The solution of stationary ODE problems in quantum mechanics by Magnus methods with stepsize control

A second-order Magnus-type integrator for quasi-linear parabolic problems

“…This is confirmed by a variety of recent works; as a small selection we mention [5,7,17,18,19,29]. Following an approach studied by Magnus [21] for a linear system of nonautonomous ordinary differential equations In order to obtain a numerical approximation to the exact solution of (1.2), the Magnus expansion (1.3) is truncated and the integrals are determined by means of a quadrature formula.…”

Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian

“…In our experiments, we choose the step size only regarding accuracy requirements and do not observe any problems with stability, even if the Hamiltonians that we consider are more complicated than those considered in [14] (which is why the results by Hochbruck and Lubich do not directly apply to our situation). Note that Wensch et al [34] have derived a locally adaptive Magnus propagator. Their reasoning is based on extrapolation and does not take the difficulties with the norm of the Hamiltonian into account (i.e., they rely on convergence of the Magnus expansion).…”

Gradient symplectic algorithms for solving the radial Schrödinger equation

“…Such a brute force approach cannot be extended to higher orders. A more systematic way of dealing with the time-ordered exponential is via the Magnus expansion 18,19,20 , but the Magnus expansion requires explicit time integration in addition to evaluating higher order commutators. A more elegant solution is Suzuki's 21 reinterpretation of the time-order exponential as reviewed in Ref.…”