Parareal Multiscale Methods for Highly Oscillatory Dynamical Systems

Abstract

We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solution, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.

A stable parareal-like method for the second order wave equation

“…In this paper, we will focus on the initial value problem of the standard second order wave equation: (1) u tt = c 2 (x)∆u, x ∈ [0, 1) d , 0 ≤ t < T, with initial conditions u(x, 0) = u 0 (x), u t (x, 0) = p 0 (x) and either periodic boundary conditions or absorbing boundary conditions from placing a perfectly matched layer around [0, 1) d . The wave equation is a physical model for seismic wave and electromagnetic wave in certain simplified setups.…”

“…The parareal method tends to suffer from slow convergence or instability when applied to hyperbolic problems. Using an oscillatory dynamical system as an example, [1,2] pointed out that the phase error between the coarse and fine propagators is the reason for the slow convergence. Analogously for advection problem, the authors in [30] observe that numerical dispersion between the solvers make the parareal method converging from above and hence causes instability.…”

“…On the other hand, it is known that the slow convergence and instability of the parareal method for hyperbolic problems can be due to some notions of phase errors [2,1] and numerical dispersion [30]. In [1], effective multiscale parareal schemes relying on elaborate phase correction are proposed for a class of highly oscillatory dynamical systems. For dynamical systems on the complex plane, correcting the phase of the coarse solution corresponds to multiplication of a complex number.…”

An Approximate Time-Parallel Method for the Fast and Accurate Computation of Particle Trajectories in a Magnetic Field*

“…But, the reason that TP methods are mentioned in this Section and in more detail in Section 1.4 is that the numerical methods employed to generate the proposed hybridization draw heavily from them. The base equation that is transitioned into a multiscale context is not Parareal, as in [9] [10] [11] [12], but a related TP equation, and further development is required to improve the "wall clock" speed of the computations as discussed in Section 1.6. Section 1.7 describes the specific physical problem that the proposed hybridization, which we refer to as the hybridization by TP approximation, will be applied to and Section 1.7.4 presents the application details.…”

“…This work explains that a modified reconstruction operator that incorporates fine scale information at a previous time is required for a sensible TP update. In (22) is given in [12].…”