Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions
Journal of Computational Physics
Scattered node compact finite difference-type formulas generated from radial basis functions
Journal of Computational Physics
Domain decomposition by radial basis functions for time dependent partial differential equations
ACST'06 Proceedings of the 2nd IASTED international conference on Advances in computer science and technology
Journal of Computational Physics
Convergence of Unsymmetric Kernel-Based Meshless Collocation Methods
SIAM Journal on Numerical Analysis
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
Meshfree explicit local radial basis function collocation method for diffusion problems
Computers & Mathematics with Applications
Preconditioning for radial basis functions with domain decomposition methods
Mathematical and Computer Modelling: An International Journal
Stabilization of RBF-generated finite difference methods for convective PDEs
Journal of Computational Physics
Optimal constant shape parameter for multiquadric based RBF-FD method
Journal of Computational Physics
Optimal variable shape parameter for multiquadric based RBF-FD method
Journal of Computational Physics
Journal of Computational Physics
Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs
Journal of Computational Physics
Stable calculation of Gaussian-based RBF-FD stencils
Computers & Mathematics with Applications
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.48 |
In this work an extension is proposed to the Local Hermitian Interpolation (LHI) method; a meshless numerical method based on interpolation with small and heavily overlapping radial basis function (RBF) systems. This extension to the LHI method uses interpolation functions which themselves satisfy the partial differential equation (PDE) to be solved. In this way, a much improved reconstruction of partial derivatives can be obtained, resulting in significantly improved accuracy in many cases. The implementation algorithm is described, and is validated via three convection-diffusion-reaction problems, for steady and transient situations. A Crank-Nicolson implicit time stepping technique is used for the time-dependent problems. In the proposed approach, a form of 'analytical upwinding' is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function, which includes the desired information about the convective velocity field. The implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one-dimensional travelling front problem at Peclet numbers of 500, 1000, 2000, 5000 and infinity, which corresponds to a shock front in the case of infinity. In addition, the accuracy of the numerical scheme is validated against a one-dimensional steady state solution exhibiting strong boundary layer effects, and also against a steady and a transient three-dimensional convection-diffusion problem on irregular datasets. All the test cases are validated against the corresponding analytical solutions. Finally, the effect of various interpolation stencil configurations is investigated, and some important limitations on local data-centre distribution are identified.