Analysis of meshless local and spectral meshless radial point interpolation (MLRPI and SMRPI) on 3-D nonlinear wave equations

Abstract In this paper, the meshless local radial point interpolation (MLRPI) method is applied to simulate three space dimensional nonlinear wave equation of the form u tt + α u t + β u = u xx + u yy + u zz + δ g ( u ) u t + f ( x , y , z , t ) , 0 x , y , z 1 , t > 0 subject to given appropriate initial and Dirichlet boundary conditions and then a new spectral meshless radial point interpolation (SMRPI) method is proposed to solve the mentioned problem. 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 spheres or cubes. In innovative SMRPI method, it is not needed to any integration. The radial point interpolation method is proposed to construct shape functions for MLRPI and basis functions for SMRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local subdomains in MLRPI whereas the operational matrices converts easily the governing equations (even high order) into linear system of equations in SMRPI. A two-step time discretization method is employed to approximate the time derivatives. To treat the nonlinearity part, a kind of predictor–corrector scheme combined with one-step time discretization and Crank–Nicolson technique is adopted. A comparison study of the efficiency and accuracy of the MLRPI and SMRPI method is given by applying on mentioned problem. Convergence studies in the numerical examples show that the SMRPI method possesses excellent rates of convergence.

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