A meshless finite point method for the improved Boussinesq equation using stabilized moving least squares approximation and Richardson extrapolation

A meshless finite point method (FPM) is developed in this paper for the numerical solution of the nonlinear improved Boussinesq equation. A time discrete technique is used to approximate time derivatives, and then a linearized procedure is presented to deal with the nonlinearity. To achieve stable convergence numerical results in space, the stabilized moving least squares approximation is used to obtain the shape function, and then the FPM is adopted to establish the linear system of discrete algebraic equations. To enhance the accuracy and convergence order in time, the Richardson extrapolation is finally incorporated into the FPM. Numerical results show that the FPM is fourth-order accuracy in both space and time and can obtain highly accurate results in simulating the propagation of a single solitary wave, the interaction of two solitary waves, the solitary wave break-up and the solution blow-up phenomena.

Automated summary

This paper proposes a new numerical method for solving the nonlinear improved Boussinesq equation. The method has high accuracy and stable convergence in both spatial and temporal dimensions.