Numerical analysis of the meshless element-free Galerkin method for hyperbolic initial-boundary value problems

Tang, Yaozong; Li, Xin

doi:10.21136/am.2017.0061-17

Cited by 1 publication

(1 citation statement)

References 23 publications

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“…The meshfree approach can be categorized into boundary-type and domain-type meshless methods depending on the basis function satisfying the governing equation or not [4]. Several domain-type meshless methods [5][6][7] have been proposed, such as the element-free Galerkin method [8,9], the meshless local Petrov-Galerkin method [10,11], the radial basis function collocation method (RBFCM) [12,13], the moving least squares approximation [14,15] and the generalized finite difference method [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

Xiao

Liu

2020

Mathematics

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In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

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