Finite Point Method Of Numerical Shock Instability

This thesis is about Finite Point Method (FPM) and numerical shock instability. The main results in this paper consist of the next parts as following:1. The formulae of the second-order directional differential's extremum at an arbitrary point in a computational domain are presented for 2-D smooth function. Given three second-order directional differentials in corresponding mutual non-parallel directions, or two second-order directional differentials in corresponding non-parallel directions and their second-order mixed differential, the maximum and minimum of the second-order directional differentials and the corresponding directions are presented. A conclusion is that the maximum direction and the minimum direction are perpendicular to each other.2. Based on the second-order directional differential formulae, the second type four-point formula about the first-order directional differential are derived; the three-point formula and four-point formula in a different sense of the difference are discussed; also, a conclusion is given that the truncation error of from two to five-point formula continues to decline.3. By FPM, the simulation to the second-order elliptic PDE on a set of 2-D scattering points in non-regular domain is proposed. Several test problems and their numerical results show that the FPM has good accuracy and convergence rate. Generally, the approximation of first-order differential is almost second-order accurate and the approximation of second-order differential is almost first-order accurate.4. In order to simulate the hyperbolic conservation laws by FPM, the method of choosing neighboring points, artificial viscosity and corresponding schemes are studied. The preliminary numerical results and their conclusions are given: for FPM, the satisfying results for many problems with weak discontinuities are obtained; as to the general problems, the better viscosity and better method for choosing neighboring points in accordance with the direction of the flow field and the form of discontinuities are under consideration. 5. A hybrid method to eliminate the shock instability in 2-D Euler equations is proposed. On one hand, this paper chooses the fluxes that are free of shock instability to be used in mass equation and one of momentum equation. On the other hand, this paper chooses the fluxes that can resolve full-wave in Riemann problems and suffer from shock instability in another momentum equation and energy equation. This hybrid method does help to eliminate shock instability of the Roe solver, HLLC solver and AUSMD scheme. Furthermore, the hybrid method has a high computational efficiency and good resolution of shock.