Essentially compact schemes for unsteady viscous incompressible flows

A new fourth-order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced, The scheme is based on the vorticity-stream function formulation. It is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions, it is also very efficient, at every time step or Runge-Kutta stage, only two Poisson-like equations have to be solved, The Poisson-like equations are amenable to standard fast Poisson solvers usually designed for second order schemes, Detailed comparison with the second-order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow. This efficient fourth-order scheme also made it possible to compute the driven cavity flow at Reynolds number 10(6) on a 1024(2) grid al a reasonable cost. Fourth-order convergence is proved under mild regularity requirements, This is the first such result to our knowledge. (C) 1996 Academic Press, Inc.