A Second‐Order Structure‐Preserving Difference Scheme for Nonlinear Space Fractional Dispersive Equations

ABSTRACT In this paper, we develop a second‐order structure‐preserving Crank–Nicolson difference scheme for a Riesz space fractional regularized long wave‐Korteweg‐de Vries (RLW‐KdV) equation. The proposed scheme can preserve mass and energy in a discrete sense. Based on the Brouwer fixed‐point theorem, the existence of the difference solution is proved. The convergence, stability, and uniqueness of the difference scheme are proved by using the energy method. It is shown to be convergent of accuracy in the discrete ‐norm with the space step size and time step size . In addition, the resulting system of nonlinear equations is solved by the Newton method. When the fractional order is two, all those results are in accord with the difference scheme proposed for the classical RLW‐KdV equation. Some numerical examples are given to confirm the theoretical results.