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In this paper, a new conservative high-order compact finite difference scheme is studied for the initial-boundary value problem of the generalized Rosenau-regularized long wave equation. We design new conservative nonlinear fourth-order compact finite difference schemes. It is proved by the discrete energy method that the compact scheme is uniquely solvable; we have the energy conservation and the mass conservation for this approach in discrete Sobolev spaces. The convergence and stability of the difference schemes are obtained, and its numerical convergence order is \(O(\tau^{2}+h^{4})\) in the \(L^{\infty}\)-norm. Furthermore, numerical results are given to support the theoretical analysis. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

A new conservative nonlinear high-order compact finite difference scheme for the general Rosenau-RLW equation