Meshless local Petrov–Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation

Abstract This paper proposes an approach based on the Galerkin weak form and moving least squares (MLS) approximation 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. The main difficulty of methods in fully three-dimensional problems is the large computational costs. In the proposed method, which is a kind of Meshless local Petrov–Galerkin (MLPG) method, meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG 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 moving least squares approximation is proposed to construct shape functions. A two-step time discretization method is employed to approximate the time derivatives. To treat the nonlinearity, a kind of predictor–corrector scheme combined with one-step time discretization and Crank–Nicolson technique is adopted. Several numerical examples are presented and satisfactory agreements are achieved.