A radial basis function partition of unity collocation method for convection-diffusion equations ⋆

Numerical solution of multi-dimensional PDEs is a challenging problem with respect to computational cost and memory requirements, as well as regarding representation of realistic geometries and adaption to solution features. Meshfree methods such as global radial basis function approximation have been successfully applied to several types of problems. However, due to the dense linear systems that need to be solved, the computational cost grows rapidly with dimension. In this paper, we instead propose to use a locally supported RBF collocation method based on a partition of unity approach to numerically solve time-dependent PDEs. We investigate the stability and accuracy of the method for convection-diffusion problems in two space dimensions as well as for an American option pricing problem. The numerical experiments show that we can achieve both spectral and high-order algebraic convergence for convection-diffusion problems, and that we can reduce the computational cost for the option pricing problem by adapting the node layout to the problem characteristics.