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- Title
Convergence analysis of three parareal solvers for impulsive differential equations.
- Authors
Zhiyong Wang; Liping Zhang
- 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) = f with stable impulsive perturbations Δu(t) = αu(t-) for t = τl, where α ∊ (-2, 0), Δu(t-) = u(t+) - u(t-), and l ∊ 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 ρ ≈ 0.3 if α = 0 and F is an L-stable numerical method. However, ifF is not L-stable, such as the widely used Trapezoidal rule, it unfortunately holds that ρ ≈ 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.
- Subjects
IMPULSIVE differential equations; EULER method; LINEAR differential equations; MAXIMAL functions; RUNGE-Kutta formulas
- Publication
ScienceAsia, 2019, Vol 45, Issue 1, p74
- ISSN
1513-1874
- Publication type
Article
- DOI
10.2306/scienceasia1513-1874.2019.45.074