High-performance computing of structure-preserving algorithm for the coupled BBM system formulated by weighted compact difference operators

Abstract

A compact difference operator is a valuable technique which has been used to develop numerical schemes capable of solving various types of differential equations. Advantages of compact procedures include obtaining high accuracy with a relatively small number of grid points. In this paper, we introduce weighted parameters in standard compact difference operators. Therefore, applications of both high-order compact finite difference operators for spatial derivatives and the Crank–Nicolson/Adams–Bashforth method for temporal derivatives have been used to solve coupled BBM equations. As the consequence system of equations is uncoupled during computing, it requires less CPU processing time. In addition, when producing numerical solutions, the system equations obtained at each time step are constant-coefficient matrices, gaining a remarkable number of computational advantages. In addition, deriving an a priori estimate of numerical solutions obtains convergence and stability analyses. Simultaneously, the algorithm is globally structure-preserving as to the negligence of nonphysical behavior. Numerical results demonstrate that the proposed weighted compact finite difference scheme can improve the computational efficiency of standard second-order methods. Additionally, appropriately weighted parameters can refine numerical accuracy. Lastly, an investigation of the relevant properties of numerically computed solutions is conducted, including a finite time blow-up of a head-on collision and a two-way wave propagation.