Compact structure-preserving algorithm with high accuracy extended to the improved Boussinesq equation

Abstract

The improved Boussinesq equation is numerically studied using a higher-order compact finite difference technique. The aim is to achieve a mass and energy preserving scheme precisely on any time–space regions. The advantage of this scheme is that we can deal with a nonlinear partial differential equation with an implicit linear algorithm. Furthermore, the characteristics of the method are its simple steps and effective clearness. In addition, the convergence and stability analysis are then conducted to search a numerical solution whose the existence and uniqueness are guaranteed. The spatial accuracy is analyzed and found to be fourth order on a uniform grid. The numerical results are compared with established available data in literature for similar test cases, and the results are seen to be in good agreement. Besides, we perform relevant numerical simulations to illustrate the faithfulness of the present method by the evidences of the solitary wave interaction as well as the rapidly depressed solitary waves generation under sufficiently instantly decaying initial data.