The Method of Fundamental Solutions for the Solution of Nonlinear Plane Potential Problems

The method of fundamental solutions is described for the solutionof elliptic boundary value problems governed by Laplace's equationin the plane subject to nonlinear radiation-type boundary conditions.The effectiveness of the method is demonstrated by examiningits performance on two problems from the literature, and comparisonsare made with published results obtained using boundary elementmethods. Portions of this work were conducted while this author wasa visiting assistant professor at the University of Kentucky. Partially supported by the National Science Foundation undergrant MCS-8303287 and grant RII-8610671, and by the Commonwealthof Kentucky through the Kentucky EPSCoR Program.     

Conforming Chebyshev Spectral Collocation Methods for the Solution of Laminar flow in a Constricted Channel

The numerical simulation of steady planar two-dimensional laminarflow of an incompressible fluid through an abruptly contractingchannel using spectral domain decomposition methods is described.The key features of the method are the decomposition of theflow region into a number of rectangular subregions and spectralapproximations that are pointwise C¹ continuous across subregioninterfaces. Spectral approximations to the solution are obtainedfor Reynolds numbers in the range [0, 500]. The size of thesalient corner vortex decreases as the Reynolds number increasesfrom 0 to around 45. As the Reynolds number is increased further,the vortex grows slowly. A vortex is detected downstream ofthe contraction at a Reynolds number of around 175 that continuesto grow as the Reynolds number is increased further.