An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations

The aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of Classical and quasi-Optimal Schwarz Waveform Relaxation (SWR) domain decomposition methods applied to the computation of the stationary states of the one-dimensional linear and nonlinear Schrödinger equations with a space-dependent potential. Although SWR methods are currently used for efficiently solving high dimensional partial differential equations, their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension, except in simple cases. In this work, we tackle this problem for linear and nonlinear one-dimensional Schrödinger equations by developing techniques which can be extended to higher dimensional problems and other types of PDEs. The approach combines themethod developed inGander andHalpern (SIAM J NumAnal 45(2):666–697, 2007) for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolic asymptotic expansions. For computing the stationary states, we consider the imaginary-time formulation of the Schrödinger equation based on the Normalized Gradient Flow (NGF) method and use a semi-implicit Euler scheme for the discretization. Some numerical results in the one-dimensional case illustrate the analysis for both the linear Schrödinger and Gross-Pitaevskii equations. B E. Lorin [email protected] X. Antoine [email protected] 1 Institut Elie Cartan de Lorraine, Université de Lorraine, F-54506 Vandoeuvre-lès-Nancy, France 2 Inria Nancy Grand-Est/IECL SPHINX team, Inria, Villers-lès-Nancy, France 3 Centre de Recherches Mathématiques, Université de Montréal, Montréal H3T 1J4, Canada 4 School of Mathematics and Statistics, Carleton University, Ottawa K1S 5B6, Canada