Adaptive radial basis function methods for the numerical solution of partial differential equations, with application to the simulation of the human tear film.

摘要

This thesis deals with meshless adaptive numerical methods for solving partial differential equations. Specifically, the type of meshless method used is the radial basis function (RBF) method. We did numerous numerical experiments, built the algorithm from scratch, analyzed it, and tested it on some common test problems. The emphasis of this thesis is on computations and not on proofs.;During our journey, we found many interesting results and ideas. Our goal in developing the adaptive RBF based method is to use it in the future as a method for solving two dimensional tear film equations in a blink cycle. The problem is challenging involving complex moving geometries, fourth-order nonlinear PDEs, and nontrivial boundary conditions. To get some insight, we begin with the one dimensional versions and solve them with spectral collocation methods. In the one dimensional case, we are able to compare them with data from in vivo observations.;Our first experiment regarding the adaptive RBF method started with one dimensional adaptive interpolation problem. We found out that the use of variable RBF shape parameters is substantial. Extending the method to handle more general problems including time-independent and time-dependent problems in one and two dimensions is straightforward. The method can be extended even more to the generalized adaptive finite difference method with no need of special stencils. This can overcome the ill-conditioning issue that is found in the adaptive global RBF methods when applied to problems that exhibit very steep slopes.