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New, Highly Accurate Propagator for the Linear and Nonlinear Schrödinger Equation

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Abstract

A propagation method for the time dependent Schrödinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde’s usually results in system of ode’s of the form

$$u_t - G u =s$$
(0.1)

where G is a operator (matrix) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modified exponential evolution operator, had been developed already for this type of problems where G is a linear, time independent matrix and s is a constant vector. In this paper we will describe a new algorithm for the more general case where s is a time-dependent r.h.s vector. An iterative version of the new algorithm can be applied to the general case where G depends on t or u. Numerical results for Schrödinger equation with time-dependent potential and to non-linear Schrödinger equation will be presented.

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Authors and Affiliations

  1. School of Computer Sciences, Academic College of Tel-Aviv Yaffo, Rabenu Yeruham St., Tel-Aviv, 61803, Israel

    Hillel Tal-Ezer

  2. Institute of Chemistry and The Fritz Haber Research Center, The Hebrew University, Jerusalem, 91904, Israel

    Ronnie Kosloff & Ido Schaefer

Authors
  1. Hillel Tal-Ezer
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  2. Ronnie Kosloff
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  3. Ido Schaefer
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Correspondence to Hillel Tal-Ezer.

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Tal-Ezer, H., Kosloff, R. & Schaefer, I. New, Highly Accurate Propagator for the Linear and Nonlinear Schrödinger Equation. J Sci Comput 53, 211–221 (2012). https://doi.org/10.1007/s10915-012-9583-x

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  • DOI: https://doi.org/10.1007/s10915-012-9583-x

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