Nonlinear Transmission Conditions for Schwarz and Dual Schur Complement Time Domain Decomposition

摘要

In this paper, we propose a right transmission condition for the time decomposition that consists to transform the initial boundary value problem into a time boundary values problem. This allows us to use the classical multiplicative Schwarz algorithm using non-overlapping time slices. It also avoids the symmetrizing of the time interval needed to set the unknown value of the solution at the end time boundary of the last time slice. We show that, for nonlinear scalar problems, we must imposed some invariant of the problem as transmission conditions between time slices. We derive a Robin transmission condition in order to break the sequentiality of the propagating of the exact solution from the first time slice to the time slices that follow. We show the purely linear behaviour of this multiplicative Schwarz and its extrapolation to the right transmission conditions using Aitken's technique to accelerate convergence. Then a dual Schur complement technique is used on the nonlinear problem. We derive a method where the nonlinear transmission conditions are used to solve on each time slice while imposing the constraint of the solution continuity between time slices.