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Parareal's discrete dispersion relation

Banff International Research Station for Mathematical Innovation and Discovery

While it has been established that Parareal has stability problems for hyperbolic and advection-dominated problems, details of how it propagates waves are less well understood. The talk will show how, starting from the interpretation of Parareal as a preconditioned fixed point iteration, one can derive a stability function for linear problems. After ``normalising'' this function to a unit time interval it is then possible to derive and analyse a discrete dispersion relation for Parareal. This allows to estimate the impact of e.g. the choice of propagators, size of fine and coarse time step, number of time slices etc. on Parareal's wave propagation characteristics. In particular, I will discuss the effects of phase and amplitude errors in the coarse method on convergence. The formulation also allows a worst case estimate for convergence through the maximum singular value of the error propagation matrix. This allows to link the number of iterations to the number of processors in the speedup model and to make better predictions about weak scaling of Parareal.

Mathematics; Numerical analysis; Partial differential equations

video/mp4

Author affiliation: University of Leeds

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61647

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/