A collocation method based on one‐dimensional RBF interpolation scheme for solving PDEs

Abstract

Purpose

To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.

Design/methodology/approach

The proposed method is based on a Cartesian grid and a 1D integrated‐radial‐basis‐function scheme. The employment of integration to construct the RBF approximations representing the field variables facilitates a fast convergence rate, while the use of a 1D interpolation scheme leads to considerable economy in forming the system matrix and improvement in the condition number of RBF matrices over a 2D interpolation scheme.

Findings

The proposed method is verified by considering several test problems governed by second‐ and fourth‐order PDEs; very accurate solutions are achieved using relatively coarse grids.

Research limitations/implications

Only 1D and 2D formulations are presented, but we believe that extension to 3D problems can be carried out straightforwardly. Further, development is needed for the case of non‐rectangular domains.

Originality/value

The contribution of this paper is a new effective collocation formulation based on RBFs for solving PDEs.

Citation

Mai‐Duy, N. and Tanner, R.I. (2007), "A collocation method based on one‐dimensional RBF interpolation scheme for solving PDEs", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 17 No. 2, pp. 165-186. https://doi.org/10.1108/09615530710723948