Solving Boussinesq equations with a meshless finite difference method

This paper mainly focus on presenting a newly-developed meshless numerical scheme, named the generalized finite difference method (GFDM), to efficiently and accurately solve the improved Boussinesq-type equations (BTEs). Based on the improved BTEs, the wave propagated over a flat or irregular bottom topography is described as a two-dimensional horizontal problem with nonlinear water waves. The GFDM and the 2nd-order Runge-Kutta method (RKM) were employed for spatial and temporal discretizations for this problem, respectively. The ramping function and the sponge layer, combing in this proposed scheme, were adopted for incident and outgoing waves, respectively. As one of domain-type meshless methods, GFDM can improve the numerical efficiency due to avoiding time-consuming meshing generation and numerical quadrature. Furthermore, the partial derivatives of Boussinesq equations can be transformed as linear combinations of nearby function values by the moving-least-squares method of the GFDM, simplifying the numerical procedures. Specifically, GFDM is suitable for complex fluid field with some irregular boundaries because of the flexible distribution of nodes. Four numerical examples were selected to verify the accuracy and applicability in the improved BTEs of the proposed meshless scheme. The results were compared with other numerical predictions and experimental observations and good agreements were depicted.