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Volume 44 Number 6 Volume 45 Number 1

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Research articles

ScienceAsia 45(2019): 74-84 |doi: 10.2306/scienceasia1513-1874.2019.45.074

Convergence analysis of three parareal solvers for impulsive differential equations

Zhiyong Wanga, Liping Zhangb,*

ABSTRACT:     We are interested in using the parareal algorithm consisting of two propagators, the fine propagator F and the coarse propagator G, to solve the linear differential equations u(t)+Au(t) = ƒ with stable impulsive perturbations Δu(t) = αu(t-) for t = τl, where α ∈ (-2,0), Δu(t) = u(t+) ? u(t-), and I ∈ N. We consider the case that A is a symmetric positive definite matrix and G is defined by the implicit Euler method. In this case, provable results show that the algorithm possesses constant convergence factor ? ≈ 0.3 if α = 0 and F is an L-stable numerical method. However, if F is not L-stable, such as the widely used Trapezoidal rule, it unfortunately holds that ? ≈ 1 if λmax>>1, where λmax is the maximal eigenvalue of A. We show that with stable impulses the parareal algorithm possesses constant convergence factors for both the L-stable and A-stable F-propagators, such as the implicit Euler method, the Trapezoidal rule and the 4th-order Gauss Runge-Kutta method. Sharp dependence of the convergence factor of the resulting three parareal algorithms on the impulsive parameter α is derived and numerical results are provided to validate the theoretical analysis.

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a School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610731, China
b School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China

* Corresponding author, E-mail: zlp640602@163.com

Received 27 Apr 2018, Accepted 21 Mar 2019