Optimized Schwarz Methods in Time for Discrete Transport Control

Abstract

We investigate optimized Schwarz domain decomposition methods in time for the control of the 1D transport equation. In the case of an internal control over the whole domain, the optimization problem can be transformed into a system of two coupled PDEs. We then apply the time-domain decomposition (without overlap) strategy on this PDE system as well as on its discretized counterpart. Under Fourier analysis, we analyse three different iterations: the fixed point iteration, the relaxed iteration and the preconditioned GMRES method. For each case, we propose parameters for the transmission conditions that lead to fast convergence of the method. We illustrate our results by numerical examples