Abstract
In this paper, we investigate the Sine-Gordon Equations with the Riesz space fractional derivative. Firstly, a fourth-order conservative difference scheme is proposed for the one-dimensional problem. The unique solvability, conservation and boundedness of the difference scheme are rigorously demonstrated. It is proved that the scheme is convergent at the order of O(τ2+h4) in the l∞ norm, where τ,h are the time and space step, respectively. Subsequently, the proposed difference scheme is extended to solve the two-dimensional problem. Combining the Revised Newton method, conjugate gradient method and fast Fourier transform, a fast method is proposed for the implementation of the proposed numerical schemes. Finally, several numerical examples are provided to verify the correctness of the theoretical results and the efficiency of the proposed fast algorithm.
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