Towards essential improvement for the Parareal-TR and Parareal-Gauss4 algorithms

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:308

Article

Wu, Shu-Lin¹ 

[1] Sichuan Univ Sci & Engn, Sch Sci, Zigong 643000, Sichuan, Peoples R China

关键词: Parareal;    Trapezoidal rule;    4th-order Gauss RK method;    Convergence analysis;   

DOI  :  10.1016/j.cam.2016.05.036

来源: Elsevier

【 摘 要 】

Parareal is an iterative algorithm and is characterized by two propagators g and F, which are respectively associated with large step size Delta T and small step size Delta t, where Delta T = J Delta t and J >= 2 is an integer. For symmetric positive definite (SPD) system u'(t) Au(t) = g(t) arising from semi-discretizing time-dependent PDEs, if we fix the g-propagator to the Backward-Euler method and choose for F some L-stable time-integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy rho approximate to 1/3,for all J >= 2 and for all sigma (A) subset of [0, + infinity), where sigma(A) is the spectrum of the matrix A. However, this result does not hold when time-integrators that lack L-stability, such as the Trapezoidal rule and the 4th-order Gauss RK method, are chosen as the F-propagator. The parareal algorithms using these two methods for the F-propagator are denoted by Parareal-TR and Parareal-Gauss4. In this paper, we propose a strategy to let these two parareal algorithms possess such a uniform convergence property. The idea is to choose an L-stable propagator (F) over tilde and on each coarse time-interval [T-n, Tn+1 we perform first two steps of (F) over tilde, then followed by J-2 steps of F. Precisely, for the Trapezoidal rule we select the 2nd-order SDIRK method as the (F) over tilde -propagator, and for the 4th-order Gauss RK method we select the 4th-order Lobatto III-C method as the (F) over tilde -propagator. Numerical results are given to support our theoretical conclusions. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

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