A second-order parareal algorithm for fractional PDEs

We are concerned with using the parareal (parallel-in-time) algorithm for large scale ODEs system U′(t)+AU(t)+dAαU(t)=F(t) arising frequently in semi-discretizing time-dependent PDEs with spatial fractional operators, where d>0 is a constant, α∈(0,1) and A is a spare and symmetric positive definite (SPD) matrix. The parareal algorithm is iterative and is characterized by two propagators F and G, which are respectively associated with small temporal mesh size Δt and large temporal mesh size ΔT. The two mesh sizes satisfy ΔT=JΔt with J≥2 being an integer, which is called mesh ratio. Let Tunitf and Tunitg be respectively the computational cost of the two propagators for moving forward one time step. Then, it is well understood that the speedup of the parareal algorithm, namely E, satisfies E=O(clog⁡(1/ρ)), where c:=Tunitf/Tunitg and ρ is the convergence factor. A larger E corresponds a more efficient parareal solver. For G = Backward-Euler and some choices of F, previous studies show that ρ can be a satisfactory quantity. Particularly, for F = 2nd-order DIRK (diagonally implicit Runge-Kutta), it holds ρ≈13 for any choice of the mesh ratio J. In this paper, we continue to consider F = 2nd-order DIRK, but with a new choice for G, the IMEX (implicit-explicit) Euler method, where the 'implicit' and 'explicit' computation is respectively associated with A and dAα. Compared to the widely used Backward-Euler method, this choice on the one hand increases c (this point is apparent), and interestingly on the other hand it can also make the convergence factor ρ smaller: ρ≈15! Numerical results are provided to support our conclusions.