A Radial Basis Function Partition of Unity Collocation Method for Convection-Diffusion Equations Arising in Financial Applications

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF-PUM) proposed here. The objective of this paper is to establish that RBF-PUM is viable for parabolic PDEs of convection-diffusion type. The stability and accuracy of RBF-PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection-diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF-PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF-PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.