Symbol-based preconditioning for riesz distributed-order space-fractional diffusion equations

In this work, we examine the numerical solution of a 1D distributed-order space-fractional diffusion equation. Discretizing the given problem by means of an implicit finite difference scheme based on the shifted Grunwald-Letnikov formula, the resulting linear systems show a Toeplitz structure. Then, by using well-known spectral tools for Toeplitz sequences, we determine the corresponding symbol describing its asymptotic eigenvalue distribution as the matrix size diverges. The spectral analysis is performed under different assumptions with the aim of estimating the intrinsic asymptotic ill-conditioning of the involved matrices. The obtained results suggest to precondition the involved linear systems with either a Laplacian-like preconditioner or with more general tau-preconditioners. Due to the symmetric positive definite nature of the coefficient matrices, we opt for the preconditioned conjugate gradient method, and we compare the performances of our proposal with a Strang circulant alternative given in the literature.