Essentially Compact Schemes for Unsteady Viscous Incompressible Flows

✍ Scribed by Weinan E; Jian-Guo Liu

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 548 KB
• Volume: 126
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.0125

✦ Synopsis

As we emphasized repeatedly in [4], a basic design principle for finite difference schemes in vorticity-stream func-A new fourth-order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced. tion formulation is to avoid coupling between the vorticity The scheme is based on the vorticity-stream function formulation. boundary condition and interior field equations. In this It is essentially compact and has the nice features of a compact regard, the nonlinear convection terms present a problem scheme with regard to the treatment of boundary conditions. It is for designing compact schemes. In [3,8], this difficulty was also very efficient, at every time step or Runge-Kutta stage, only overcome by using an appropriate change of variables.

two Poisson-like equations have to be solved. The Poisson-like equations are amenable to standard fast Poisson solvers usually de-While accomplishing the task of obtaining compact differsigned for second order schemes. Detailed comparison with the encing formulas for the convective terms, this trick also second-order scheme shows the clear superiority of this new fourthintroduces a considerable amount of complexity into the order scheme in resolving both the boundary layers and the gross scheme. This greatly limits the feasibility of these schemes.

features of the flow. This efficient fourth-order scheme also made

The main purpose of this paper is to introduce a simple it possible to compute the driven cavity flow at Reynolds number 10 6 on a 1024 2 grid at a reasonable cost. Fourth-order convergence and efficient fourth-order scheme which overcomes all is proved under mild regularity requirements. This is the first such these difficulties. One main idea is the following. Since the result to our knowledge.