Abstract
In [SIAM J. Sci. Comput., 36 (2014), pp. A693--A713] the authors present a new coarse propagator for the parareal method applied to oscillatory PDEs that exhibit time-scale separation and show, und...
Highlights
IntroductionIn this paper we are interested in the convergence of the asymptotic parallel-in-time (APinT) [14] parareal method [21, 22] for oscillatory systems of equations with the following form: (1.1) (1.2) du dt + 1 \varepsi Lu \scrN (u) =
In this paper we are interested in the convergence of the asymptotic parallel-in-time (APinT) [14] parareal method [21, 22] for oscillatory systems of equations with the following form: (1.1) (1.2) du dt + 1 \varepsi Lu \scrN (u) =0, u(t)| t=0 = u0.Here, u is the vector of unknowns, L is a skew-Hermitian matrix with purely imaginary eigenvalues, and \scrN (\cdot ) is a nonlinear operator
Our motivation for studying (1.1) comes from the development of efficient time-stepping schemes for solving spatially discretized PDEs that arise in geophysical fluid applications, where it is important that the time step \Delta T can be chosen on a time scale that is independent of \varepsi
Summary
In this paper we are interested in the convergence of the asymptotic parallel-in-time (APinT) [14] parareal method [21, 22] for oscillatory systems of equations with the following form: (1.1) (1.2) du dt + 1 \varepsi Lu \scrN (u) =. The linear term induces temporal oscillations on an \scrO (\varepsi) time scale, which can require the use of prohibitively small time steps for standard numerical integrators if \varepsi is small and if the temporal oscillations of u(t) are significant (e.g., Lu(t) is not small). Convergence of the parareal method for such highly oscillatory problems can require a time step that scales like \scrO (\varepsi) [12]. The analysis in the current paper assumes that PDEs can be written as a system of ODEs as in (1.1)
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