An Efficient Time-splitting Method for the Ehrenfest Dynamics

Abstract

The Ehrenfest dynamics, representing a quantum-classical mean-field type coupling, is a widely used approximation in quantum molecular dynamics. In this paper, we propose a timesplitting method for an Ehrenfest dynamics, in the form of a nonlinearly coupled Schrödinger-Liouville system. We prove that our splitting scheme is stable uniformly with respect to the semiclassical parameter, and, moreover, that it preserves a discrete semiclassical limit. Thus one can accurately compute physical observables using time steps induced only by the classical Liouville equation, i.e., independent of the small semiclassical parameter -in addition to classical mesh sizes for the Liouville equation. Numerical examples illustrate the validity of our meshing strategy.

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

“…The splitting methods form an important group of methods which are quite accurate and efficient [57]. Actually, they have been widely applied for dealing with highly oscillatory systems such as the Schrödinger/nonlinear Schrödinger equations [1,8,9,22,23,55,67], the Dirac/nonlinear Dirac equations [5,6,14,54], the Maxwell-Dirac system [10,49], the Zakharov system [12,13,41,50], the Gross-Pitaevskii equation for Bose-Einstein condensation (BEC) [11], the Stokes equation [21], and the Enrenfest dynamics [32], etc.…”

Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

“…For instance, in order to obtain "correct" observables of the Schrödinger equation in the semiclassical limit regime, the time-splitting spectral method requires much weaker constraints on time step size and mesh size than the finite difference methods [9]. Similar properties have been observed for the nonlinear Schrödinger equation (NLSE)/Gross-Pitaevskii equation (GPE) in the semiclassical limit regime [2] and the Enrenfest dynamics [25]. However, in general, splitting methods still suffer from the mesh size/time step constraints related to the high frequencies in the aforementioned problems, i.e.…”

“…In the Hamiltonian system and general ordinary differential equations (ODEs), the splitting approach has been shown to preserve the structural/geometric properties [31,47] and are superior in many applications. Developments of splitting type methods in solving partial differential equations (PDEs) include utilization in Schrödinger/nonlinear Schrödinger equations [2,9,10,19,20,37,45], Dirac/nonlinear Dirac equations [7,8,14,36], Maxwell-Dirac system [11,32], Zakharov system [12,13,28,34,35], Stokes equation [18], and Enrenfest dynamics [25], etc.…”

Computing quantum dynamics in the semiclassical regime

“…Time-splitting for the above semiclassical non-linear Schrödinger equation was studied by Carles (2013) and Carles and Gallo (2017) using WKB analysis, and adaptive splitting methods has been developed by Auzinger, Kassebacher, Koch and Thalhammer (2016). Splitting methods for non-linear Schrödinger-type systems in the context of coupled Ehrenfest dynamics were considered by Jin, Sparber and Zhou (2017) and Fang, Jin and Sparber (2018).…”