Domain decomposition method for the N-body time-independent and time-dependent Schrödinger equations

Abstract

This paper is devoted to the derivation of a pleasingly parallel Galerkin method for the timeindependent N -body Schrödinger equation, and its time-dependent version modeling molecules subject to an external electric field [12,13,16]. In this goal, we develop a Schwarz Waveform Relaxation (SWR) Domain Decomposition Method (DDM) for the N -body Schrödinger equation. In order to optimize the efficiency and accuracy of the overall algorithm, i) we use mollifiers to regularize the singular potentials and to approximate the Schrödinger Hamiltonian, ii) we select appropriate orbitals, and iii) we carefully derive and approximate the SWR transmission conditions. Some low dimensional numerical experiments are presented to illustrate the methodology.

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

“…where Λ ±,p = Op(λ ±,p ), for p = 1/2, 0, −1/2... as similarly done in (26). Following [17], we set in Ω ±…”

“…Although these methods have received much attention over the past decades, the first application to the Schrödinger equation can be found in [23], where the authors consider the real-time linear one-dimensional Schrödinger equation. In another recent paper [12], some algorithms are analyzed for the one-dimensional time-dependent Linear Schrödinger Equation (LSE) where are included ionization and recombination processes by an intense electric field, and in [26] a SWR methodology for solving the N -body Schrödinger equation is considered. In [14], the authors study the numerical performance with a GPU implementation of Schwarz waveform relaxation methods for the one-dimensional dynamical solution of the LSE with a general potential.…”