A three-dimensional finite volume method based on radial basis functions for the accurate computational modelling of nonlinear diffusion equations

We investigate the effectiveness of a finite volume method incorporating radial basis functions for simulating nonlinear diffusion processes. Past work conducted in two dimensions is extended to produce a three-dimensional discretisation that employs radial basis functions (RBFs) as a means of local interpolation. When combined with Gaussian quadrature integration methods, the resulting finite volume discretisation leads to accurate numerical solutions without the need for very fine meshes, and the additional overheads they entail. The resulting nonlinear, algebraic system is solved efficiently using a Jacobian-free Newton–Krylov method. By employing the method as an extension of existing shape function-based approaches, the number of nonlinear iterations required to achieve convergence can be reduced while also permitting an effective preconditioning technique. Results highlight the improved accuracy offered by the new method when applied to three test problems. By successively refining the meshes, we are also able to demonstrate the increased order of the new method, when compared to a traditional shape function-based method. Comparing the resources required for both methods reveals that the new approach can be many times more efficient at producing a solution of a given accuracy.