In this paper, fast numerical methods are established for solving a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by the weighted and shifted Gr$\ddot{\rm{u}}$nwald formula in time and the 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 the preconditioned Krylov subspace method is developed to solve the symmetric positive definite Toeplitz linear systems 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 discretized Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around one, 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 numerical methods.

翻译：在本文中,为了解决某一类时间分布顺序和Riesz空间片分扩散波方程式,我们制定了快速数字方法。我们通过加权和移动的Gr$\dddot=rm{u ⁇ $nwald公式和空间分偏偏差公式,得出了新的差别方案。分析了时间、空间和差异方案分级的无条件稳定性和二级趋同。在一维的案例中,开发了哥伯格-塞门库尔公式,利用先决条件的Krylov子空间方法解决正对称肯定的托普利茨线性系统。在二维的案例中,我们还设计了一种全球先决条件的同源梯度方法,配有一条截断的前提条件,用以解决离散的Sylvester矩阵方程式。我们证明,两种情况下的前提条件矩阵的频谱都围绕着一个,因此,拟议的与先决条件的数值方法会很快汇合在一起。进行了一些数字实验,以证明拟议的差别方案的有效性,并表明拟议的快速解算法的性比其他方法要好。