We consider the two-dimensional Navier-Stokes equations in pure streamfunction formulation. This formulation has the advantage that the differential equation as well as the boundary conditions and the initial condition are formulated in terms of the streamfunction alone. In [1] we provided a compact fourth-order scheme for this system for a rectangle domain in two dimensions. Here we are interested in approximating the same problem, but now in a domain with complex geometry. To do this, we embed the computational domain in a square and lay out a uniform Cartesian grid. We have grid points of several categories. Some points lay outside the physical domain, thus neither the differential equation nor the boundary conditions are implemented for these points. For points laying on the boundary of the domain we apply only the boundary conditions. Points which are inside the physical domain are called computational points. For these points (unless they are too close to the boundary) we apply only the differential equation. Most of the computational points are in the center of a regular element, formed by the rectangular Cartesian grid. For these points we apply the high-order scheme suggested in [1]. Some of the computational points are in the center of an irregular element. Those points are close to the boundary (but not too close). For these points we approximate the spatial derivatives on an irregular mesh. Pure derivatives are approximated via one-dimensional irregular discretizations. The mixed derivatives are written in terms of pure derivatives along the Cartesian coordinates and along the diagonals. Then, these pure derivatives are approximated via one-dimensional approximations on an irregular grid. Numerical results demonstrate the high-order accuracy of the suggested scheme. [1] D. Fishelov, M. Ben-Artzi and J.P. Croisille, “A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations, http://www.springerlink.com/openurl.asp?genre=article id=doi:10.1007/s10915-009-9322-0"J. Scientific Computing, Vol. 42, Issue 2. pp. 216--250, 2010.
