Abstract
We focus on a time-dependent one-dimensional space-fractional diffusion equation with constant diffusion coefficients. An all-at-once rephrasing of the discretized problem, obtained by considering the time as an additional dimension, yields a large block linear system and paves the way for parallelization. In particular, in case of uniform space–time meshes, the coefficient matrix shows a two-level Toeplitz structure, and such structure can be leveraged to build ad-hoc iterative solvers that aim at ensuring an overall computational cost independent of time. In this direction, we study the behavior of certain multigrid strategies with both semi- and full-coarsening that properly take into account the sources of anisotropy of the problem caused by the grid choice and the diffusion coefficients. The performances of the aforementioned multigrid methods reveal sensitive to the choice of the time discretization scheme. Many tests show that Crank–Nicolson prevents the multigrid to yield good convergence results, while second-order backward-difference scheme is shown to be unconditionally stable and that it allows good convergence under certain conditions on the grid and the diffusion coefficients. The effectiveness of our proposal is numerically confirmed in the case of variable coefficients too and a two-dimensional example is given.
Highlights
Fractional diffusion equations (FDEs) generalize classical partial differential equations (PDEs), and their recent success is due to the non-local behavior of fractional operators that translates in an appropriate modeling of anomalous diffusion phenomena appearing in several applicative fields, like imaging or electrophysiology [2, 4]
A proof of unconditional stability of the resulting method was given. We extend this result to the case where the space scheme is weighted and shifted Grünwald difference (WSGD) and the diffusion coefficients are not necessarily equal to each other
In this work we focused on an all-at-once rephrasing of a time-dependent onedimensional space-FDE with constant diffusion coefficients discretized with WSGD in space and CN or BDF2 in time
Summary
Fractional diffusion equations (FDEs) generalize classical partial differential equations (PDEs), and their recent success is due to the non-local behavior of fractional operators that translates in an appropriate modeling of anomalous diffusion phenomena appearing in several applicative fields, like imaging or electrophysiology [2, 4]. In presence of uniform grids, the discretization matrices show a Toeplitz-like structure and this paves the way for the design of iterative solvers specialized for structured linear systems In this regard, for one-dimensional space-FDE problems we mention the circulant preconditioning in [14], the multigrid method in [20], and the structure preserving tridiagonal preconditioners in [6]. An all-at-once rephrasing of the discretized problem over a uniform space–time grid, obtained by considering the time as an additional dimension, yields large (multilevel) Toeplitz linear systems and opens to parallelization In this regard, we mention the banded Toeplitz preconditioner proposed in [27] for solving non-linear space-FDEs, and the block structured preconditioner given in [3] for dealing with arbitrary dimensional space problems.
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