Essentially compact schemes for unsteady viscous incompressible flows

Abstract: A new 4th 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 4th order scheme in resolving both the boundary layers and the gross features of the flow. This efficient 4th order scheme also made it possible to compute the driven cavity flow at Reynolds number 106 on a 10242 grid at a reasonable cost. 4th order convergence is proved under mild regularity requirements. This is the first such result to our knowledge. Compact schemes have attracted a considerable amount of attention recently in the area of viscous incompressible flow calculations. Ideally these schemes offer two attractiv