Article ID Journal Published Year Pages File Type
4967908 Journal of Computational Physics 2017 22 Pages PDF
Abstract
Numerical methods for fractional PDEs is a hot topic recently. This work is concerned with the parareal algorithm for system of ODEs u′(t)+Au(t)=f that arising from semi-discretizations of time-dependent fractional diffusion equations with nonsymmetric Riemann-Liouville fractional derivatives. The spatial semi-discretization of this kind of fractional derivatives often results in a coefficient matrix A with spectrum σ(A) satisfyingσ(A)⊆S(η):={λ∈C:ℜ(λ)≥η,ℑ(λ)∈R}, where η>0 is a measure of dissipativity of the differential equations. To accelerate the parareal algorithm, we propose a scaled model u′(t)+1αAu(t)=f (with α>0) to serve the coarse grid correction, which is an important component of our parareal algorithm. Given η and α, we derive a sharp bound of the convergence factor of the parareal iterations. Moreover, by minimizing such a bound we get optimized scaling factor αopt. It is shown that, compared to α=1 (i.e., the classical implementation pattern of the coarse grid correction), the optimized scaling factor significantly improves the convergence rate. Numerical examples are presented to support the theoretical finding.
Related Topics
Physical Sciences and Engineering Computer Science Computer Science Applications
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