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

This paper proposes a temporal two-grid compact difference scheme for solving the Benjamin-Bona-Mahony-Burgers (BBMB) equation with initial and periodic boundary conditions. The method first solves a nonlinear system on a coarse temporal grid, then interpolates the coarse solution onto a fine grid, and finally performs a linearized correction on the fine grid, thereby reducing 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. Under the maximum norm, the method is shown to be second-order accurate in time and fourth-order accurate in space. Finally, some numerical experiments are provided to demonstrate the effectiveness and feasibility of the proposed strategy.