Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

Cheng-Yu Ku,Chih-Yu Liu,Shih-Meng Hsu,Jing-En Xiao

doi:10.3390/sym12111813

Abstract

In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

Highlights

The meshfree methods recognized as powerful tools to solve practical problems governed by partial differential equations (PDEs) have attracted considerable attention [1,2,3]
It is clear that the maximum absolute error (MAE) and the root mean square error (RMSE) associated with inverse multiquadric (IMQ) radial basis functions (RBFs) with the optimum shape parameter are in the order of 10−5 and 10−7, respectively
We presented RBFs, including MQ and IMQ, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme

Summary

Introduction

The meshfree methods recognized as powerful tools to solve practical problems governed by partial differential equations (PDEs) have attracted considerable attention [1,2,3]. These methods have the advantage that it does not require mesh construction [4]. Engineering problems involving irregular geometry are usually intractable For such problems, the use of the meshfree methods to acquire approximate solutions is advantageous. The complexities involved in the solution of the governing equations require advanced mathematical approaches, such as the radial basis function collocation method (RBFCM) [7,8].

The Collocation Method of the Multiquadric Radial Basis Function

Accuracy and Convergence Analysis

The exterior source collocation differentpositions positions source points

Method

A Two-Dimensional Wave Problem

A Two-Dimensional Groundwater Flow Problem

An Unsaturated Flow Problem

17. Convergence

Conclusions