Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

Abstract

A shallow-water wave propagation model can be described as a generalized Rosenau–Burgers equation with strong nonlinearity and high-order dispersion terms. In this paper, we propose two generalized high-order (up to eighth-order) compact finite difference schemes for solving the generalized Rosenau–Burgers equation. The first scheme is a two-level nonlinear Crank–Nicolson difference scheme and the second is a three-level linearized difference scheme. We derive the discrete mass and energy properties, and provide rigorous proofs for the boundedness, existence, and convergence with order \(O(\tau ^2 + h^s)\, (s = 4, 6, 8)\) of these proposed generalized compact difference schemes, where \(\tau \) and h denote the time- and space-steps, respectively. Finally, the validity of the theoretical analysis is verified through numerical experiments, confirming the effectiveness of the proposed schemes.