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High Accurate Finite Differences Based on RBF Interpolation and its Application in Solving Differential Equations

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Abstract

The paper uses the Lagrange’s form of radial basis function (RBF) interpolation with zero-degree algebraic precision to give arbitrary order’s finite difference (RBF-FD) of interpolated function at nodes. In particular, we are interested in analyzing the approximation errors of first and second order differences based on three equidistant nodes. Then we give the best parameter values of RBF to guarantee that these two differences have the highest approximation order. As the application of those RBF formulas, the methods of solving initial value problem of first order ordinary differential equation, two-point boundary value problem and the boundary value problem of Poisson equation are investigated. Through ingeniously utilizing the differential equations to give the best parameters, the convergence order of the RBF-FD schemes constructed in this paper is two times of the polynomial finite difference schemes under the same node stencil, while the calculating time of the RBF-FD schemes has no significant increase.

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Acknowledgements

The work is supported by National Natural Science Foundation (Nos. 11271041 and 91630203) and Special Project for civil aircraft (MJ-F-2012-04).

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Authors and Affiliations

  1. School of Mathematics and Systematic Science and Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Renzhong Feng & Junna Duan

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  1. Renzhong Feng
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  2. Junna Duan
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Correspondence to Renzhong Feng.

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Feng, R., Duan, J. High Accurate Finite Differences Based on RBF Interpolation and its Application in Solving Differential Equations. J Sci Comput 76, 1785–1812 (2018). https://doi.org/10.1007/s10915-018-0684-z

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  • DOI: https://doi.org/10.1007/s10915-018-0684-z

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