A second-order parareal algorithm for fractional PDEs

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• 作者：Shu-Lin Wu ^{wushulin_ylp@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Parareal algorithms ; Fractional PDEs ; IMEX/Backward-Euler method ; 2nd-order DIRK method
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：307
• 期：Complete
• 页码：280-290
• 全文大小：470 K

文摘

We are concerned with using the parareal (parallel-in-time) algorithm for large scale ODEs system U^′(t)+AU(t)+dA^αU(t)=F(t)${\mathbf{U}}^{\prime }(t)+\mathbf{A}\mathbf{U}(t)+d{\mathbf{A}}^{\alpha }\mathbf{U}(t)=F(t)$ arising frequently in semi-discretizing time-dependent PDEs with spatial fractional operators, where d>0$d>0$ is a constant, α∈(0,1)$\alpha \in (0,1)$ and A is a spare and symmetric positive definite (SPD) matrix. The parareal algorithm is iterative and is characterized by two propagators F$\mathcal{F}$ and G$\mathcal{G}$, which are respectively associated with small temporal mesh size Δt and large temporal mesh size ΔT  . The two mesh sizes satisfy ΔT=JΔt$\mathrm{\Delta }T=J\mathrm{\Delta }t$ with J≥2$J\ge 2$ being an integer, which is called mesh ratio. Let ${\mathbb{T}}_{\text{unit}}^f$ and ${\mathbb{T}}_{\text{unit}}^g$ 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$\mathbb{E}$, satisfies E=O(clog⁡(1/ρ))$\mathbb{E}=\mathcal{O}(c\log (1/\rho ))$, where $c:={\mathbb{T}}_{\text{unit}}^f/{\mathbb{T}}_{\text{unit}}^g$ and ρ   is the convergence factor. A larger E$\mathbb{E}$ corresponds a more efficient parareal solver. For G$\mathcal{G}$ = Backward-Euler and some choices of F$\mathcal{F}$, previous studies show that ρ   can be a satisfactory quantity. Particularly, for F$\mathcal{F}$ = 2nd-order DIRK (diagonally implicit Runge–Kutta), it holds $\rho \approx \frac{1}{3}$ for any choice of the mesh ratio J  . In this paper, we continue to consider F$\mathcal{F}$ = 2nd-order DIRK, but with a new choice for G$\mathcal{G}$, the IMEX (implicit–explicit) Euler method, where the ‘implicit’ and ‘explicit’ computation is respectively associated with A and dA^α$d{\mathbf{A}}^{\alpha }$. 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: $\rho \approx \frac{1}{5}$! Numerical results are provided to support our conclusions.