A Linearized High-Order Compact Numerical Method in Space and Time for Solving Two-Dimensional Nonlinear Generalized Benjamin–Bona–Mahony–Burgers Equations

To address the issues of mismatched temporal and spatial accuracy as well as the high computational cost associated with nonlinear iterative procedures in existing high-order numerical methods for the two-dimensional Generalized Benjamin–Bona–Mahony–Burgers equation, this paper proposes a novel linearized high-order finite difference scheme. This method integrates the fourth-order backward difference formula in time with the fourth-order compact scheme in space, and constructs a fully linear numerical scheme through the interpolation-based linearization technique. The main innovations are: (1) achieving uniform fourth-order convergence in both time and space, thereby maintaining high accuracy and numerical stability even under relatively large time step sizes; (2) significantly enhancing computational efficiency for high-dimensional problems through the proposed linearization strategy; (3) providing rigorous theoretical analysis, including proofs of the existence and uniqueness of the numerical solution and the linear stability of the scheme. Comprehensive numerical experiments are conducted to demonstrate the effectiveness and reliability of the proposed method.