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 ( τ 2 + h 4 ) 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.

中文翻译：

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Riesz空间分数Sine-Gordon方程的四阶保守差分格式及其快速实现

摘要 在本文中，我们研究了具有 Riesz 空间分数阶导数的 Sine-Gordon 方程。首先，针对一维问题提出了四阶保守差分格式。严格证明了差分方案的独特可解性、守恒性和有界性。证明该方案在 l ∞ 范数的 O ( τ 2 + h 4 ) 阶上收敛，其中 τ , h 分别是时间和空间步长。随后，将提出的差分方案扩展到解决二维问题。结合修正牛顿法、共轭梯度法和快速傅立叶变换，提出了一种快速方法来实现所提出的数值方案。最后，