Two high-order compact finite difference schemes for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation

Abstract

In this paper, two numerical methods for solving the initial boundary value problem of one-dimensional nonlinear Generalized Benjamin-Borne-Mahony-Burgers equation are presented. Both methods utilize a fourth-order backward difference scheme for the discretization of the first-order derivative in the time direction, and apply a fourth-order compact difference scheme and a fourth-order Padé scheme to discretize the second-order and first-order spatial derivatives, respectively. The primary difference between the two methods lies in their distinct linearization strategies for the nonlinear term, which results in the formation of two linear systems. Both methods achieve fourth-order convergence in time and space. Subsequently, theoretical proofs are provided for the conservation property, stability and the existence and uniqueness of the numerical solution of the proposed numerical scheme. Finally, numerical experiments are conducted to verify the reliability and effectiveness of both methods.