Xing and Wen Advances in Difference Equations (2018)2018:238 https://doi.org/10.1186/s13662-018-1689-5 R E S E A R C H Open Access A conservative difference scheme for the Riesz space-fractional sine-Gordon equation Zhiyong Xing1,2* and Liping Wen1 *Correspondence:
[email protected] Abstract 1School of Mathematics and Computational Science, Xiangtan In this paper, we study a conservative difference scheme for the sine-Gordon University, Xiangtan, P.R. China equation (SGE) with the Riesz space fractional derivative. We rigorously establish the 2Department of Mathematics, conservation property and solvability of the difference scheme. We discuss the Shaoyang University, Shaoyang, P.R. China stability and convergence of the difference scheme in the L∞ norm. To reduce the computational complexity, we introduce a revised Newton method for implementing the difference scheme. Finally, we provide several numerical experiments to support the theoretical results. Keywords: Space-fractional sine-Gordon equation; Riesz fractional derivative; Conservation law; Newton method; Convergence and stability 1 Introduction The nonlinear SGE ∂2u – u + sin u =0, x =(x ,...,x ) ∈ Rn, (1.1) ∂t2 1 n arises in many different areas, such as stability of fluid motions, differential geometry, Josephson junctions, models of particle physics [1], the propagation of fluxon [2], the motion of a rigid pendulum attached to a stretched wire [3], the phenomenon of supra- transmission in nonlinear media [4], and so on. Therefore the investigation for the SGE has attracted attention of some researchers, and many significant achievements have been made. Aremarkablepropertyof(1.1) is the energy conservation: 1 ∂u 2 ∂u 2 + +2P(u) dx =const, 2 R ∂t ∂x where P(u)=1–cos u; it was studied by many researchers [5–9].