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• W2040549599 abstract "We analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem $mathbf{u}'+Amathbf{u}=g$. The coarse propagator $mathcal{G}$ is fixed to the backward-Euler method and three time integrators are chosen for the $mathcal{F}$-propagator: the trapezoidal rule, the third-order diagonal implicit Runge--Kutta (RK) (DIRK) method, and the fourth-order Gauss RK method. It is well known that the Parareal-Euler algorithm using the backward-Euler method for $mathcal{F}$ and $mathcal{G}$ converges rapidly, but less is known when one uses for $mathcal{F}$ the trapezoidal rule, or the fourth-order Gauss RK method, especially when the mesh ratio $J(=Delta T/Delta t)$ is small. We show that for a specified $lambda_{max}$(the maximal eigenvalue of $A$ or its upper bound), there exists some critical $J_{rm cri}$ such that the parareal solvers derived from these three choices of $mathcal{F}$ converge as fast as Parareal-Euler, provided $Jgeq J_{rm cri}$. Precisely, for $mathcal{F}$ the trapezoidal rule and the fourth-order Gauss RK method, $J_{rm cri}$ depends on $Delta T, Delta t$, and $lambda_{max}$ and we present concise formulas to calculate $J_{cri}$, while for $mathcal{F}$ the third-order DIRK method, $J_{cri}=4$, independently of these parameters. Numerical examples with applications in fractional PDEs and uncertainty quantification are presented to support the theoretical predictions." @default.
• W2040549599 created "2016-06-24" @default.
• W2040549599 creator A5021118595 @default.
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• W2040549599 date "2015-01-01" @default.
• W2040549599 modified "2023-09-29" @default.
• W2040549599 title "Convergence Analysis for Three Parareal Solvers" @default.
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• W2040549599 doi "https://doi.org/10.1137/140970756" @default.
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