Multilevel preconditioning techniques for Schwarz waveform relaxation domain decomposition methods for real-and imaginary-time nonlinear Schrödinger equations

X Antoine (1, 2) , E Lorin (3, 4)

1 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
2 IECL - Institut Élie Cartan de Lorraine
3 CRM - Centre de Recherches Mathématiques [Montréal]
4 School of Mathematics and Statistics [Ottawa]

This paper is dedicated to the derivation of a multilevel Schwarz Waveform Relaxation (SWR) Domain Decomposition Method (DDM) in real-and imaginary-time for the NonLinear Schrödinger Equation (NLSE). In imaginary-time, it is shown that the use of 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 approach.

Mots clés: Schwarz waveform relaxation algorithm; multilevel preconditioning; Domain decomposition method; stationary states; nonlinear Schrödinger equation; dynamics

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

X Antoine, E Lorin. Multilevel preconditioning techniques for Schwarz waveform relaxation domain decomposition methods for real-and imaginary-time nonlinear Schrödinger equations. Applied Mathematics and Computation, 2018, 336 (1), pp.403-417. ⟨10.1016/j.amc.2018.04.075⟩. ⟨hal-01266021⟩