This paper is dedicated to the derivation of multilevel Schwarz Waveform Relaxation (SWR) Domain Decomposition Methods (DDM) in real- and imaginary-time for the NonLinear Schrödinger Equation (NLSE). In imaginary-time, it is shown that the multilevel SWR-DDM accelerates the convergence compared to the one-level SWR-DDM, resulting in an important reduction of the computational time and memory storage. In real-time, the method requires in addition the storage of the solution in overlapping zones at any time, but on coarser discretization levels. The method is numerically validated on the Classical SWR and Robin-based SWR methods, but can however be applied to any SWR-DDM approach.

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Keywords Domain decomposition method, Dynamics, Multilevel preconditioning, Nonlinear Schrödinger equation, Schwarz waveform relaxation algorithm, Stationary states
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Journal Applied Mathematics and Computation
Antoine, X. (X.), & Lorin, E. (2018). Multilevel preconditioning technique for Schwarz waveform relaxation domain decomposition method for real- and imaginary-time nonlinear Schrödinger equation. Applied Mathematics and Computation, 336, 403–417. doi:10.1016/j.amc.2018.04.075