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Journal of Scientific Computing

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09-03-2018

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

Authors: Renzhong Feng, Junna Duan

Published in: Journal of Scientific Computing | Issue 3/2018

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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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Title: High Accurate Finite Differences Based on RBF Interpolation and its Application in Solving Differential Equations
Authors: Renzhong Feng
Junna Duan
Publication date: 09-03-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0684-z

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