Waveform Iteration with asynchronous communication

Abstract

Consider the coupling of two multi-physics systems of time-dependent ODEs. We propose a new Waveform iteration type method for coupling which uses asynchronous communication to be both parallel in time and fast in convergence. Analytical results show convergence in the continuous setting and numerical results for two coupled head equations show good performance of this method, which is both parallel and converging fast.

Waveform Relaxation with asynchronous time-integration

“…All numerical experiments were run on an Intel i5-2500K 3.30 GHz CPU with Python 3.6.9, Open MPI 2.1.1, FEniCS 2019.2.0.dev0 [16]. The code is available at [18]. For GS we denote the different orderings of GS WR by "GS DN" and "GS ND".…”

“…In the previous section, we considered the convergence of the fully discrete WR iteration (14) for ∆t fixed for k → ∞, which gives (13). Here, we instead discuss the convergence of the continuous iteration (18) for k → ∞. Before doing so, we would like to point out that we cannot guarantee that we obtain (18) from ( 14) in the limit ∆t → 0.…”

“…We thus assume that the limit has only a finite number of jumps and consider (18) in a piecewise sense with piece-wise Lipschitz-continuous data. This guarantees existence of a piece-wise solution of ( 18) for all k > 0.…”

“…We can apply this solution formula to (18). Replacing u(k) via integration by parts and performing lengthy, but straight-forward rearrangements, yield the solution:…”