  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.
