Meshless local radial point interpolation to three-dimensional wave equation with Neumann's boundary conditions

Abstract

In this article, the meshless local radial point interpolation (MLRPI) method is applied to simulate three-dimensional wave equation subject to given appropriate initial and Neumann's boundary conditions. The main drawback of methods in fully 3-D problems is the large computational costs. In the MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as a cube or a sphere. The point interpolation method with the help of radial basis functions is proposed to form shape functions in the frame of MLRPI. The local weak formulation using Heaviside step function converts the set of governing equations into local integral equations on local subdomains where Neumann's boundary condition is imposed naturally. A two-step time discretization technique with the help of the Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that the MLRPI method possesses reliable rates of convergence.

Keywords:

• meshless local radial point interpolation (MLRPI)
• local weak formulation
• radial basis function
• 3-D wave equation
• Neumann's boundary conditions

2010 AMS Subject Classification:

• 65M99

Acknowledgements

The authors are very grateful to two anonymous reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper very much.

Disclosure Statement

No potential conflict of interest was reported by the authors.