On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three-dimensional wave equations

• Published in: International journal for numerical methods in engineering Vol. 105; no. 2; pp. 83 - 110
• Main Author: Shivanian, Elyas
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 13.01.2016; Wiley Subscription Services, Inc
• Subjects: 3-D wave equation; convergence and stability analysis; Dirichlet problem; Finite element method; Interpolation; local weak formulation; Mathematical analysis; Mathematical models; meshless local radial point interpolation (MLRPI) method; Meshless methods; radial basis function; Three dimensional; Wave equations
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.4960

Summary In this article, the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt+αut+βu=uxx+uyy+uzz+f(x,y,z,t),xT=(x,y,z)∈Ω⊆Rd,(d=2,3),t>0 subject to given initial and Dirichlet boundary conditions. The main difficulty of the great number of methods in full 3‐D problems is the large computational costs. In meshless local radial point interpolation method, it does not require any background integration cells, so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method with the help of radial basis functions is proposed to construct shape functions that have Kronecker delta function property. A weak formulation with the Heaviside step function converts the set of governing equations into local integral equations on local subdomains. A two‐step time discretization method is employed to evaluate the time derivatives. This suggests Crank‐Nicolson technique to be applied on the right hand side of the equation. The convergence analysis and stability of the method are fully discussed. Three illustrative examples are presented, and satisfactory agreements are achieved. It is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones. Copyright © 2015 John Wiley & Sons, Ltd.