On symmetric-conjugate composition methods in the numerical integration of differential equations

Blanes, S.; Casas, F.; Chartier, P.; Escorihuela-Tomàs, A.

doi:10.1090/mcom/3715

On symmetric-conjugate composition methods in the numerical integration of differential equations
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by S. Blanes, F. Casas, P. Chartier and A. Escorihuela-Tomàs HTML | PDF

Math. Comp. 91 (2022), 1739-1761 Request permission

Abstract:

We analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and their efficiency is tested on several examples. Our analysis shows that higher-order schemes are more efficient even when time step sizes are relatively large.

References

V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295, DOI 10.1007/978-1-4757-2063-1
A. Aubry and P. Chartier, Pseudo-symplectic Runge-Kutta methods, BIT 38 (1998), no. 3, 439–461. MR 1652824, DOI 10.1007/BF02510253
A. Bandrauk, E. Dehghanian, and H. Lu, Complex integration steps in decomposition of quantum exponential evolution operators, Chem. Phys. Lett. 419 (2006), pp. 346–350.
Sergio Blanes and Fernando Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Math. 54 (2005), no. 1, 23–37. MR 2134093, DOI 10.1016/j.apnum.2004.10.005
Sergio Blanes and Fernando Casas, A concise introduction to geometric numerical integration, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. MR 3642447
S. Blanes, F. Casas, P. Chartier, and A. Murua, Optimized high-order splitting methods for some classes of parabolic equations, Math. Comp. 82 (2013), no. 283, 1559–1576. MR 3042575, DOI 10.1090/S0025-5718-2012-02657-3
S. Blanes, F. Casas, and A. Escorihuela-Tomàs, Applying splitting methods with complex coefficients to the numerical integration of unitary problems, arXiv:2104.02412, 2021 (to appear in J. Comput. Dyn.).
Sergio Blanes, Fernando Casas, and Ander Murua, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl. SeMA 45 (2008), 89–145. MR 2477860
Sergio Blanes, Fernando Casas, and Ander Murua, Splitting methods with complex coefficients, Bol. Soc. Esp. Mat. Apl. SeMA 50 (2010), 47–60. MR 2664321, DOI 10.1007/bf03322541
S. Blanes, F. Casas, and J. Ros, Extrapolation of symplectic integrators, Celestial Mech. Dynam. Astronom. 75 (1999), no. 2, 149–161. MR 1750213, DOI 10.1023/A:1008364504014
Fernando Casas, Philippe Chartier, Alejandro Escorihuela-Tomàs, and Yong Zhang, Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations, J. Comput. Appl. Math. 381 (2021), Paper No. 113006, 14. MR 4105675, DOI 10.1016/j.cam.2020.113006
F. Castella, P. Chartier, S. Descombes, and G. Vilmart, Splitting methods with complex times for parabolic equations, BIT 49 (2009), no. 3, 487–508. MR 2545817, DOI 10.1007/s10543-009-0235-y
J. Chambers, Symplectic integrators with complex time steps, Astron. J. 126 (2003), 1119–1126.
F. Goth, Higher order auxiliary field Monte Carlo methods, arXiv:2009.04491, 2020.
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006. Structure-preserving algorithms for ordinary differential equations. MR 2221614
Eskil Hansen and Alexander Ostermann, High order splitting methods for analytic semigroups exist, BIT 49 (2009), no. 3, 527–542. MR 2545819, DOI 10.1007/s10543-009-0236-x
Robert I. McLachlan, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comput. 16 (1995), no. 1, 151–168. MR 1311683, DOI 10.1137/0916010
Robert I. McLachlan, Families of high-order composition methods, Numer. Algorithms 31 (2002), no. 1-4, 233–246. Numerical methods for ordinary differential equations (Auckland, 2001). MR 1950923, DOI 10.1023/A:1021195019574
Robert I. McLachlan and G. Reinout W. Quispel, Splitting methods, Acta Numer. 11 (2002), 341–434. MR 2009376, DOI 10.1017/S0962492902000053
Hans Munthe-Kaas and Brynjulf Owren, Computations in a free Lie algebra, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999), no. 1754, 957–981. MR 1694699, DOI 10.1098/rsta.1999.0361
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation, vol. 7, Chapman & Hall, London, 1994. MR 1270017, DOI 10.1007/978-1-4899-3093-4
Haruo Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A 150 (1990), no. 5-7, 262–268. MR 1078768, DOI 10.1016/0375-9601(90)90092-3