An Effective Dissipation-Preserving Fourth-Order Difference Solver for Fractional-in-Space Nonlinear Wave Equations

In this paper, we devise an efficient dissipation-preserving fourth-order difference solver for the fractional-in-space nonlinear wave equations. First of all, we present a detailed derivation of the discrete energy dissipation property of the system. Then, with the help of the mathematical induction and Brouwer fixed point theorem, it is shown that the proposed scheme is uniquely solvable. Subsequently, by virtue of utilizing the discrete energy method, it is proven that the proposed solver achieves the convergence rates of O(Δt2+h4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(\Delta t^2+h^{4})$$\end{document} in the discrete L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document}- norm, and is unconditionally stable. And moreover, the exhibited convergence analysis is unconditional for the time step and space size, in comparison with the restrictive conditions required in the existing works. Finally, numerical results confirm the efficiency of the proposed scheme and exhibit the correctness of theoretical results.