Computers & Mathematics with Applications ( IF 2.9 ) Pub Date : 2021-05-13 , DOI: 10.1016/j.camwa.2021.05.003 Huan-Yan Jian , Ting-Zhu Huang , Xian-Ming Gu , Xi-Le Zhao , Yong-Liang Zhao
In this paper, fast numerical methods are established to solve a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by a weighted and shifted Grünwald formula in time and a fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing a preconditioned Krylov subspace method is developed to solve the Toeplitz linear system derived from the proposed difference scheme. In the two-dimensional case, we also design a global preconditioned conjugate gradient method with a truncated preconditioner to solve the resulting Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around 1, such that the proposed numerical methods with preconditioners converge very quickly. Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference schemes and show that the performances of the proposed fast solution algorithms are better than other testing methods.
中文翻译:
时间分布阶和Riesz空间分数阶扩散波方程的快速二阶隐式差分格式
本文建立了快速的数值方法来求解一类时间分布阶和Riesz空间分数阶扩散波方程。我们通过时间上经过加权和移位的Grünwald公式以及空间上的分数中心差分公式得出新的差分方案。分析了差分方案在时间,空间和分布阶上的无条件稳定性和二阶收敛性。在一维情况下,利用预处理的Krylov子空间方法开发了Gohberg-Semencul公式,以解决从所提出的差分方案导出的Toeplitz线性系统。在二维情况下,我们还设计了带有截断的预处理器的全局预处理共轭梯度方法,以求解所得的Sylvester矩阵方程。我们证明,在两种情况下,预处理矩阵的谱都聚集在1附近,从而使带有预处理器的所提出的数值方法收敛非常快。进行了一些数值实验,证明了所提出的差分方案的有效性,并表明所提出的快速求解算法的性能优于其他测试方法。