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 one-dimensional integrated-radial-basis-function (1D-IRBF) 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.