A Fourth-Order Compact and Conservative Difference Scheme for the Generalized Rosenau-Korteweg de Vries Equation in Two Dimensions

Abstract

In this paper, a conservative difference scheme for the Rosenau-Korteweg de Vries (RKdV) equation in 2D is proposed. The system satisfies the conservative laws in energy and mass. Existence and uniqueness of its difference solution have been shown. The order of $O(τ^2 +h^4)$ in the discrete $L^∞$-norm with time step $τ$ and mesh size $h$ is obtained. Some important lemmas are proposed to prove the high order convergence. We prove that the present scheme is unconditionally stable. Numerical results are also given in order to check the properties of analytical solution.