Dynamic analysis of wave scenarios based on enhanced numerical models for the good Boussinesq equation

The good Boussinesq equation, a modification of the Boussinesq equation, aims to enhance predictions about shallow water wave behavior. This paper introduces two finite difference schemes for solving the good Boussinesq equation including linear and nonlinear implicit finite difference methods. Both schemes utilize the pseudo-compact difference approach, delivering second-order precision with an additional term to boost numerical simulation accuracy while maintaining the grid points of the standard scheme. These schemes rigorously preserve the critical physical characteristics of the good Boussinesq equation, ensuring more precise representation. We establish the existence of solutions with discrete differences and demonstrate, through the discrete energy method, their uniqueness, stability, and second-order convergence in the maximum norm. Furthermore, we propose an iterative algorithm tailored for the nonlinear implicit finite difference scheme, resulting in significant reductions in computational costs compared to the linear scheme. The results of our numerical experiments demonstrate that our methods are competitive and efficient when compared to difference schemes and previously used methods, while maintaining crucial physical qualities. Furthermore, we run relevant numerical simulations to demonstrate the accuracy of the current methods using evidence from the solitary wave interaction with the initial amplitudes of the wave. It is also suggested that the issue has a critical initial wave amplitude for the interaction of two solitary waves, where blow up occurs in a finite amount of time for initial wave amplitudes greater than the new blow-up criteria value.