An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

Suranon Yensiri,Ruth Skulkhu

doi:10.3390/math5040054

Abstract

The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

Highlights

For several decades, numerical solutions of partial differential equations (PDEs) have been studied by researchers in many different areas of science, engineering and mathematics
Considering the global Radial basis function (RBF) scheme, all four of RBF seem to provide the same level of accuracy and fluctuated graph of max error for small values of shape parameter, while there is a little bit of difference in accuracy and smooth curve of the graph for the large values of shape parameter
The results show that the accuracy of RBF and the finite difference (RBF-FD) is less than the accuracy of the global RBF scheme for the same number of nodes

Summary

Introduction

Numerical solutions of partial differential equations (PDEs) have been studied by researchers in many different areas of science, engineering and mathematics. The main idea of the RBF method is approximating the solution in terms of linear combination of infinitely differentiable RBF φ, which is the function that depends only on the distance to a center point x and shape parameter ε. Similar to the finite difference approach, the key idea is approximating the differential operator of solutions at each interior node by using a linear combination of the function values at the neighboring node locations and determining the FD-weights so that the approximations become exact for all the RBFs that are centered at the neighboring nodes. The RBF-FD method is algebraically accurate in exchange for low computational cost and needs more additional techniques to improve the accuracy [10,14] In the latter part of this work, we will aim to research improving the accuracy of the RBF-FD method by using different ways to formulate the RBF-FD method. The study of the RBF-FD method with these techniques will be included in this work as well

RBF Collocation Method

RBF-FD Method

Solution Accuracy for Elliptic PDEs

Future Work

Discussion

Conclusions