An efficient algorithm based on splitting for the time integration of the Schrödinger equation
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An efficient algorithm based on splitting for the time integration of the Schrödinger equationFecha de publicación
2015-12Editor
ElsevierCita bibliográfica
BLANES, Sergio; CASAS, Fernando; MURUA, Ander. An efficient algorithm based on splitting for the time integration of the Schrödinger equation. Journal of Computational Physics, 2015, vol. 303, p. 396-412.Tipo de documento
info:eu-repo/semantics/articleVersión de la editorial
http://www.sciencedirect.com/science/article/pii/S0021999115006555Versión
info:eu-repo/semantics/sumittedVersionPalabras clave / Materias
Resumen
We present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the Schrödinger equation when the Hamiltonian operator is either time-independent or changes ... [+]
We present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the Schrödinger equation when the Hamiltonian operator is either time-independent or changes slowly with time. In the later case, the evolution operator can be effectively approximated in a step-by-step manner: first divide the time integration interval in sufficiently short subintervals, and then successively solve a Schrödinger equation with a different time-independent Hamiltonian operator in each of these subintervals. When discretized in space, the Schrödinger equation with the time-independent Hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous Hamiltonian system corresponding to a system of coupled harmonic oscillators. The particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals. Sharp local error bounds are obtained for the solution of the linear autonomous Hamiltonian system considered in each time subinterval. Our schemes can be considered, in this setting, as polynomial approximations to the matrix exponential in a similar way as methods based on Chebyshev and Taylor polynomials. The theoretical analysis, supported by numerical experiments performed for different time-independent Hamiltonians, indicates that the new methods are more efficient than schemes based on Chebyshev polynomials for all tolerances and time interval lengths. The algorithm we present automatically selects, for each time subinterval, the most efficient splitting scheme (among several new optimized splitting methods) for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the Hamiltonian operator. [-]
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Journal of Computational Physics Volume 303, 15 December 2015Derechos de acceso
Copyright © 2015 Elsevier Inc. All rights reserved.
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