Applying splitting methods with complex coefficients to the numerical integration of unitary problems

Sergio Blanes; Fernando Casas; Alejandro Escorihuela-Tomàs

doi:10.3934/jcd.2021022

Article Contents

2022, Volume 9, Issue 2: 85-101. Doi: 10.3934/jcd.2021022

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Applying splitting methods with complex coefficients to the numerical integration of unitary problems

1.
Universitat Politècnica de València, Instituto de Matemática Multidisciplinar, 46022-Valencia, Spain
2.
Departament de Matemàtiques and IMAC, Universitat Jaume I, 12071-Castellón, Spain
3.
Departament de Matemàtiques, Universitat Jaume I, 12071-Castellón, Spain

^*Corresponding author: Sergio Blanes
^*Corresponding author: Sergio Blanes

Received: February 28, 2021

Revised: July 31, 2021

Published: April 2022

Work supported by Ministerio de Ciencia e Innovación (Spain) through project PID2019-104927GB-C21/AEI/10.13039/501100011033. A.E.-T. has been additionally funded by the predoctoral contract BES-2017-079697 (Spain)

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Abstract

We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schrödinger equation. We prove that a particular class of integrators are conjugate to unitary methods for sufficiently small step sizes when applied to problems defined in the group $ \mathrm{SU}(2) $. In the general case, the error in both the energy and the norm of the numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral discretization techniques in space.

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Mathematics Subject Classification: Primary: 65L05, 37M15; Secondary: 65P10, 65M22.

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Figure 1. Left: 2-norm error vs. computational cost (number of exponentials) for $ \Psi_{SC,c}^{[3]} $ (dotted line), $ \mathcal{S}^{[4]} $ (real coefficients, solid line), $ \Psi_{P,c}^{[4]} $ (complex coefficients, dash-dotted line) and $ \Psi_{SC,c}^{[4]} $ (dashed line). Right: Error in unitarity for $ \Psi_{P,c}^{[4]} $ (solid line) and the symmetric-conjugate methods $ \Psi_{SC,c}^{[3]} $ (dotted line) and $ \Psi_{SC,c}^{[4]} $ (dashed line)

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Figure 2. Absolute value of the eigenvalues of the approximate solution matrix obtained with $ \Psi_{P,c}^{[4]} $ with complex coefficients ($ k = 1 $, black dashed line), $ \Psi_{SC,c}^{[3]} $ (blue dotted line) and $ \Psi_{SC,c}^{[4]} $ (red, solid line)

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Figure 3. Error in norm of the approximate solution (left) and error in energy (37) (right) for the quartic potential (36) obtained by the palindromic schemes $ \Psi_{P,r}^{[4]} $ (magenta, dashed line), $ \Psi_{P,c}^{[4]} $ (blue dotted line) and the symmetric-conjugate method $ \Psi_{SC,c}^{[3]} $ (black solid line) along the integration interval. The step size is chosen so that all methods have the same computational cost

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Figure 4. Error in norm of the approximate solution (left) and error in energy (37) (right) for the quartic potential (36) obtained by the palindromic scheme $ \Xi_{P,r}^{[4]} $ (blue dotted line), and the symmetric-conjugate schemes $ \Psi_{SC,r}^{[3]} $ (black solid line) and $ \Xi_{SC,r}^{[4]} $ (magenta dashed line) along the integration interval. The step size is chosen so that all methods have the same computational cost

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Figure 5. Error in norm of the approximate solution (left) and error in energy (37) (right) for the Pöschl–Teller potential (38) obtained by the palindromic scheme $ \Psi_{P,c}^{[4]} $ (blue dotted line) and the symmetric-conjugate method $ \Psi_{SC,c}^{[3]} $ (black solid line) along the integration interval. The result achieved by $ \Psi_{P,r}^{[4]} $ is out of the scale

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Figure 6. Error in norm of the approximate solution (left) and error in energy (37) (right) for the Pöschl–Teller potential (38) obtained by the palindromic scheme $ \Xi_{P,r}^{[4]} $ (blue dotted line), and the symmetric-conjugate schemes $ \Psi_{SC,r}^{[3]} $ (black solid line) and $ \Xi_{SC,r}^{[4]} $ (magenta dashed line) along the integration interval

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Figure 7. Maximum of error in the expected value of the energy in the interval $ t \in [0,100] $ as a function of the time step (left) and the computational cost (number of FFTs, right) for several splitting schemes. Pöschl–Teller potential

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