Mathematics and Computers in Simulation ( IF 4.6 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.matcom.2021.03.034 Dongdong Hu , Wenjun Cai , Zhuangzhi Xu , Yonghui Bo , Yushun Wang
In this paper, an efficient numerical scheme is presented for solving the space fractional nonlinear damped sine–Gordon equation with periodic boundary condition. To obtain the fully-discrete scheme, the modified Crank–Nicolson scheme is considered in temporal direction, and Fourier pseudo-spectral method is used to discretize the spatial variable. Then the dissipative properties and spectral-accuracy convergence of the proposed scheme in norm in one-dimensional (1D) space are derived. In order to effectively solve the nonlinear system, a linearized iteration based on the fast Fourier transform algorithm is constructed. The resulting algorithm is computationally efficient in long-time computations due to the fact that it does not involve matrix inversion. Extensive numerical comparisons of one- and two-dimensional (2D) cases are reported to verify the effectiveness of the proposed algorithm and the correctness of the theoretical analysis.
中文翻译:
具有阻尼的空间分数阶非线性正弦-Gordon方程的耗散保持傅立叶拟谱方法
在本文中,提出了一种有效的数值方案,用于求解具有周期边界条件的空间分数阶非线性阻尼正弦-Gordon方程。为了获得全离散方案,在时间方向上考虑了改进的Crank-Nicolson方案,并使用傅里叶伪谱方法离散化了空间变量。然后,所提方案的耗散特性和频谱精度收敛性得到了验证。推导一维(1D)空间中的范数。为了有效地求解非线性系统,构造了基于快速傅里叶变换算法的线性化迭代。由于该算法不涉及矩阵求逆,因此在长时间计算中具有很高的计算效率。报告了一维和二维(2D)案例的大量数值比较,以验证所提出算法的有效性和理论分析的正确性。