The Study Of Construction And Convergence Of Difference Schemes For Fluid Dynamics Equations

In this paper, a class of the second order accurate explicit Gauss schemes with staggered grids for the computation of solutions of hyperbolic conservation laws are presented, the advantages of these schemes are: Riemann solver-free, faster and programming is much simple, no complete set of eigenvectors is needed and hence weakly hyperbolic system can be solved. In one dimensional case, these schemes are and total variation diminishing and convergence under the restriction of CFL condition, the convergence rate is the first order, and a pointwise error bound is presented. In the multidimensional, these schemes are Maximum and Minimum Bounds under the restriction of CFL condition. These schemes are extended to system of hyperbolic conservation laws. The numerical solutions obtained in computation of Riemann problem are satisfied.Hyperbolic conservation laws with stiff source terms could describe the effect of relaxation as in the kinetic theory of gases, water waves and traffic flows, etc. The Gauss schemes with staggered grids for hyperbolic conservation laws are applied to solve hyperbolic conservation laws with stiff source terms, a class high resolution schemes for hyperbolic conservation laws with stiff source terms are presented. These schemes are the second order accurate and TVD under the restriction of CFL condition, convergence of these schemes are proved. The numerical solutions obtained in computation of Riemann problem are satisfied.Hamilton-Jacobi (HJ) equations are frequently encountered in applications,e. g. , in differential games, and control theory, are closely related to hyperbolic conservation laws. This is helpful for us to construct difference approximation schemes for HJ equation with aids of schemes for conservation laws. In this paper we present a class GAUSS schemes with staggered grids for HJ equation, based on GAUSS schemes with staggered grids for conservation laws. The schemes are numerically tested on a variety of ID and 2D problems; solutions obtained in computationThe results of the numerical solutions for Euler equation and shallow water equations by GAUSS schemes presented in the paper under distributed memory parallel multiprocessor are presented. The efficient of parallel computation are satisfied.