An Effective Dissipation-Preserving

An Effective Dissipation-Preserving Fourth-Order Difference Solver for Fractional-in-Space Nonlinear Wave EquationsJianqiang Xie · Zhiyue Zhang1Received: 2 August 2018 / Revised: 17 December 2018 / Accepted: 30 January 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019AbstractIn 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 math- ematical 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 ( t +h ) in the discrete L - 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 condi- tions required in the existing works. Finally, numerical results confirm the efficiency of the proposed scheme and exhibit the correctness of theoretical results.Keywords Dissipation-preserving scheme · Finite difference methods · Solvability · Convergence · StabilityMathematics Subject Classification 65M06 · 35R11 · 65M121 IntroductionIn this paper, we focus on the high-order numerical solution of the following initial and boundary value problems (IBVPs) of space fractional nonlinear wave equations