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doi: 10.3934/jcd.2021022
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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 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, 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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