A Finite Volume Method Based on Radial Basis Functions for Two-Dimensional Nonlinear Diffusion Equations

The finite volume method is the favoured numerical technique for solving (possibly coupled, nonlinear, anisotropic) diffusion equations. The method transforms the original problem into a system of nonlinear, algebraic equations through the process of discretisation. The accuracy of this discretisation determines to a large extent the accuracy of the final solution. A new method of discretisation is presented, designed to achieve high accuracy without imposing excessive computational requirements. In particular, the method employs radial basis functions as a means of local gradient interpolation. When combined with high order Gaussian quadrature integration methods, the interpolation based on radial basis functions produces an efficient and accurate discretisation. The resulting nonlinear, algebraic system is solved efficiently using a Jacobian-free Newton–Krylov method. Information obtained from the Newton–Krylov iterations is used to construct an effective preconditioner in order to reduce the number of nonlinear iterations required to achieve an accurate solution. Results to date have been promising, with the method giving accuracy several orders of magnitude better than simpler methods based on shape functions for both linear and nonlinear diffusion problems.