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A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards' equation
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 |
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.
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] | |
[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] | |
[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. |







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