Convergence of two conservative high-order accurate difference schemes for the generalized Rosenau–Kawahara-RLW equation

In this paper, we present two high-order accurate difference schemes for the generalized Rosenau–Kawahara-RLW equation. The proposed schemes guarantee the conservation of the discrete energy. The unique solvability of the difference solution is proved. A priori error estimates for the numerical solution is derived. Convergence and stability of the difference schemes are proved. The convergence order is O(h4+k2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^{4} + k^{2} )$$\end{document} in the uniform norm is discussed without any restrictions on the mesh sizes. Finally, numerical experiments are carried out to support the theoretical claims.