Numerical simulation of wave flow : Integrating the BBM-KdV equation using compact difference schemes

Abstract

The nonlinear convection term $u{u}_x$ plays a critical role in scientific and engineering contexts, capturing the complex interaction between a function and its spatial derivative. In numerical analysis, this term significantly impacts the stability of computational methods and requires careful treatment for accurate solutions. This study presents efficient, high-order linear numerical schemes for solving the Benjamin–Bona–Mahony-KdV equation, incorporating three strategies to approximate the nonlinear term while preserving mass and/or energy. The effectiveness and precision of the proposed methods are demonstrated through rigorous testing in comprehensive numerical experiments, providing clear insight into their performance. Our observations show that these schemes preserve conservative properties while offering improved accuracy and stability compared to the standard second-order scheme. These findings underscore the potential to advance numerical methods for differential equations and provide strong evidence for the effectiveness of the proposed high-order approach in accurately modeling complex wave behavior.