Uniform convergence and stability of linearized fourth-order conservative compact scheme for Benjamin-Bona-Mahony-Burgers' equation

TL;DR

This paper introduces a high-order compact finite difference scheme for the BBMB equation, proving its convergence, stability, and efficiency through theoretical analysis and numerical validation.

Contribution

A novel fourth-order compact scheme for the BBMB equation with proven uniform convergence, stability, and practical efficiency in solving linear systems.

Findings

Optimal convergence order of O(τ^2 + h^4) in maximum norm.

Almost unconditional stability of the scheme.

Numerical results confirm theoretical accuracy and efficiency.

Abstract

In the paper, a newly developed three-point fourth-order compact operator is utilized to construct an efficient compact finite difference scheme for the Benjamin-Bona-Mahony-Burgers' (BBMB) equation. Detailed derivation is carried out based on the reduction order method together with a three-level linearized technique. The conservative invariant, boundedness and unique solvability are studied at length. The uniform convergence is proved by the technical energy argument with the optimal convergence order $\mathcal{O}(\tau^2+h^4)$ in the sense of the maximum norm. The almost unconditional stability can be achieved based on the uniform boundedness of the numerical solution. The present scheme is very efficient in practical computation since only a system of linear equations with a symmetric circulant matrix needing to be solved at each time step. The extensive numerical examples verify our theoretical results and demonstrate the superiority of the scheme when compared with state-of-the-art those in the references.