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

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.