Fast parareal iterations for fractional diffusion equations

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.