Local integration of population dynamics via moving least squares approximation

Abstract

This paper applies an approach based on the Galerkin and collocation methods so-called meshless local Petrov---Galerkin (MLPG) method to treat a nonlinear partial integro-differential equation arising in population dynamics. In the proposed method, the MLPG method 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 one-step time discretization method is employed to approximate the time derivative. To treat the nonlinearity, a simple predictor---corrector scheme is performed. Also the integral term, which is a kind of convolution, is treated by the cubic spline interpolation. Convergence in both time and spatial discretizations is shown and more, stability of the method is illustrated.