Multi-product expansion, Suzuki's method and the Magnus integrator for solving time-dependent problems.

Juergen Geiser , Siu A. Chin

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 19 pp.

MSC 2000

65M15 Error bounds

65L05 Initial value problems

Abstract
In this paper we discuss the extention to exponential splitting methods with respect to time-dependent operators. For such extensions, the Magnus integration, see Blanes et al 2006. and the Suzuki's method are incorporating ideas to the time-ordered exponential, see Suzuki 1993. We formulate each methods and present their advantages to special time-dependent harmonic oscillator problems. An decisive and comprehensive comparison on the Magnus expansion with Suzuki's method on some problems are given. Here classical and also quantum mechanical can be treated to present the solving in time-dependent problems. We choose a radial Schrodinger equation as a classical time-dependent harmonic oscillation which combine classical and quantum calculations simultaneously. Here we present the different schemes of the integrator based Magnus scheme and the differential based Suzuki's method. Based on the spiked harmonic oscillator case we could analyze the differences.