Application of radial basis functions method to advection-diffusion and viscoelastic flow problems

Abstract

The objective of the thesis is to demonstrate an application of radial basis functions method to linear and nonlinear advection-diffusion and viscoelastic flow problems. As far as the radial basis functions method is considered, multiquadrics and thin-plate splines types of functions are used in the study. Firstly, the study is handling the linear advection-diffusion type of equation in one and two dimensional cases with specific examples in order to compare with the analytical and the numerical solutions existing in the literature. Then, one and two dimensional Burgers’ equations and nonlinear advection-diffusion equation are solved to demonstrate the efficiency of the method. In addition to that, two models of viscoelastic flow in one dimension, Upper-Convected Maxwell and Oldroyd-B fluid models for mode one are investigated using radial basis functions collocation method considering start-up flow between parallel plates. Especially, for multiquadric radial basis functions solutions, the shape parameter effect is investigated and shape parameter optimization is carried out with the known exact solutions. It can be claimed from all implementations that this meshless radial basis functions collocation method is very easy to code, flexible with respect to high-dimensional geometries and efficient in comparison with the other methods.