Discrete compatibility in finite difference methods for viscous incompressible fluid flow

Thom's vorticity condition for solving the incompressible Navier-Stokes equations is generally known as a first-order method since the local truncation error for the value of boundary vorticity is first-order accurate. In the present paper, it is shown that convergence in the boundary vorticity is actually second order for steady problems and for time-dependent problems when t > 0. The result is proved by looking carefully at error expansions for the discretization which have been previously used to show second-order convergence of interior vorticity. Numerical convergence studies confirm the results. At t = 0 the computed boundary vorticity is first-order accurate as predicted by the local truncation error. Using simple model problems for insight we predict that the size of the second-order error term in the boundary condition blows up like C/root t as t --> 0. This is confirmed by careful numerical experiments. A similar phenomenon is observed for boundary vorticity computed using a primitive method based on the staggered marker-and-cell grid. (C) 1996 Academic Press, Inc.