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A collocation method based on one-dimensional RBF interpolation scheme for solving PDEs. (English) Zbl 1231.76188
Summary: To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains. 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. 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. 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. The contribution of this paper is a new effective collocation formulation based on RBFs for solving PDEs.
MSC:
76M15Boundary element methods (fluid mechanics)
65N35Spectral, collocation and related methods (BVP of PDE)
76M25Other numerical methods (fluid mechanics)