A Robust Layer { Resolving Numerical Method for a Freeconvection

We consider free convection near a semi-innnite vertical at plate. This problem is singularly perturbed with perturbation parameter Gr, the Grashof number. Our aim is to nd numerical approximations of the solution in a bounded domain, which does not include the leading edge of the plate, for arbitrary values of Gr 1. Thus, we need to determine values of the velocity components and temperature with errors that are Gr{independent. We use the Blasius approach to reformulate the problem in terms of two coupled non-linear ordinary diierential equations on a semi{ innnite interval. A novel iterative numerical method for the solution of the transformed problem is described and numerical approximations are obtained for the Blasius solution functions, their derivatives and the corresponding physical velocities and temperature. The numerical method is Gr{ uniform in the sense that error bounds of the form C p N ?p , where C p and p are independent of the Gr, are valid for the interpolated numerical solutions. The numerical approximations are therefore of controllable accuracy. A free or natural convection ow occurs when a uid at rest, subjected to a body force such as gravity, is near an object at a diierent temperature. The heat transfer between the object and the uid causes an increase or a decrease in the uid density at the surface of the object, and thus generates an unbalancing body force. The uid near the surface is accelerated, and a boundary layer develops. We study this problem for a two-dimensional, steady ow near a semi-innnite at plate. This involves an interesting and typical system of singularly perturbed partial diierential equations.