Journal of Computational Mathematics Vol.27, No.4, 2009, 441-458. http://www.global-sci.orghcm doi: 1O.4208hcm.2009.27.4.012 STABLE FOURTH-ORDER STREAM-FUNCTION METHODS FOR INCOMPRESSIBLE FLOWS WITH BOUNDARIES* Thomas Y. Hou Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA, 91125 USA Email: hou@acm.caltech.edu Brian R. Wet ton Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada Email: wetton@math.ubc.ca Abstract Fourth-order stream-function methods are proposed for the time dependent, incom- pressible N avier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the no- slip condition. Formal error analy;;is is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically. Mathematics subject classification: 65M12, 76D05. Key words: Incompressible flow, Stream-function formulation, Finite difference methods. 1. Introduction Numerical methods based on the vorticity stream-function formulation of the Navier-Stokes equations have been used with success for flows in two-dimensions (2D). There are many refer- enCes to second-order finite difference methods applied to different problems: the driven cavity problem [20]; flow past a cylinder [16]; and flow in tubes with occlusions [14] for example. Considerable effort has been spent in developing higher order finite difference methods for the vorticity stream-function formulation, including the early work in [1,7,10,11]. These authors use some combination of compact differencing and one-sided differencing near the boundary to develop fourth-order methods. More modern approaches [3,9] use the stream-function directly, without reference to vorticity. These methods also use compact differencing. In this paper, we develop a new fourth-order method for the time dependent Navier-Stokes and Boussinesq Equations that uses a wide stencil rather than compact differencing. This approach offers additional flexibility in choice of time stepping and applicability in mapped grids over other methods. Formal analysis of a simple model problem indicates that the method will give fourth- order accuracy in computed velocities. This is demonstrated for the full problem in numerical convergence studies. There are several reasons to consider higher order methods. Of course, the underlying reason is that we wish to get an approximation to a flow to a given accuracy or resolution using less computational time and less storage space. This can manifest itself in many ways. In some cases, high accurate benchmark solutions are required [11]. In other cases, the higher * Received March 11, 2008 / Revised version received May 3, 2008/ Accepted August 6, 2008 /