Two fourth-order conservative compact difference schemes for the generalized Korteweg–de Vries–Benjamin Bona Mahony equation

In this paper, the generalized Korteweg-de Vries–Benjamin Bona Mahony(GKdV-BBM) equation is investigated by two compact finite difference methods. One is a two-level-nonlinear difference scheme and another is a three-level-linearized difference scheme. Both of the schemes provide second and fourth-order accuracy in time and space, respectively. It is important that they preserve certain properties of the original equation, such as conservative properties. The solvability of the proposed numerical schemes is proved by Brouwer's fixed point theorem and mathematical induction, respectively. The convergence and unconditional stability of the proposed difference schemes are also established through the discrete energy method, without imposing any restrictions on the grid ratios. Finally, numerical results are presented to confirm the theoretical findings, and they also demonstrate the efficiency and reliability of the proposed compact approaches.