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

Abstract

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.