American Institute of Mathematical Sciences

Advanced
April  2022, 9(2): 85-101. doi: 10.3934/jcd.2021022

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

Sergio Blanes 1,, , Fernando Casas 2, and  Alejandro Escorihuela-Tomàs 3,

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

Received  March 2021 Revised  August 2021 Published  April 2022 Early access  December 2021

Fund Project: 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)

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.

Keywords: Splitting methods, unitary problems, complex coefficients, preservation properties.
Mathematics Subject Classification: Primary: 65L05, 37M15; Secondary: 65P10, 65M22.
Citation: Sergio Blanes, Fernando Casas, Alejandro Escorihuela-Tomàs. Applying splitting methods with complex coefficients to the numerical integration of unitary problems. Journal of Computational Dynamics, 2022, 9 (2) : 85-101. doi: 10.3934/jcd.2021022
References:
[1]

A. Bandrauk, E. Dehghanian and H. Lu, Complex integration steps in decomposition of quantum exponential evolution operators, Chem. Phys. Lett., 419 (2006), 346-350. 

[2]

S. Blanes and F. Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Math., 54 (2005), 23-37.  doi: 10.1016/j.apnum.2004.10.005.

[3]

S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016.

[4]

S. Blanes, F. Casas, P. Chartier and A. Escorihuela-Tomàs, On symmetric-conjugate composition methods in the numerical integration of differential equations, arXiv: 2101.04100 (to appear in Math. Comput.). doi: 10.1090/mcom/3715.

[5]

S. Blanes, F. Casas, P. Chartier and A. Murua, Optimized high-order splitting methods for some classes of parabolic equations, Math. Comput., 82 (2013), 1559-1576.  doi: 10.1090/S0025-5718-2012-02657-3.

[6]

S. Blanes, F. Casas and A. Murua, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl., 45 (2008), 89-145. 

[7]

S. Blanes, F. Casas and A. Murua, Splitting methods with complex coefficients, Bol. Soc. Esp. Mat. Apl., 50 (2010), 47-60.  doi: 10.1007/bf03322541.

[8]

S. Blanes and P. Moan, Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods, J. Comput. Appl. Math., 142 (2002), 313-330.  doi: 10.1016/S0377-0427(01)00492-7.

[9]

F. Casas, P. Chartier, A. Escorihuela-Tomàs and Y. Zhang, Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations, J. Comput. Appl. Math., 381 (2021), 113006.  doi: 10.1016/j.cam.2020.113006.

[10]

F. Castella, P. Chartier, S. Descombes and G. Vilmart, Splitting methods with complex times for parabolic equations, BIT Numer. Math., 49 (2009), 487-508.  doi: 10.1007/s10543-009-0235-y.

[11]

J. Chambers, Symplectic integrators with complex time steps, Astron. J., 126 (2003), 1119-1126.  doi: 10.1086/376844.

[12]

S. Flügge, Practical Quantum Mechanics, Springer, 1971.

[13]

A. Galindo and P. Pascual, Quantum Mechanics. I., Texts and Monographs in Physics. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-83854-5.

[14]

F. Goth, Higher order auxiliary field quantum Monte Carlo methods, arXiv: 2009.04491.

[15]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, 2006.

[16]

E. Hansen and A. Ostermann, Exponential splitting for unbounded operators, Math. Comput., 78 (2009), 1485-1496.  doi: 10.1090/S0025-5718-09-02213-3.

[17]

E. Hansen and A. Ostermann, High order splitting methods for analytic semigroups exist, BIT Numer. Math., 49 (2009), 527-542.  doi: 10.1007/s10543-009-0236-x.

[18]

C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, European Mathematical Society, 2008. doi: 10.4171/067.

[19]

R. McLachlan and R. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434.  doi: 10.1017/S0962492902000053.

[20]

T. Prosen and I. Pizorn, High order non-unitary split-step decomposition of unitary operators, J. Phys. A: Math. Gen., 39 (2006), 5957-5964.  doi: 10.1088/0305-4470/39/20/021.

[21]

Q. Sheng, Solving partial differential equations by exponential splitting, IMA J. Numer. Anal., 9 (1989), 199-212. doi: 10.1093/imanum/9.2.199.

[22]

M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A, 146 (1990), 319-323.  doi: 10.1016/0375-9601(90)90962-N.

[23]

M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys., 32 (1991), 400-407.  doi: 10.1063/1.529425.

show all references

References:
[1]

A. Bandrauk, E. Dehghanian and H. Lu, Complex integration steps in decomposition of quantum exponential evolution operators, Chem. Phys. Lett., 419 (2006), 346-350. 

[2]

S. Blanes and F. Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Math., 54 (2005), 23-37.  doi: 10.1016/j.apnum.2004.10.005.

[3]

S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016.

[4]

S. Blanes, F. Casas, P. Chartier and A. Escorihuela-Tomàs, On symmetric-conjugate composition methods in the numerical integration of differential equations, arXiv: 2101.04100 (to appear in Math. Comput.). doi: 10.1090/mcom/3715.

[5]

S. Blanes, F. Casas, P. Chartier and A. Murua, Optimized high-order splitting methods for some classes of parabolic equations, Math. Comput., 82 (2013), 1559-1576.  doi: 10.1090/S0025-5718-2012-02657-3.

[6]

S. Blanes, F. Casas and A. Murua, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl., 45 (2008), 89-145. 

[7]

S. Blanes, F. Casas and A. Murua, Splitting methods with complex coefficients, Bol. Soc. Esp. Mat. Apl., 50 (2010), 47-60.  doi: 10.1007/bf03322541.

[8]

S. Blanes and P. Moan, Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods, J. Comput. Appl. Math., 142 (2002), 313-330.  doi: 10.1016/S0377-0427(01)00492-7.

[9]

F. Casas, P. Chartier, A. Escorihuela-Tomàs and Y. Zhang, Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations, J. Comput. Appl. Math., 381 (2021), 113006.  doi: 10.1016/j.cam.2020.113006.

[10]

F. Castella, P. Chartier, S. Descombes and G. Vilmart, Splitting methods with complex times for parabolic equations, BIT Numer. Math., 49 (2009), 487-508.  doi: 10.1007/s10543-009-0235-y.

[11]

J. Chambers, Symplectic integrators with complex time steps, Astron. J., 126 (2003), 1119-1126.  doi: 10.1086/376844.

[12]

S. Flügge, Practical Quantum Mechanics, Springer, 1971.

[13]

A. Galindo and P. Pascual, Quantum Mechanics. I., Texts and Monographs in Physics. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-83854-5.

[14]

F. Goth, Higher order auxiliary field quantum Monte Carlo methods, arXiv: 2009.04491.

[15]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, 2006.

[16]

E. Hansen and A. Ostermann, Exponential splitting for unbounded operators, Math. Comput., 78 (2009), 1485-1496.  doi: 10.1090/S0025-5718-09-02213-3.

[17]

E. Hansen and A. Ostermann, High order splitting methods for analytic semigroups exist, BIT Numer. Math., 49 (2009), 527-542.  doi: 10.1007/s10543-009-0236-x.

[18]

C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, European Mathematical Society, 2008. doi: 10.4171/067.

[19]

R. McLachlan and R. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434.  doi: 10.1017/S0962492902000053.

[20]

T. Prosen and I. Pizorn, High order non-unitary split-step decomposition of unitary operators, J. Phys. A: Math. Gen., 39 (2006), 5957-5964.  doi: 10.1088/0305-4470/39/20/021.

[21]

Q. Sheng, Solving partial differential equations by exponential splitting, IMA J. Numer. Anal., 9 (1989), 199-212. doi: 10.1093/imanum/9.2.199.

[22]

M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A, 146 (1990), 319-323.  doi: 10.1016/0375-9601(90)90962-N.

[23]

M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys., 32 (1991), 400-407.  doi: 10.1063/1.529425.

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jana Rodriguez Hertz. Some advances on generic properties of the Oseledets splitting. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4323-4339. doi: 10.3934/dcds.2013.33.4323
[2]

Lukas Einkemmer, Alexander Ostermann. A comparison of boundary correction methods for Strang splitting. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2641-2660. doi: 10.3934/dcdsb.2018081
[3]

Raffaele D'Ambrosio, Martina Moccaldi, Beatrice Paternoster. Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2763-2773. doi: 10.3934/dcdsb.2018105
[4]

Raffaele D'Ambrosio, Stefano Di Giovacchino. Numerical preservation issues in stochastic dynamical systems by $ \vartheta $-methods. Journal of Computational Dynamics, 2022, 9 (2) : 123-131. doi: 10.3934/jcd.2021023
[5]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics and Games, 2021, 8 (4) : 467-486. doi: 10.3934/jdg.2021014
[6]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic and Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[7]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[8]

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella. Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinetic and Related Models, 2017, 10 (1) : 1-32. doi: 10.3934/krm.2017001
[9]

Jana Kopfová. Nonlinear semigroup methods in problems with hysteresis. Conference Publications, 2007, 2007 (Special) : 580-589. doi: 10.3934/proc.2007.2007.580
[10]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control and Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[11]

Zhong Tan, Huaqiao Wang, Yucong Wang. Time-splitting methods to solve the Hall-MHD systems with Lévy noises. Kinetic and Related Models, 2019, 12 (1) : 243-267. doi: 10.3934/krm.2019011
[12]

Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic and Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573
[13]

Maria Assunta Pozio, Fabio Punzo, Alberto Tesei. Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 891-916. doi: 10.3934/dcds.2011.30.891
[14]

Matthias Eller. Loss of derivatives for hyperbolic boundary problems with constant coefficients. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1347-1361. doi: 10.3934/dcdsb.2018154
[15]

Wataru Nakamura, Yasushi Narushima, Hiroshi Yabe. Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization. Journal of Industrial and Management Optimization, 2013, 9 (3) : 595-619. doi: 10.3934/jimo.2013.9.595
[16]

Alfonso Castro, Jorge Cossio, Carlos Vélez. Existence and qualitative properties of solutions for nonlinear Dirichlet problems. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 123-140. doi: 10.3934/dcds.2013.33.123
[17]

Lucio Boccardo, Maria Michaela Porzio. Some degenerate parabolic problems: Existence and decay properties. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 617-629. doi: 10.3934/dcdss.2014.7.617
[18]

Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11
[19]

Jie Sun. On methods for solving nonlinear semidefinite optimization problems. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 1-14. doi: 10.3934/naco.2011.1.1
[20]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53

 Impact Factor: 

Tools

Metrics

  • PDF downloads (258)
  • HTML views (198)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy