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

作者：Mai Duy N^*; Tanner R I

来源：International Journal of Numerical Methods for Heat and Fluid Flow, 2007, 17(2): 165-186.

DOI：10.1108/09615530710723948

• differential equations
• numerical analysis
• boundary-elements methods

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 ID 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.

• 出版日期2007