Radial Basis Function Interpolation And Its Application In Computational Electromagnetics

Radial basis function (RBF) is one of the most popular tools in large scaledscattered data approximation theory. The advantage of radial basis function method issolving the multivariate problem with scattered date easily as much as that in onedimension. It is simple and easy implement on the computer. Numerical methods forpartial differential equations (PDEs) using RBF is classified into the RBF interpolationmethod and quasi-interpolation method. In this paper, we discuss the application of theRBF quasi-interpolation for solving partial differential equations numerically.The research of the RBF begins with RBF interpolation. The early study of theRBF interpolation used to focus on scattered data approximation. In1990, Kansamodified Hardy’s MQ method to solve partial differential equations. Since then, solvingPDEs using RBFs collocation method has attracted many researchers’ attentions,because no tedious mesh generation is required. However, the interpolation methodneed to solve the inversion of matrix.Therefore the system may become very large andill-conditioned in order to achieve a certain prescribed accuracy. To avoid encounteringthe large scaled system of equations, some people turn to study the radial basisfunctions quasi-interpolation. As long as the scheme of the interpolation is proper, it canachieve the desired precision.This dissertation is concerned with some applications of the radial basis functionsquasi-interpolation for solving partial differential equations numerically. In practicalapplications, we know the function is usually defined in a bounded domain. Thereforehow to deal with the boundary conditions is very important. We present a newquasi-interpolation operator by constructing a boundary polynomial.Numericalexperiments verify that this new quasi-interpolation scheme have a very goodapproximation of the function itself and the higher derivatives. Based on this format, wepropose a numerical scheme for solving the one-dimensional nonlinear Schr dingerequation and the Wave equation which is the special electromagnetic field equations.The underlying idea of our means is that: the temporal derivative is approximated byfinite difference method, while employing the derivative of the quasi-interpolation to approximate the spatial derivative of the PDE, and then we obtain the discrete schemefor the original equation. At the end of this paper, the numerical experiments arepresented to verify the accuracy and efficiency of the presented schemes. The MQfunction and the Gauss function are selected about the radial basis function in thenumerical experiments.The results of numerical experiments are compared withanalytical solution and finite difference method; we see that this method is valid. Finally,we make a conclusion of this paper and advise some future works.