On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

Xavier Antoine (1, 2) , Emmanuel 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 [Carleton University]

This paper is dedicated to the analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided.

Mots clés: domain decomposition method; Schrödinger equation; pseudo-differential calculus

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

Xavier Antoine, Emmanuel Lorin. On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation. Journal of Computational and Applied Mathematics, 2019, 354, pp.15-30. ⟨10.1016/j.cam.2018.12.006⟩. ⟨hal-01649736⟩