This thesis contains results on convergence studies for different stencils of radial basis function generated finite difference (RBF-FD) method applied to solving Black-Scholes equation for pricing European call options. The results experimentally confirm the theoretical convergence rates for smooth payoff functions with stencils of size 3, 5 and 7 in one- dimensional problems, and 9, 13 and 25 in two- dimensional problems. Moreover, it is shown how different terms in the equation can be approximated individually using the proposed method and then combined into a discrete approximation of the entire spatial differential operator. This new version of the RBF-FD method, where each term has been approximated individually, has been compared to the classical method and the outcome did not show any significant performance advantages. Nevertheless, the results also showed that the second order derivative was the hardest one to approximate accurately and this poses an important finding for the future development of the method.

In this report the regular finite differences method (FDM) and a least-squares radial basis function-generated finite differences method (RBF-FD-LS) is used to solve the two-dimensional incompressible Navier-Stokes equations for the lid driven cavity problem. The Navier-Stokes equations is solved using stream function-vorticity formulation. The purpose of the report is to compare FDM and RBF-FD-LS with respect to accuracy and computational cost. Both methods were implemented in MATLAB and the problem was solved for Reynolds numbers equal to 100, 400 and 1000. In the report we present the solutions obtained as well as the results from the comparison. The results are discussed and conclusions are drawn. We came to the conclusion that RBF-FD-LS is more accurate when the stepsize of the grids used is held constant, while RBF-FD-LS costs more than FDM for similar accuracy.