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

Abstract

The aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of Classical and Optimized 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 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 tacke 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 the method developed in [24] for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolical asymptotic expansions. For computing the stationary states, we consider the imaginary-time formulation of the Schrödinger equation based on the Continuous Normalized Gradient Flow (CNGF) method and use a semiimplicit 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.

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

“…This will also be explicitly stated in the following sections. Following a similar approach as in the one-dimensional case [12], we will first factorize the Schrödinger operator in term of outgoing and incoming wave operators at the subdomains interfaces. We limit the analysis to two domains with smooth boundary, and defined as follows: 0 ∈ Ω ds, so that the curvilinear abscissa varies between 0 and s ε .…”

“…In fact, it is possible to partially relax this restriction by using Padé's approximants [6]. Notice however that, and as in the one-dimensional case, the contraction factors analytically derived below are good approximations of the numerical ones, see Section 4 as well [12]. The following symbols are defined [6] by replacing τ by iτ that Proposition 3.3.…”

“…The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [12] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.…”

“…Thanks to pseudodifferential calculus, we study the SWR-DDM for the Schrödinger equation with variable potentials and with non-flat subdomain interfaces. This paper is the sequel of [12] where the one-dimensional algorithm was analyzed in details. Domain decomposition methods are particularly well-adapted for the parallel solution of linear systems that appear in finite difference and finite element methods.…”

“…Then, we derive in Section 3 some analytical estimates of the convergence rate for the CSWR two-domains decomposition method for the linear Schrödinger (with variable potential) and the Gross-Pitaevskii equations by using the CNGF method (3) in 2-d. To this aim, we propose an extension of the techniques developed e.g. in [12,25,32] to variable coefficient equations. In particular, we make an intensive use of the theory of fractional pseudodifferential operators [33] and asymptotic symbolical calculus (see e.g.…”