Structure-Preserving Integration for Magnetic Gaussian Wave Packet Dynamics

TL;DR

This paper introduces structure-preserving numerical integrators for magnetic Gaussian wave packet dynamics, ensuring long-term stability and conservation properties in simulations of the magnetic Schrödinger equation.

Contribution

It develops novel Boris-type and symplectic integrators tailored for magnetic wave packet dynamics, preserving key invariants and geometric structures.

Findings

Methods conserve quadratic invariants of the wave packet parameters.

Integrators exhibit near-conservation of the averaged Hamiltonian over long times.

Numerical experiments confirm improved long-term stability and structure preservation.

Abstract

We develop structure-preserving time integration schemes for Gaussian wave packet dynamics associated with the magnetic Schr\"odinger equation. The variational Dirac--Frenkel formulation yields a finite-dimensional Hamiltonian system for the wave packet parameters, where the presence of a magnetic vector potential leads to a non-separable structure and a modified symplectic geometry. By introducing kinetic momenta through a minimal substitution, we reformulate the averaged dynamics as a Poisson system that closely parallels the classical equations of charged particle motion. This representation enables the construction of Boris-type integrators adapted to the variational setting. In addition, we propose explicit high-order symplectic schemes based on splitting methods and partitioned Runge--Kutta integrators. The proposed methods conserve the quadratic invariants characterizing the Hagedorn parametrization, preserve linear and angular momentum under symmetry assumptions, and exhibit near-conservation of the averaged Hamiltonian over long time intervals. Rigorous error estimates are derived for both the wave packet parameters and observable quantities, with bounds uniform in the semiclassical parameter. Numerical experiments demonstrate the favorable long-time behavior and structure preservation of the integrators.