Abstract

This study presents a fourth-order implicit compact finite difference scheme for the Benjamin–Bona–Mahony–Burgers (BBMB) equation, a nonlinear long-wave equation describing the dynamics of various wave phenomena. By employing an order-reduction framework via an auxiliary variable, we construct a compact difference scheme that yields a nonlinear algebraic system with a narrowly banded structure. Because the continuous BBMB model is governed by an underlying conservation law, the proposed numerical method is designed to preserve this structural property in the discrete sense. The discrete conservation, boundedness, and unique solvability of the scheme are firmly established, and an optimal discrete maximum norm error estimate is derived. Finally, comprehensive numerical experiments are conducted to validate both the conservative properties and the theoretical order of accuracy of the proposed scheme.

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