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
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