GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Hybrid Krylov methods for nonlinear systems of equations
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
Choosing the forcing terms in an inexact Newton method
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
NITSOL: A Newton Iterative Solver for Nonlinear Systems
SIAM Journal on Scientific Computing
Solving partial differential equations by collocation using radial basis functions
Applied Mathematics and Computation
Discretisation procedures for multi-physics phenomena
Journal of Computational and Applied Mathematics - Special issue on applied and computational topics in partial differential equations
A finite volume method for the approximation of diffusion operators on distorted meshes
Journal of Computational Physics
Iterative solution of linear systems in the 20th century
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. III: linear algebra
A Globally Convergent Newton-GMRES Subspace Method for Systems of Nonlinear Equations
SIAM Journal on Scientific Computing
Third Order Accurate Large-Particle Finite Volume Method on Unstructured Triangular Meshes
SIAM Journal on Scientific Computing
A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation
Journal of Computational Physics
The Korean Journal of Computational & Applied Mathematics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Jacobian-free Newton-Krylov methods: a survey of approaches and applications
Journal of Computational Physics
Applied Numerical Mathematics
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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.