Radial basis function (RBF) methods are a fairly new method for the numerical solution of PDEs. When compared to existing numerical PDE schemes, RBF methods are very accurate, easy to implement, and extremely flexible. The methods are grid free which allow them to be easily implemented in very complex geometries.

Despite showing extreme promise, several issues must be resolved before the RBF methods reach their potential. The issues include ill-conditioning, the presence of relatively large boundary region errors, and efficient implementation. In this talk, we discuss strategies for coping with the ill-conditioning problem and a method to reduce boundary region errors in PDE problems. Successfully dealing with the ill-conditioning problem and suppressing boundary region errors has a dramatic positive effect on the accuracy of RBF methods for derivative approximation. Numerical results are given for time dependent PDE problems.

Spring 2003 Colloquium Schedule | Mathematical Sciences | The University of Montana