Research articles
ScienceAsia 45 (2019): 7484 doi:
10.2306/scienceasia15131874.2019.45.074
Convergence analysis of three parareal solvers for
impulsive differential equations
Zhiyong Wang_{a}, Liping Zhang_{b,*}
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 = τ_{l}, 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 Lstable numerical method. However,
if F is not Lstable, such as the widely used Trapezoidal rule, it unfortunately holds that ? ≈ 1 if λ_{max>>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 Lstable and Astable Fpropagators, such as the implicit Euler method, the Trapezoidal rule and
the 4thorder Gauss RungeKutta 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, Email: zlp640602@163.com
Received 27 Apr 2018, Accepted 21 Mar 2019
