detonation waves are studied.?3?The hybrid compact-WENO scheme is designed by comjugating the central scheme and upwind conserved compact scheme with the WENO scheme.The positive-preserving limiters are designed for solving the Euler equations and detonation equa-tions respectively,in the framework of the hybrid compact-WENO scheme.We prove the positivity-preserving property of the hybrid scheme to guarantee the positivity of the density and pressure in the detonation equations.Moreover,the mass fraction is always constrained in[0,1].The numerical results demonstrate the high order accuracy and efficiency of the proposed hybrid method.?4?The general and iterative forms of the conjugate Fourier shock detection meth-ods are constructed based on the triangle functional polynomials.The algorithms and the computational complexity,such as MXM,EOD and CFT,are studied in de-tail.Furthermore,the conjugate Fourier discontinuous method is used in the hybrid compact-WENO scheme.The numerical results are presented to illustrate the high order accuracy and efficiency of the compact-WENO hybrid scheme,which bases on the conjugate Fourier shock detection method.