5 research outputs found
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations
The -step backwards difference formula (BDF) for solving the system of
ODEs can result in a kind of all-at-once linear systems, which are solved via
the parallel-in-time preconditioned Krylov subspace solvers (see McDonald,
Pestana, and Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] and Lin
and Ng [arXiv:2002.01108, 17 pages]. However, these studies ignored that the
-step BDF () is not selfstarting, when they are exploited to solve
time-dependent PDEs. In this note, we focus on the 2-step BDF which is often
superior to the trapezoidal rule for solving the Riesz fractional diffusion
equations, but its resultant all-at-once discretized system is a block
triangular Toeplitz system with a low-rank perturbation. Meanwhile, we first
give an estimation of the condition number of the all-at-once systems and then
adapt the previous work to construct two block circulant (BC) preconditioners.
Both the invertibility of these two BC preconditioners and the eigenvalue
distributions of preconditioned matrices are discussed in details. The
efficient implementation of these BC preconditioners is also presented
especially for handling the computation of dense structured Jacobi matrices.
Finally, numerical experiments involving both the one- and two-dimensional
Riesz fractional diffusion equations are reported to support our theoretical
findings.Comment: 18 pages. 2 figures. 6 Table. Tech. Rep.: Institute of Mathematics,
Southwestern University of Finance and Economics. Revised-1: refine/shorten
the contexts and correct some typos; Revised-2: correct some reference