In order to reduce the numerical viscosities of TVD schemes and improve their resolution, this paper suggests a new TVD scheme of Harten form with less numerical dissipation. Numerical experiments show that such a TVD scheme is better than Harten TVD scheme[1] and modified Harten scheme (Harten-Yee TVD scheme[2]) in simulating boundary layer and vortices.

Based on the theory of Sweby an implicit TVD scheme is obtained. The new method is first-order accurate with a truncation error of O The efficiency of this new scheme is demonstrated by some numerical computations of single Burgers equation and two-dimensional SNS equations.

This paper presents some preliminary work on the application of Schwarz alternating method, combined with a multigrid solver, for solving the steady 2D viscous incompressible Navier-Stokes equations. The difficulty caused by the violation of the compatibility condition imposed on the normal velocity at the boundary, which arises when subdomain problems are not solved exactly at each iteration, is surrounded by adding artificial compressiblity constants to the continuity equation each time when a subdomain problem is to be solved. These constants permit to maintain the compatibility condition of the subdomain problems and one can easily show that they tend to zero when the iteration number tends to infinity (in fact, they can be expressed explicitly as functions of the iteration number and some geometrical parameters of the computational domain). Numerical experiences have been made for flows in a channel with a facing step.

Accourding to the thought of TVD, we propose the TVD-type second-order CSCM^[1], from a siugle eqution, then extend it to system of equtions. To test the method and computer code, we compare our results for sphere with those of the reference [2]. Using our code, we simulate the supersonic viscous separated flow over indented nosetips. We compare the first-order results with the second-order results, and viscous results with inviscid results. Computational grids are generated by a simpler and faster quasi-orthogonal algebraic generation method.