Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

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.