Temporal Two-Grid Compact Difference Scheme for Benjamin-Bona-Mahony-Burgers Equation

TL;DR

This paper introduces a two-grid compact difference scheme for efficiently solving the BBMB equation, achieving high accuracy and conservation properties through a multi-step approach validated by rigorous analysis and numerical experiments.

Contribution

The paper presents a novel temporal two-grid compact difference scheme that reduces computational cost while maintaining high accuracy for the BBMB equation.

Findings

Achieves convergence order of O(τ_c^2 + τ_f^2 + h^4)

Proves conservation, stability, and unique solvability of the scheme

Numerical experiments confirm effectiveness and accuracy

Abstract

This paper proposes a temporal two-grid compact difference (TTCD) scheme for solving the Benjamin-Bona-Mahony-Burgers (BBMB) equation with initial and periodic boundary conditions. The method consists of three main steps: first, solving a nonlinear system on a coarse time grid of size $\tau_c$; then obtaining a coarse approximation on the fine time grid of size $\tau_f$ via linear Lagrange interpolation; and finally solving a linearized scheme on the fine grid to obtain the corrected solution. The TTCD scheme reduces computational cost without sacrificing accuracy. Moreover, using the energy method, we rigorously prove the conservation property, unique solvability, convergence, and stability of the proposed scheme. It is shown that the method achieves convergence of order $\mathcal{O}(\tau_c^2 + \tau_f^2 + h^4)$ in the maximum norm, where $h$ is space step size. Finally, some numerical experiments are provided to demonstrate the effectiveness and feasibility of the proposed strategy.