A new finite element implicit scheme is constructed by using the Hybrid Finite Difference-Finite Element Method in. The implicit scheme overcomes the diffculty of solving large sparse matrix and demanding huge capacity in traditional finite element implicit schemes and at the same time, makes use of the approximate factorization and diagonalization techniques in finite difference method to acquire high calculation efficiency.

The present paper investigates the problems of improving the numerical accuracy and the long-time computation for two-dimensional vortex methods, which are used to simulate the time-developing process of a unsteady separated flow.

A class of second-order explicit and implicit symmetric total variation diminishing (TVD) schemes is constructed for the computation of weak solutions of hyperbolic conservation laws.The resulting scheme can be viewed as a limited anti-diffusive flux, that is centered weighted flux, to add to a first-order GCIR scheme. Numerical experiments for solving the Euler equations for capturing shock waves show that the symmetric TVD schemes are quite robust and accurate.