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Splitting methods for differential equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical analysis: Ordinary differential equations Numerical problems in dynamical systems

Published online by Cambridge University Press:  04 September 2024

Sergio Blanes Open the ORCID record for Sergio Blanes [Opens in a new window] ,
Fernando Casas  and
Ander Murua
Sergio Blanes
Affiliation:
Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022-Valencia, Spain E-mail: [email protected]
Fernando Casas
Affiliation:
Departament de Matemàtiques and IMAC, Universitat Jaume I, 12071-Castellón, Spain E-mail: [email protected]
Ander Murua
Affiliation:
Konputazio Zientziak eta A.A. Saila, Informatika Fakultatea, EHU/UPV, Donostia/San Sebastián, Spain E-mail: [email protected]
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Abstract

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This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution. We analyse in detail the order conditions that have to be satisfied by these classes of methods to achieve a given order, and provide some insight about their qualitative properties in connection with geometric numerical integration and the treatment of highly oscillatory problems. Since splitting methods have received considerable attention in the realm of partial differential equations, we also cover this subject in the present survey, with special attention to parabolic equations and their problems. An exhaustive list of methods of different orders is collected and tested on simple examples. Finally, some applications of splitting methods in different areas, ranging from celestial mechanics to statistics, are also provided.

MSC classification

Primary: 65L05: Initial value problems 65L20: Stability and convergence of numerical methods 65P10: Hamiltonian systems including symplectic integrators
Secondary: 65M12: Stability and convergence of numerical methods 65M70: Spectral, collocation and related methods
Type
Research Article
Information
Acta Numerica , Volume 33 , July 2024 , pp. 1 - 161
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

References

Aichinger, M. and Krotscheck, E. (2005), A fast configuration space method for solving local Kohn–Sham equations, Comput. Mater. Sci. 34, 188–212.10.1016/j.commatsci.2004.11.002CrossRefGoogle Scholar
Akhmatskaya, E., Fernández-Pendás, M., Radivojević, T. and Sanz-Serna, J. M. (2017), Adaptive splitting integrators for enhancing sampling efficiency of modified Hamiltonian Monte Carlo methods in molecular simulations, Langmuir 33, 11530–11542.10.1021/acs.langmuir.7b01372CrossRefGoogle Scholar
Alamo, A. and Sanz-Serna, J. M. (2016), A technique for studying strong and weak local errors of splitting stochastic integrators, SIAM J. Numer. Anal. 54, 3239–3257.10.1137/16M1058765CrossRefGoogle Scholar
Alberdi, E., Antoñana, M., Makazaga, J. and Murua, A. (2019), An algorithm based on continuation techniques for minimization problems with highly non-linear equality constraints. Available at arXiv:1909.07263.Google Scholar
Alonso-Mallo, I., Cano, B. and Reguera, N. (2018), Avoiding order reduction when integrating linear initial boundary value problems with exponential splitting methods, IMA J. Numer. Anal. 38, 1294–1323.10.1093/imanum/drx047CrossRefGoogle Scholar
Alonso-Mallo, I., Cano, B. and Reguera, N. (2019), Avoiding order reduction when integrating reaction–diffusion boundary value problems with exponential splitting methods, J. Comput. Appl. Math. 357, 228–250.10.1016/j.cam.2019.02.023CrossRefGoogle Scholar
Alvermann, A. and Fehske, H. (2011), High-order commutator-free exponential time-propagation of driven quantum systems, J. Comput. Phys. 230, 5930–5956.10.1016/j.jcp.2011.04.006CrossRefGoogle Scholar
Andersen, H. C. (1983), Rattle: A ‘velocity’ version of the Shake algorithm for molecular dynamics calculations, J. Comput. Phys. 52, 24–34.10.1016/0021-9991(83)90014-1CrossRefGoogle Scholar
Anderson, M. H., Ensher, J. R., Mathews, M. R., Wieman, C. E. and Cornell, E. A. (1995), Observations of Bose–Einstein condensation in a dilute atomic vapor, Science 269, 198–201.10.1126/science.269.5221.198CrossRefGoogle Scholar
Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, second edition, Springer.10.1007/978-1-4757-2063-1CrossRefGoogle Scholar
Ascher, U., Ruuth, S. and Spiteri, R. (1997), Implicit–explicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25, 151–167.10.1016/S0168-9274(97)00056-1CrossRefGoogle Scholar
Ascher, U., Ruuth, S. and Wetton, B. (1995), Implicit–explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal. 32, 797–823.10.1137/0732037CrossRefGoogle Scholar
Auer, J., Krotscheck, E. and Chin, S. A. (2001), A fourth-order real-space algorithm for solving local Schrödinger equations, J. Chem. Phys. 115, 6841–6846.10.1063/1.1404142CrossRefGoogle Scholar
Bader, P. and Blanes, S. (2011), Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schrödinger equations, Phys. Rev. E 83, art. 046711.10.1103/PhysRevE.83.046711CrossRefGoogle ScholarPubMed
Bader, P., Blanes, S. and Casas, F. (2013), Solving the Schrödinger eigenvalue problem by the imaginary time propagation technique using splitting methods with complex coefficients, J. Chem. Phys. 139, art. 124117.10.1063/1.4821126CrossRefGoogle ScholarPubMed
Bader, P., Iserles, A., Kropielnicka, K. and Singh, P. (2014), Effective approximation for the semiclassical Schrödinger equation, Found. Comput. Math. 14, 689–720.10.1007/s10208-013-9182-8CrossRefGoogle Scholar
Bader, P., Iserles, A., Kropielnicka, K. and Singh, P. (2016), Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential, Proc. R. Soc. A 472, art. 20150733.10.1098/rspa.2015.0733CrossRefGoogle Scholar
Bandrauk, A. D. and Shen, H. (1991), Improved exponential split operator method for solving the time-dependent Schrödinger equation, Chem. Phys. Lett. 176, 428–432.10.1016/0009-2614(91)90232-XCrossRefGoogle Scholar
Bandrauk, A. D., Dehghanian, E. and Lu, H. (2006), Complex integration steps in decomposition of quantum exponential evolution operators, Chem. Phys. Lett. 419, 346–350.10.1016/j.cplett.2005.12.006CrossRefGoogle Scholar
Bao, W., Cai, Y. and Feng, Y. (2022), Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein–Gordon equation with weak nonlinearity, SIAM J. Numer. Anal. 60, 1962–1984.10.1137/21M1449774CrossRefGoogle Scholar
Bao, W., Cai, Y. and Yin, J. (2020), Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime, Math. Comp. 89, 2141–2173.10.1090/mcom/3536CrossRefGoogle Scholar
Bao, W., Jaksch, D. and Markowich, P. A. (2003a), Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation, J. Comput. Phys. 187, 318–342.10.1016/S0021-9991(03)00102-5CrossRefGoogle Scholar
Bao, W., Jaksch, D. and Markowich, P. A. (2004), Three dimensional simulation of jet formation in collapsing condensates, J. Phys. B 37, 329–343.10.1088/0953-4075/37/2/003CrossRefGoogle Scholar
Bao, W., Jin, S. and Markowich, P. A. (2002), On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys. 175, 487–524.10.1006/jcph.2001.6956CrossRefGoogle Scholar
Bao, W., Jin, S. and Markowich, P. A. (2003b), Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput. 25, 27–64.10.1137/S1064827501393253CrossRefGoogle Scholar
Bellman, R. (1970), Introduction to Matrix Analysis, second edition, McGraw-Hill.Google Scholar
Bernier, J. (2021), Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators, Found. Comput. Math. 21, 1401–1439.10.1007/s10208-020-09487-4CrossRefGoogle Scholar
Bernier, J. and Grébert, B. (2022), Birkhoff normal forms for Hamiltonian PDEs in their energy space, J. Éc. Polytech. Math. 9, 681–745.10.5802/jep.193CrossRefGoogle Scholar
Bernier, J., Blanes, S., Casas, F. and Escorihuela-Tomàs, A. (2023), Symmetric-conjugate splitting methods for linear unitary problems, BIT 63, art. 58.10.1007/s10543-023-00998-4CrossRefGoogle Scholar
Bernier, J., Crouseilles, N. and Li, Y. (2021), Exact splitting methods for kinetic and Schrödinger equations, J. Sci. Comput. 86, art. 10.10.1007/s10915-020-01369-9CrossRefGoogle Scholar
Berry, D. W., Childs, A. M., Cleve, R., Kothari, R. and Somma, R. D. (2015), Simulating Hamiltonian dynamics with a truncated Taylor series, Phys. Rev. Lett. 114, art. 090502.10.1103/PhysRevLett.114.090502CrossRefGoogle ScholarPubMed
Blanes, S. and Budd, C. J. (2005), Adaptive geometric integrators for Hamiltonian problems with approximate scale invariance, SIAM J. Sci. Comput. 26, 1089–1113.10.1137/S1064827502416630CrossRefGoogle Scholar
Blanes, S. and Casas, F. (2005), On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Math. 54, 23–37.10.1016/j.apnum.2004.10.005CrossRefGoogle Scholar
Blanes, S. and Casas, F. (2006), Splitting methods for non-autonomous separable dynamical systems, J. Phys. A 39, 5405–5423.10.1088/0305-4470/39/19/S05CrossRefGoogle Scholar
Blanes, S. and Casas, F. (2016), A Concise Introduction to Geometric Numerical Integration, CRC Press.Google Scholar
Blanes, S. and Gradinaru, V. (2020), High order efficient splittings for the semiclassical time-dependent Schrödinger equation, J. Comput. Phys. 405, art. 109157.10.1016/j.jcp.2019.109157CrossRefGoogle Scholar
Blanes, S. and Iserles, A. (2012), Explicit adaptive symplectic integrators for solving Hamiltonian systems, Celestial Mech. Dynam. Astronom. 114, 297–317.10.1007/s10569-012-9441-zCrossRefGoogle Scholar
Blanes, S. and Moan, P. C. (2002), Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods, J. Comput. Appl. Math. 142, 313–330.10.1016/S0377-0427(01)00492-7CrossRefGoogle Scholar
Blanes, S., Calvo, M. P., Casas, F. and Sanz-Serna, J. M. (2021a), Symmetrically processed splitting integrators for enhanced Hamiltonian Monte Carlo sampling, SIAM J. Sci. Comput. 43, A3357–A3371.10.1137/20M137940XCrossRefGoogle Scholar
Blanes, S., Casas, F. and Escorihuela-Tomàs, A. (2022a), Applying splitting methods with complex coefficients to the numerical integration of unitary problems, J. Comput. Dyn. 9, 85–101.10.3934/jcd.2021022CrossRefGoogle Scholar
Blanes, S., Casas, F. and Escorihuela-Tomàs, A. (2022b), Runge–Kutta–Nyström symplectic splitting methods of order 8, Appl. Numer. Math. 182, 14–27.10.1016/j.apnum.2022.07.010CrossRefGoogle Scholar
Blanes, S., Casas, F. and Murua, A. (2004), On the numerical integration of ordinary differential equations by processed methods, SIAM J. Numer. Anal. 42, 531–552.10.1137/S0036142902417029CrossRefGoogle Scholar
Blanes, S., Casas, F. and Murua, A. (2006a), Composition methods for differential equations with processing, SIAM J. Sci. Comput. 27, 1817–1843.10.1137/030601223CrossRefGoogle Scholar
Blanes, S., Casas, F. and Murua, A. (2006b), Symplectic splitting operator methods tailored for the time-dependent Schrödinger equation, J. Chem. Phys. 124, art. 234105.10.1063/1.2203609CrossRefGoogle ScholarPubMed
Blanes, S., Casas, F. and Murua, A. (2008a), On the linear stability of splitting methods, Found. Comput. Math. 8, 357–393.10.1007/s10208-007-9007-8CrossRefGoogle Scholar
Blanes, S., Casas, F. and Murua, A. (2008b), Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl. 45, 89–145.Google Scholar
Blanes, S., Casas, F. and Murua, A. (2011), Error analysis of splitting methods for the time dependent Schrödinger equation, SIAM J. Sci. Comput. 33, 1525–1548.10.1137/100794535CrossRefGoogle Scholar
Blanes, S., Casas, F. and Murua, A. (2015), An efficient algorithm based on splitting for the time integration of the Schrödinger equation, J. Comput. Phys. 303, 396–412.10.1016/j.jcp.2015.09.047CrossRefGoogle Scholar
Blanes, S., Casas, F. and Murua, A. (2017a), Symplectic time-average propagators for the Schrödinger equation with a time-dependent Hamiltonian, J. Chem. Phys. 146, art. 114109.10.1063/1.4978410CrossRefGoogle ScholarPubMed
Blanes, S., Casas, F. and Ros, J. (1999a), Extrapolation of symplectic integrators, Celestial Mech. Dynam. Astronom. 75, 149–161.10.1023/A:1008364504014CrossRefGoogle Scholar
Blanes, S., Casas, F. and Ros, J. (1999b), Symplectic integrators with processing: A general study, SIAM J. Sci. Comput. 21, 711–727.10.1137/S1064827598332497CrossRefGoogle Scholar
Blanes, S., Casas, F. and Ros, J. (2000), Processing symplectic methods for near-integrable Hamiltonian systems, Celestial Mech. Dynam. Astronom. 77, 17–35.10.1023/A:1008311025472CrossRefGoogle Scholar
Blanes, S., Casas, F. and Ros, J. (2001a), High-order Runge–Kutta–Nyström geometric methods with processing, Appl. Numer. Math. 39, 245–259.10.1016/S0168-9274(00)00035-0CrossRefGoogle Scholar
Blanes, S., Casas, F. and Ros, J. (2001b), New families of symplectic Runge–Kutta–Nyström integration methods, in Numerical Analysis and its Applications, Vol. 1988 of Lecture Notes in Computer Science, Springer, pp. 102–109.10.1007/3-540-45262-1_13CrossRefGoogle Scholar
Blanes, S., Casas, F. and Sanz-Serna, J. M. (2014), Numerical integrators for the Hybrid Monte Carlo method, SIAM J. Sci. Comput. 36, A1556–A1580.10.1137/130932740CrossRefGoogle Scholar
Blanes, S., Casas, F. and Thalhammer, M. (2017b), High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations, Comput. Phys. Commun. 220, 243–262.10.1016/j.cpc.2017.07.016CrossRefGoogle Scholar
Blanes, S., Casas, F. and Thalhammer, M. (2018), Convergence analysis of high-order commutator-free quasi-Magnus integrators for nonautonomous linear evolution equations, IMA J. Numer. Anal. 38, 743–778.10.1093/imanum/drx012CrossRefGoogle Scholar
Blanes, S., Casas, F., Chartier, P. and Escorihuela-Tomàs, A. (2022c), On symmetric-conjugate composition methods in the numerical integration of differential equations, Math. Comp. 91, 1739–1761.10.1090/mcom/3715CrossRefGoogle Scholar
Blanes, S., Casas, F., Chartier, P. and Murua, A. (2013a), Optimized high-order splitting methods for some classes of parabolic equations, Math. Comp. 82, 1559–1576.10.1090/S0025-5718-2012-02657-3CrossRefGoogle Scholar
Blanes, S., Casas, F., Farrés, A., Laskar, J., Makazaga, J. and Murua, A. (2013b), New families of symplectic splitting methods for numerical integration in dynamical astronomy, Appl. Numer. Math. 68, 58–72.10.1016/j.apnum.2013.01.003CrossRefGoogle Scholar
Blanes, S., Casas, F., González, C. and Thalhammer, M. (2021b), Convergence analysis of high-order commutator-free quasi-Magnus integrators for nonautonomous linear Schrödinger equations, IMA J. Numer. Anal. 41, 594–617.10.1093/imanum/drz058CrossRefGoogle Scholar
Blanes, S., Casas, F., González, C. and Thalhammer, M. (2023), Efficient splitting methods based on modified potentials: Numerical integration of linear parabolic problems and imaginary time propagation of the Schrödinger equation, Commun. Comput. Phys. 33, 937–961.10.4208/cicp.OA-2022-0247CrossRefGoogle Scholar
Blanes, S., Casas, F., Oteo, J. A. and Ros, J. (2009), The Magnus expansion and some of its applications, Phys. Rep. 470, 151–238.10.1016/j.physrep.2008.11.001CrossRefGoogle Scholar
Blanes, S., Iserles, A. and Macnamara, S. (2022d), Positivity-preserving methods for ordinary differential equations, ESAIM Math. Model. Numer. Anal. 56, 1843–1870.10.1051/m2an/2022042CrossRefGoogle Scholar
Bolley, C. and Crouzeix, M. (1978), Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques, RAIRO Anal. Numér. 12, 237–245.10.1051/m2an/1978120302371CrossRefGoogle Scholar
Bou-Rabee, N. (2017), Cayley splitting for second-order Langevin stochastic partial differential equations. Available at arXiv:1707.05603.Google Scholar
Bou-Rabee, N. and Sanz-Serna, J. M. (2018), Geometric integrators and the Hamiltonian Monte Carlo method, Acta Numer. 27, 113–206.10.1017/S0962492917000101CrossRefGoogle Scholar
Bréhier, C.-E., Cohen, D. and Giordano, G. (2024), Splitting schemes for Fitzhugh–Nagumo stochastic partial differential equations, Discrete Contin. Dyn. Syst. Ser. B 29, 214–244.10.3934/dcdsb.2023094CrossRefGoogle Scholar
Bréhier, C.-E., Cohen, D. and Jahnke, T. (2023), Splitting integrators for stochastic Lie–Poisson systems, Math. Comp. 92, 2167–2216.10.1090/mcom/3829CrossRefGoogle Scholar
Brooks, S., Gelman, A., Jones, G. L. and Meng, X.-L., eds (2011), Handbook of Markov Chain Monte Carlo, CRC Press.10.1201/b10905CrossRefGoogle Scholar
Butcher, J. C. (1969), The effective order of Runge–Kutta methods, in Proceedings of the Conference on the Numerical Solution of Differential Equations (Morris, J. L., ed.), Vol. 109 of Lecture Notes in Mathematics, Springer, pp. 133–139.10.1007/BFb0060019CrossRefGoogle Scholar
Butcher, J. C. and Sanz-Serna, J. M. (1996), The number of conditions for a Runge–Kutta method to have effective order p, Appl. Numer. Math. 22, 103–111.10.1016/S0168-9274(96)00028-1CrossRefGoogle Scholar
Butler, S. T. and Friedman, M. H. (1955), Partition function for a system of interacting Bose–Einstein particles, Phys. Rev. 98, 287–293.10.1103/PhysRev.98.287CrossRefGoogle Scholar
Calvo, M. P. and Hairer, E. (1995), Accurate long-term integration of dynamical systems, Appl. Numer. Math. 18, 95–105.10.1016/0168-9274(95)00046-WCrossRefGoogle Scholar
Calvo, M. P. and Sanz-Serna, J. M. (1993a), The development of variable-step symplectic integrators, with application to the two-body problem, SIAM J. Sci. Comput. 14, 936–952.10.1137/0914057CrossRefGoogle Scholar
Calvo, M. P. and Sanz-Serna, J. M. (1993b), High-order symplectic Runge–Kutta–Nyström methods, SIAM J. Sci. Comput. 14, 1237–1252.10.1137/0914073CrossRefGoogle Scholar
Calvo, M. P., López-Marcos, M. A. and Sanz-Serna, J. M. (1998), Variable step implementation of geometric integrators, Appl. Numer. Math. 28, 1–16.10.1016/S0168-9274(98)00035-XCrossRefGoogle Scholar
Campostrini, M. and Rossi, P. (1990), A comparison of numerical algorithms for dynamical fermions, Nuclear Phys. B 329, 753–764.10.1016/0550-3213(90)90081-NCrossRefGoogle Scholar
Candy, J. and Rozmus, W. (1991), A symplectic integration algorithm for separable Hamiltonian functions, J. Comput. Phys. 92, 230–256.10.1016/0021-9991(91)90299-ZCrossRefGoogle Scholar
Carles, R. (2008), Semi-Classical Analysis for Nonlinear Schrödinger Equations, World Scientific.10.1142/6753CrossRefGoogle Scholar
Casas, F. and Murua, A. (2009), An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications, J. Math. Phys. 50, art. 033513.10.1063/1.3078418CrossRefGoogle Scholar
Casas, F., Crouseilles, N., Faou, E. and Mehrenberger, M. (2017), High-order Hamiltonian splitting for the Vlasov–Poisson equations, Numer. Math. 135, 769–801.10.1007/s00211-016-0816-zCrossRefGoogle Scholar
Casas, F., Sanz-Serna, J. M. and Shaw, L. (2022), Split Hamiltonian Monte Carlo revisited, Statist. Comput. 32, art. 86.10.1007/s11222-022-10149-4CrossRefGoogle Scholar
Casas, F., Sanz-Serna, J. M. and Shaw, L. (2023), A new optimality property of Strang’s splitting, SIAM J. Numer. Anal. 61, 1369–1385.10.1137/22M1528690CrossRefGoogle Scholar
Castella, F., Chartier, P., Descombes, S. and Vilmart, G. (2009), Splitting methods with complex times for parabolic equations, BIT 49, 487–508.10.1007/s10543-009-0235-yCrossRefGoogle Scholar
Cazenave, T. (2003), Semilinear Schrödinger Equations, American Mathematical Society.10.1090/cln/010CrossRefGoogle Scholar
Ceperley, D. M. (1995), Path integrals in the theory of condensed helium, Rev. Modern Phys. 67, 279–355.10.1103/RevModPhys.67.279CrossRefGoogle Scholar
Chambers, J. E. (2003), Symplectic integrators with complex time steps, Astron. J. 126, 1119–1126.10.1086/376844CrossRefGoogle Scholar
Chan, R. and Murua, A. (2000), Extrapolation of symplectic methods for Hamiltonian problems, Appl. Numer. Math. 34, 189–205.10.1016/S0168-9274(99)00127-0CrossRefGoogle Scholar
Channell, P. J. and Scovel, J. C. (1990), Symplectic integration of Hamiltonian systems, Nonlinearity 3, 231–259.10.1088/0951-7715/3/2/001CrossRefGoogle Scholar
Chartier, P. and Murua, A. (2009), An algebraic theory of order, ESAIM Math. Model. Numer. Anal. 43, 607–630.10.1051/m2an/2009029CrossRefGoogle Scholar
Chen, K.-T. (1957), Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula, Ann. of Math. 65, 163–178.10.2307/1969671CrossRefGoogle Scholar
Chen, Y.-A., Childs, A. M., Hafezi, M., Jiang, Z., Kim, H. and Xu, Y. (2022), Efficient product formulas for commutators and applications to quantum simulations, Phys. Rev. Res. 4, art. 013191.10.1103/PhysRevResearch.4.013191CrossRefGoogle Scholar
Childs, A. M. and Su, Y. (2019), Nearly optimal lattice simulation by product formulas, Phys. Rev. Lett. 123, art. 050503.10.1103/PhysRevLett.123.050503CrossRefGoogle ScholarPubMed
Childs, A. M., Su, Y., Tran, M. C., Wiebe, N. and Zhu, S. (2021), Theory of Trotter error with commutator scaling, Phys. Rev. X 11, art. 011020.Google Scholar
Chin, S. A. (1997), Symplectic integrators from composite operator factorizations, Phys. Lett. A 226, 344–348.10.1016/S0375-9601(97)00003-0CrossRefGoogle Scholar
Chin, S. A. (2005), Structure of positive decomposition of exponential operators, Phys. Rev. E 71, art. 016703.10.1103/PhysRevE.71.016703CrossRefGoogle ScholarPubMed
Chin, S. A. (2006), Complete characterization of fourth-order symplectic integrators with extended-linear coefficients, Phys. Rev. E 73, art. 026705.10.1103/PhysRevE.73.026705CrossRefGoogle ScholarPubMed
Chin, S. A. (2023), Anatomy of path integral Monte Carlo: Algebraic derivation of the harmonic oscillator’s universal discrete imaginary-time propagator and its sequential optimization, J. Chem. Phys. 159, art. 134109.10.1063/5.0164086CrossRefGoogle ScholarPubMed
Chin, S. A. and Chen, C. R. (2002), Gradient symplectic algorithms for solving the Schrödinger equation with time-dependent potentials, J. Chem. Phys. 117, 1409–1415.10.1063/1.1485725CrossRefGoogle Scholar
Chin, S. A. and Geiser, J. (2011), Multi-product operator splitting as a general method of solving autonomous and nonautonomous equations, IMA J. Numer. Anal. 31, 1552–1577.10.1093/imanum/drq022CrossRefGoogle Scholar
Chin, S. A. and Krotscheck, E. (2005), Fourth-order algorithms for solving the imaginary-time Gross–Pitaevskii equation in a rotating anisotropic trap, Phys. Rev. E 72, art. 036705.10.1103/PhysRevE.72.036705CrossRefGoogle Scholar
Chorin, A., Huges, T. J. R., Marsden, J. E. and McCracken, M. (1978), Product formulas and numerical algorithms, Commun. Pure Appl. Math. 31, 205–256.10.1002/cpa.3160310205CrossRefGoogle Scholar
Chou, L. Y. and Sharp, P. W. (2000), Order 5 symplectic explicit Runge–Kutta–Nyström methods, J. Appl. Math. Decision Sci. 4, 143–150.10.1155/S1173912600000109CrossRefGoogle Scholar
Ciarlet, P. G. and Lions, J. L., eds (1990), Handbook of Numerical Analysis, Vol. I, North-Holland.Google Scholar
Cohen, J. E., Friedland, S., Kato, T. and Kelly, F. P. (1982), Eigenvalue inequalities for products of matrix exponentials, Linear Algebra Appl. 45, 55–95.10.1016/0024-3795(82)90211-7CrossRefGoogle Scholar
Creutz, M. and Gocksch, A. (1989), Higher-order hybrid Monte Carlo algorithms, Phys. Rev. Lett. 63, 9–12.10.1103/PhysRevLett.63.9CrossRefGoogle ScholarPubMed
Crouch, P. E. and Grossman, R. (1993), Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci. 3, 1–33.10.1007/BF02429858CrossRefGoogle Scholar
Danby, J. M. A. (1988), Fundamentals of Celestial Mechanics, Willmann-Bell. Google Scholar
de Raedt, H. and de Raedt, B. (1983), Applications of the generalized Trotter formula, Phys. Rev. A 28, 3575–3580.10.1103/PhysRevA.28.3575CrossRefGoogle Scholar
de Vogelaere, R. (1956), Methods of integration which preserve the contact transformation property of the Hamiltonian equations. Report no. 4, Department of Mathematics, University of Notre Dame, IN.Google Scholar
del Valle, J. C. and Kropielnicka, K. (2023), Family of Strang-type exponential splitting in the presence of unbounded and time dependent operators. Available at arXiv:2310.01556.Google Scholar
Douglas, J. and Rachford, H. H. (1956), On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. 82, 421–439.10.1090/S0002-9947-1956-0084194-4CrossRefGoogle Scholar
Duane, S., Kennedy, A. D., Pendleton, B. J. and Roweth, D. (1987), Hybrid Monte-Carlo, Phys. Lett. B 195, 216–222.10.1016/0370-2693(87)91197-XCrossRefGoogle Scholar
Einkemmer, L. and Ostermann, A. (2015), Overcoming order reduction in diffusion–reaction splitting, Part I: Dirichlet boundary conditions, SIAM J. Sci. Comput. 37, A1577–A1592.10.1137/140994204CrossRefGoogle Scholar
Einkemmer, L. and Ostermann, A. (2016), Overcoming order reduction in diffusion–reaction splitting, Part II: Oblique boundary conditions, SIAM J. Sci. Comput. 38, A3741–A3757.10.1137/16M1056250CrossRefGoogle Scholar
Einkemmer, L., Moccaldi, M. and Ostermann, A. (2018), Efficient boundary corrected Strang splitting, Appl. Math. Comput. 332, 76–89.Google Scholar
Engel, K.-J. and Nagel, R. (2006), A Short Course on Operator Semigroups, Springer.Google Scholar
Faou, E. (2012), Geometric Numerical Integration and Schrödinger Equations, European Mathematical Society.10.4171/100CrossRefGoogle Scholar
Faou, E. and Grébert, B. (2011), Hamiltonian interpolation of splitting approximation for Hamiltonian PDEs, Found. Comput. Math. 11, 381–415.10.1007/s10208-011-9094-4CrossRefGoogle Scholar
Faou, E. and Lubich, C. (2006), A Poisson integrator for Gaussian wavepacket dynamics, Comput. Vis. Sci. 9, 45–55.10.1007/s00791-006-0019-8CrossRefGoogle Scholar
Faou, E., Gradinaru, V. and Lubich, C. (2009), Computing semi-classical quantum dynamics with Hagedorn wavepackets, SIAM J. Sci. Comput. 31, 3027–3041.10.1137/080729724CrossRefGoogle Scholar
Feit, M. D., Fleck, J. A. Jr and Steiger, A. (1982), Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47, 412–433.10.1016/0021-9991(82)90091-2CrossRefGoogle Scholar
Feng, K. (1992), Formal power series and numerical algorithms for dynamical systems, in Proc. Conf. Scientific Computation Hangzhou, 1991 (Chan, T. and Shi, Z.-C., eds), World Scientific, pp. 28–35.Google Scholar
Feng, K. and Qin, M. Z. (1987), The symplectic methods for the computation of Hamiltonian equations, in Numerical Methods for Partial Differential Equations (Zhu, Y. L. and Gao, B. Y., eds), Vol. 1297 of Lecture Notes in Mathematics, Springer, pp. 1–37.10.1007/BFb0078537CrossRefGoogle Scholar
Fernández-Pendás, M., Akhmatskaya, E. and Sanz-Serna, J. M. (2016), Adaptive multi-stage integrators for optimal energy conservation in molecular simulations, J. Comput. Phys. 327, 434–449.10.1016/j.jcp.2016.09.035CrossRefGoogle Scholar
Feynman, R. P. (1972), Statistical Mechanics: A Set of Lectures, Addison-Wesley.Google Scholar
Feynman, R. P. (1982), Simulating physics with computers, Internat. J. Theoret. Phys. 21, 467–488.10.1007/BF02650179CrossRefGoogle Scholar
Folkner, W., Williams, J., Boggs, D., Park, R. and Kuchynka, P. (2014), The planetary and lunar ephemerides DE430 and DE431. IPN Progress Report 42-196.Google Scholar
Forest, E. (1992), Sixth-order Lie group integrators, J. Comput. Phys. 99, 209–213.10.1016/0021-9991(92)90203-BCrossRefGoogle Scholar
Forest, E. and Ruth, R. (1990), Fourth-order symplectic integration, Phys. D 43, 105–117.10.1016/0167-2789(90)90019-LCrossRefGoogle Scholar
Foster, J., Reis, G. and Strange, C. (2024), High order splitting methods for SDEs satisfying a commutativity condition, SIAM J. Numer. Anal. 62, 500–532.10.1137/23M155147XCrossRefGoogle Scholar
García-Archilla, B., Sanz-Serna, J. M. and Skeel, R. D. (1999), Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput. 20, 930–963.10.1137/S1064827596313851CrossRefGoogle Scholar
Glowinski, R., Osher, S. J. and Yin, W., eds (2016a), Splitting Methods in Communication, Imaging, Science, and Engineering, Springer.10.1007/978-3-319-41589-5CrossRefGoogle Scholar
Glowinski, R., Pan, T.-W. and Tai, X.-C. (2016b), Some facts about operator-splitting and alternating direction methods, in Splitting Methods in Communication, Imaging, Science, and Engineering (Glowinski, R., Osher, S. J. and Yin, W., eds), Springer, pp. 19–94.10.1007/978-3-319-41589-5_2CrossRefGoogle Scholar
Golden, S. (1957), Statistical theory of many-electron systems: Discrete bases of representation, Phys. Rev. 107, 1283–1290.10.1103/PhysRev.107.1283CrossRefGoogle Scholar
Goldman, D. and Kaper, T. J. (1996), Nth-order operator splitting schemes and nonreversible systems, SIAM J. Numer. Anal. 33, 349–367.10.1137/0733018CrossRefGoogle Scholar
Goldstein, H. (1980), Classical Mechanics, second edition, Addison-Wesley.Google Scholar
Gragg, W. B. (1965), On extrapolation algorithms for ordinary differential equations, SIAM J. Numer. Anal. 2, 384–403.Google Scholar
Gray, S. and Manolopoulos, D. E. (1996), Symplectic integrators tailored to the time-dependent Schrödinger equation, J. Chem. Phys. 104, 7099–7112.10.1063/1.471428CrossRefGoogle Scholar
Gray, S. and Verosky, J. M. (1994), Classical Hamiltonian structures in wave packet dynamics, J. Chem. Phys. 100, 5011–5022.10.1063/1.467219CrossRefGoogle Scholar
Griffiths, D. F. and Sanz-Serna, J. M. (1986), On the scope of the method of modified equations, SIAM J. Sci. Statist. Comput. 7, 994–1008.10.1137/0907067CrossRefGoogle Scholar
Hairer, E. (1994), Backward analysis of numerical integrators and symplectic methods, Ann. Numer. Math. 1, 107–132.Google Scholar
Hairer, E., Lubich, C. and Wanner, G. (2003), Geometric numerical integration illustrated by the Störmer–Verlet method, Acta Numer. 12, 399–450.10.1017/S0962492902000144CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G. (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, second edition, Springer.Google Scholar
Hairer, E., Nørsett, S. P. and Wanner, G. (1993), Solving Ordinary Differential Equations I: Nonstiff Problems, second revised edition, Springer.Google Scholar
Hansen, E. and Ostermann, A. (2009a), Exponential splitting for unbounded operators, Math. Comp. 78, 1485–1496.10.1090/S0025-5718-09-02213-3CrossRefGoogle Scholar
Hansen, E. and Ostermann, A. (2009b), High order splitting methods for analytic semigroups exist, BIT 49, 527–542.10.1007/s10543-009-0236-xCrossRefGoogle Scholar
He, Y., Sun, Y., Liu, J. and Qin, H. (2015), Volume-preserving algorithms for charged particle dynamics, J. Comput. Phys. 281, 135–147.10.1016/j.jcp.2014.10.032CrossRefGoogle Scholar
He, Y., Sun, Y., Liu, J. and Qin, H. (2016), Higher order volume-preserving schemes for charged particle dynamics, J. Comput. Phys. 305, 172–184.10.1016/j.jcp.2015.10.032CrossRefGoogle Scholar
Hochbruck, M. and Lubich, C. (2003), On Magnus integrators for time-dependent Schrödinger equations, SIAM J. Numer. Anal. 41, 945–963.10.1137/S0036142902403875CrossRefGoogle Scholar
Hoffman, M. (2000), Quasi-shuffle products, J. Algebraic Combin. 11, 49–68.10.1023/A:1008791603281CrossRefGoogle Scholar
Holden, H., Karlsen, K. H., Lie, K.-A. and Risebro, N. H. (2010), Splitting Methods for Partial Differential Equations with Rough Solutions, European Mathematical Society.10.4171/078CrossRefGoogle Scholar
Holden, H., Karlsen, K. H., Risebro, N. E. and Tao, T. (2011), Operator splitting for the KdV equation, Math. Comp. 80, 821–846.10.1090/S0025-5718-2010-02402-0CrossRefGoogle Scholar
Holden, H., Lubich, C. and Risebro, N. E. (2013), Operator splitting for partial differential equations with Burgers nonlinearity, Math. Comp. 82, 173–185.10.1090/S0025-5718-2012-02624-XCrossRefGoogle Scholar
Hundsdorfer, W. and Verwer, J. G. (1995), A note on splitting errors for advection–reaction equations, Appl. Numer. Math. 18, 191–199.10.1016/0168-9274(95)00069-7CrossRefGoogle Scholar
Hundsdorfer, W. and Verwer, J. G. (2003), Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations, Springer.10.1007/978-3-662-09017-6CrossRefGoogle Scholar
Iserles, A. (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press.Google Scholar
Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. and Zanna, A. (2000), Lie-group methods, Acta Numer. 9, 215–365.10.1017/S0962492900002154CrossRefGoogle Scholar
Jahnke, T. and Lubich, C. (2000), Error bounds for exponential operator splittings, BIT 40, 735–744.10.1023/A:1022396519656CrossRefGoogle Scholar
Jin, S., Markowich, P. and Sparber, C. (2011), Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numer. 20, 121–209.10.1017/S0962492911000031CrossRefGoogle Scholar
Kahan, W. and Li, R. C. (1997), Composition constants for raising the order of unconventional schemes for ordinary differential equations, Math. Comp. 66, 1089–1099.10.1090/S0025-5718-97-00873-9CrossRefGoogle Scholar
Kieri, E. (2015), Stiff convergence of force-gradient operator splitting methods, Appl. Numer. Math. 94, 33–45.10.1016/j.apnum.2015.03.005CrossRefGoogle Scholar
Koseleff, P.-V. (1993), Calcul formel pour les méthodes de Lie en mécanique hamiltonienne. PhD thesis, École Polytechnique.Google Scholar
Landau, D. P. and Binder, K., eds (2005), A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press.10.1017/CBO9780511614460CrossRefGoogle Scholar
Lasagni, F. M. (1988), Canonical Runge–Kutta methods, Z. Angew. Math. Phys. 39, 952–953.10.1007/BF00945133CrossRefGoogle Scholar
Laskar, J. (1989), A numerical experiment on the chaotic behaviour of the solar system, Nature 338, 237–238.10.1038/338237a0CrossRefGoogle Scholar
Laskar, J. and Robutel, P. (2001), High order symplectic integrators for perturbed Hamiltonian systems, Celestial Mech. Dynam. Astronom. 80, 39–62.10.1023/A:1012098603882CrossRefGoogle Scholar
Laskar, J., Robutel, P., Joutel, F., Gastineau, M., Correia, A. C. M. and Levrard, B. (2004), A long-term numerical solution for the insolation quantities of the Earth, Astron. Astrophys. 428, 261–285.10.1051/0004-6361:20041335CrossRefGoogle Scholar
Laslett, L. J. (1986), Nonlinear dynamics: A personal perspective, in Nonlinear Dynamics Aspects of Particle Accelerators (Jowett, J. M., Month, M. and Turner, S., eds), Vol. 247 of Lecture Notes in Physics, Springer, pp. 519–580.10.1007/BFb0107360CrossRefGoogle Scholar
Lasser, C. and Lubich, C. (2020), Computing quantum dynamics in the semiclassical regime, Acta Numer. 29, 229–401.10.1017/S0962492920000033CrossRefGoogle Scholar
Lehtovaara, L., Toivanen, J. and Eloranta, J. (2007), Solution of time-independent Schrödinger equation by the imaginary time propagation method, J. Comput. Phys. 221, 148–157.10.1016/j.jcp.2006.06.006CrossRefGoogle Scholar
Leimkuhler, B. and Matthews, C. (2013a), Rational construction of stochastic numerical methods for molecular sampling, Appl. Math. Res. Express 2013, 34–56.Google Scholar
Leimkuhler, B. and Matthews, C. (2013b), Robust and efficient configurational molecular sampling via Langevin dynamics, J. Chem. Phys. 138, art. 174102.10.1063/1.4802990CrossRefGoogle ScholarPubMed
Leimkuhler, B. and Matthews, C. (2015), Molecular Dynamics, Springer.10.1007/978-3-319-16375-8CrossRefGoogle Scholar
Leimkuhler, B. and Reich, S. (2004), Simulating Hamiltonian Dynamics, Cambridge University Press.Google Scholar
Leimkuhler, B., Matthews, C. and Stolz, G. (2016), The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics, IMA J. Numer. Anal. 36, 13–79.Google Scholar
Leimkuhler, B., Reich, S. and Skeel, R. D. (1996), Integration methods for molecular dynamics, in Mathematical Approaches to Biomolecular Structure and Dynamics (Mesirov, J. P., Schulten, K. and Sumners, D. W., eds), Springer, pp. 161–185.10.1007/978-1-4612-4066-2_10CrossRefGoogle Scholar
Lie, S. (1888), Theorie der Transformationsgruppen, Druck un Verlag.Google Scholar
Liu, J. S. (2008), Monte Carlo Strategies for Scientific Computing, second edition, Springer.Google Scholar
Liu, X., Ding, P., Hong, J. and Wang, L. (2005), Optimization of symplectic schemes for time-dependent Schrödinger equations, Comput. Math. Appl. 50, 637–644.10.1016/j.camwa.2004.09.015CrossRefGoogle Scholar
Lloyd, S. (1996), Universal quantum simulators, Science 273, 1073–1078.10.1126/science.273.5278.1073CrossRefGoogle ScholarPubMed
López-Marcos, M. A., Sanz-Serna, J. M. and Skeel, R. D. (1996a), Cheap enhancement of symplectic integrators, in Proceedings of the Dundee Conference on Numerical Analysis (Griffiths, D. F. and Watson, G. A., eds), Longman, pp. 107–122.Google Scholar
López-Marcos, M. A., Sanz-Serna, J. M. and Skeel, R. D. (1996b), An explicit symplectic integrator with maximal stability interval, in Numerical Analysis: A.R. Mitchell 75th Birthday Volume (Griffiths, D. F. and Watson, G. A., eds), World Scientific, pp. 163–176.10.1142/9789812812872_0012CrossRefGoogle Scholar
López-Marcos, M. A., Sanz-Serna, J. M. and Skeel, R. D. (1997), Explicit symplectic integrators using Hessian-vector products, SIAM J. Sci. Comput. 18, 223–238.10.1137/S1064827595288085CrossRefGoogle Scholar
Lubich, C. (2008), From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, European Mathematical Society.10.4171/067CrossRefGoogle Scholar
MacNamara, S. and Strang, G. (2016), Operator splitting, in Splitting Methods in Communication, Imaging, Science, and Engineering (Glowinski, R., Osher, S. J. and Yin, W., eds), Springer, pp. 95–114.10.1007/978-3-319-41589-5_3CrossRefGoogle Scholar
Magnus, W. (1954), On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. VII, 649–673.10.1002/cpa.3160070404CrossRefGoogle Scholar
Marchuk, G. I. (1990), Splitting and alternating direction methods, in Handbook of Numerical Analysis (Ciarlet, P. G. and Lions, J. L., eds), Vol. 1, North-Holland, pp. 197–462.Google Scholar
McLachlan, R. I. (1995a), Composition methods in the presence of small parameters, BIT 35, 258–268.10.1007/BF01737165CrossRefGoogle Scholar
McLachlan, R. I. (1995b), On the numerical integration of ODE’s by symmetric composition methods, SIAM J. Sci. Comput. 16, 151–168.10.1137/0916010CrossRefGoogle Scholar
McLachlan, R. I. (1996), More on symplectic correctors, in Integration Algorithms and Classical Mechanics (Marsden, J. E., Patrick, G. W. and Shadwick, W. F., eds), Vol. 10 of Fields Institute Communications, American Mathematical Society, pp. 141–149.Google Scholar
McLachlan, R. I. (2002), Families of high-order composition methods, Numer. Algorithms 31, 233–246.10.1023/A:1021195019574CrossRefGoogle Scholar
McLachlan, R. I. and Atela, P. (1992), The accuracy of symplectic integrators, Nonlinearity 5, 541–562.10.1088/0951-7715/5/2/011CrossRefGoogle Scholar
McLachlan, R. I. and Gray, S. K. (1997), Optimal stability polynomials for splitting methods, with applications to the time-dependent Schrödinger equation, Appl. Numer. Math. 25, 275–286.10.1016/S0168-9274(97)00064-0CrossRefGoogle Scholar
McLachlan, R. I. and Murua, A. (2019), The Lie algebra of classical mechanics, J. Comput. Dyn. 6, 198–213.Google Scholar
McLachlan, R. I. and Quispel, R. (2002), Splitting methods, Acta Numer. 11, 341–434.10.1017/S0962492902000053CrossRefGoogle Scholar
McLachlan, R. I. and Quispel, R. (2006), Geometric integrators for ODEs, J. Phys. A 39, 5251–5285.10.1088/0305-4470/39/19/S01CrossRefGoogle Scholar
Messiah, A. (1999), Quantum Mechanics, Dover.Google Scholar
Miessen, A., Ollitrault, P. J., Tacchino, F. and Tavernelli, I. (2023), Quantum algorithms for quantum dynamics, Nat. Comput. Sci. 3, 25–37.10.1038/s43588-022-00374-2CrossRefGoogle ScholarPubMed
Mikkola, S. (1997), Practical symplectic methods with time transformation for the few-body problem, Celestial Mech. Dynam. Astronom. 67, 145–165.10.1023/A:1008217427749CrossRefGoogle Scholar
Moan, P. C. (2002), On backward error analysis and Nekhoroshev stability in the numerical analysis of conservative systems of ODEs. PhD thesis, University of Cambridge.Google Scholar
Moan, P. C. (2004), On the KAM and Nekhorosev theorems for symplectic integrators and implications for error growth, Nonlinearity 17, 67–83.10.1088/0951-7715/17/1/005CrossRefGoogle Scholar
Munthe-Kaas, H. and Owren, B. (1999), Computations in a free Lie algebra, Philos. Trans. Roy. Soc. A 357, 957–981.10.1098/rsta.1999.0361CrossRefGoogle Scholar
Murua, A. (2006), The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math. 6, 387–426.10.1007/s10208-003-0111-0CrossRefGoogle Scholar
Murua, A. and Sanz-Serna, J. M. (1999), Order conditions for numerical integrators obtained by composing simpler integrators, Philos. Trans. Roy. Soc. A 357, 1079–1100.10.1098/rsta.1999.0365CrossRefGoogle Scholar
Murua, A. and Sanz-Serna, J. M. (2016), Computing normal forms and formal invariants of dynamical systems by means of word series, Nonlinear Anal. 138, 326–345.10.1016/j.na.2015.10.013CrossRefGoogle Scholar
Murua, A. and Sanz-Serna, J. M. (2017), Word series for dynamical systems and their numerical integrators, Found. Comput. Math. 17, 675–712.10.1007/s10208-015-9295-3CrossRefGoogle Scholar
Neal, R. M. (2011), MCMC using Hamiltonian dynamics, in Handbook of Markov Chain Monte Carlo (Brooks, S. et al., eds), CRC Press, pp. 113–174.10.1201/b10905-6CrossRefGoogle Scholar
Neri, F. (1988), Lie algebras and canonical integration. Technical report, University of Maryland.Google Scholar
Nielsen, M. A. and Chuang, I. L. (2010), Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge University Press.Google Scholar
Okunbor, D. I. and Lu, E. J. (1994), Eighth-order explicit symplectic Runge–Kutta–Nyström integrators. Technical report CSC 94-21, University of Missouri-Rolla.Google Scholar
Okunbor, D. I. and Skeel, R. D. (1994), Canonical Runge–Kutta–Nyström methods of orders five and six, J. Comput. Appl. Math. 51, 375–382.10.1016/0377-0427(92)00119-TCrossRefGoogle Scholar
Omelyan, I. P., Mryglod, I. M. and Folk, R. (2002), On the construction of high order force gradient algorithms for integration of motion in classical and quantum systems, Phys. Rev. E 66, art. 026701.10.1103/PhysRevE.66.026701CrossRefGoogle ScholarPubMed
Omelyan, I. P., Mryglod, I. M. and Folk, R. (2003), Symplectic analytically integrable decomposition algorithms: Classification, derivation, and application to molecular dynamics, quantum and celestial mechanics simulations, Comput. Phys. Commun. 151, 272–314.10.1016/S0010-4655(02)00754-3CrossRefGoogle Scholar
Owren, B. and Marthinsen, A. (1999), Runge–Kutta methods adapted to manifolds and based on rigid frames, BIT 39, 116–142.10.1023/A:1022325426017CrossRefGoogle Scholar
Painlevé, P. (1900), Analyse des Travaux Scientifiques, Gauthier-Villars.Google Scholar
Pars, L. A. (1979), A Treatise on Analytical Dynamics, Ox Bow Press.Google Scholar
Partington, J. (2004), Linear Operators and Linear Systems: An Analytical Approach to Control Theory, Cambridge University Press.10.1017/CBO9780511616693CrossRefGoogle Scholar
Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer.10.1007/978-1-4612-5561-1CrossRefGoogle Scholar
Peaceman, D. W. and Rachford, H. H. (1955), The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3, 28–41.10.1137/0103003CrossRefGoogle Scholar
Prosen, T. and Pizorn, I. (2006), High order non-unitary split-step decomposition of unitary operators, J. Phys. A 39, 5957–5964.10.1088/0305-4470/39/20/021CrossRefGoogle Scholar
Reed, M. and Simon, B. (1980), Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press.Google Scholar
Reich, S. (1999), Backward error analysis for numerical integrators, SIAM J. Numer. Anal. 36, 1549–1570.10.1137/S0036142997329797CrossRefGoogle Scholar
Reutenauer, C. (1993), Free Lie Algebras, Oxford University Press.10.1093/oso/9780198536796.001.0001CrossRefGoogle Scholar
Rousset, F. and Schratz, K. (2022), Convergence error estimates at low regularity for time discretizations of KdV, Pure Appl. Anal. 4, 127–152.10.2140/paa.2022.4.127CrossRefGoogle Scholar
Rowlands, G. (1991), A numerical algorithm for Hamiltonian systems, J. Comput. Phys. 97, 235–239.10.1016/0021-9991(91)90046-NCrossRefGoogle Scholar
Ruth, R. (1983), A canonical integration technique, IEEE Trans. Nucl. Sci. 30, 26–69.10.1109/TNS.1983.4332919CrossRefGoogle Scholar
Ryckaert, J.-P., Ciccotti, G. and Berendsen, H. J. C. (1977), Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comput. Phys. 23, 327–341.10.1016/0021-9991(77)90098-5CrossRefGoogle Scholar
Sakkos, K., Casulleras, J. and Boronat, J. (2009), High order Chin actions in path integral Monte Carlo, J. Chem. Phys. 130, art. 204109.10.1063/1.3143522CrossRefGoogle ScholarPubMed
Sanz-Serna, J. M. (1988), Runge–Kutta schemes for Hamiltonian systems, BIT 28, 877–883.10.1007/BF01954907CrossRefGoogle Scholar
Sanz-Serna, J. M. (1992), Symplectic integrators for Hamiltonian problems: An overview, Acta Numer. 1, 243–286.10.1017/S0962492900002282CrossRefGoogle Scholar
Sanz-Serna, J. M. (1997), Geometric integration, in The State of the Art in Numerical Analysis (Duff, I. S. and Watson, G. A., eds), Vol. 63 of Institute of Mathematics and its Applications Conference Series: New Ser., Oxford University Press, pp. 121–143.10.1093/oso/9780198500148.003.0005CrossRefGoogle Scholar
Sanz-Serna, J. M. and Calvo, M. P. (1994), Numerical Hamiltonian Problems, Chapman & Hall.10.1007/978-1-4899-3093-4CrossRefGoogle Scholar
Sanz-Serna, J. M. and Portillo, A. (1996), Classical numerical integrators for wave-packet dynamics, J. Chem. Phys. 104, 2349–2355.10.1063/1.470930CrossRefGoogle Scholar
Schlick, T. (2010), Molecular Modelling and Simulation: An Interdisciplinary Guide, second edition, Springer.10.1007/978-1-4419-6351-2CrossRefGoogle Scholar
Sheng, Q. (1989), Solving linear partial differential equations by exponential splitting, IMA J. Numer. Anal. 9, 199–212.10.1093/imanum/9.2.199CrossRefGoogle Scholar
Sinitsyn, A., Dulov, E. and Vedenyapin, V. (2011), Kinetic Boltzmann, Vlasov and Related Equations, Elsevier.Google Scholar
Skeel, R. D. and Cieśliński, J. L. (2020), On the famous unpublished preprint ‘methods of integration which preserve the contact transformation property of the Hamilton equations’. Available at arXiv:2003.12268.Google Scholar
Sofroniou, M. and Spaletta, G. (2005), Derivation of symmetric composition constants for symmetric integrators, Optim. Methods Softw. 20, 597–613.10.1080/10556780500140664CrossRefGoogle Scholar
Speth, R., Green, W., MacNamara, S. and Strang, G. (2013), Balanced splitting and rebalanced splitting, SIAM J. Numer. Anal. 51, 3084–3105.10.1137/120878641CrossRefGoogle Scholar
Stetter, H. J. (1970), Symmetric two-step algorithms for ordinary differential equations, Computing 5, 267–280.10.1007/BF02248027CrossRefGoogle Scholar
Störmer, C. (1907), Sur les trajectoires des corpuscules électrisés, Arch. Sci. Phys. Nat. Genève 24, 5–18, 113–158, 221–247.Google Scholar
Strang, G. (1968), On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5, 506–517.10.1137/0705041CrossRefGoogle Scholar
Sulem, C. and Sulem, P.-L. (1999), The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Springer.Google Scholar
Suris, Y. B. (1988), Preservation of symplectic structure in the numerical solution of Hamiltonian systems, in Numerical Solution of Differential Equations (Filippov, S. S., ed.), pp. 148–160. In Russian.Google Scholar
Sussman, G. and Wisdom, J. (1992), Chaotic evolution of the Solar System, Science 257, 56–62.10.1126/science.257.5066.56CrossRefGoogle ScholarPubMed
Suzuki, M. (1990), Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A 146, 319–323.10.1016/0375-9601(90)90962-NCrossRefGoogle Scholar
Suzuki, M. (1991), General theory of fractal path integrals with applications to many-body theories and statistical physics, J. Math. Phys. 32, 400–407.10.1063/1.529425CrossRefGoogle Scholar
Suzuki, M. (1992), General nonsymmetric higher-order decomposition of exponential operators and symplectic integrators, J. Phys. Soc. Japan 61, 3015–3019.10.1143/JPSJ.61.3015CrossRefGoogle Scholar
Suzuki, M. (1995), Hybrid exponential product formulas for unbounded operators with possible applications to Monte Carlo simulations, Phys. Lett. A 201, 425–428.10.1016/0375-9601(95)00266-6CrossRefGoogle Scholar
Suzuki, M. and Umeno, K. (1993), High-order decomposition theory of exponential operators and its applications to QMC and nonlinear dynamics, in Computer Simulation Studies in Condensed Matter Physics VI (Landau, D. P., Mon, K. K. and Schüttler, H. B., eds), Springer, pp. 74–86.10.1007/978-3-642-78448-4_7CrossRefGoogle Scholar
Takahashi, M. and Imada, M. (1984), Monte carlo calculation of quantum system II. Higher order correction, J. Phys. Soc. Japan 53, 3765–3769.10.1143/JPSJ.53.3765CrossRefGoogle Scholar
Thalhammer, M. (2008), High-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal. 46, 2022–2038.10.1137/060674636CrossRefGoogle Scholar
Thalhammer, M., Caliari, M. and Neuhauser, C. (2009), High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys. 228, 822–832.10.1016/j.jcp.2008.10.008CrossRefGoogle Scholar
Trefethen, L. N. (2000), Spectral Methods in MATLAB, SIAM.10.1137/1.9780898719598CrossRefGoogle Scholar
Trotter, H. F. (1959), On the product of semi-groups of operators, Proc. Amer. Math. Soc. 10, 545–551.10.1090/S0002-9939-1959-0108732-6CrossRefGoogle Scholar
Tsitouras, C. (1999), A tenth order symplectic Runge–Kutta–Nyström method, Celestial Mech. Dynam. Astronom 74, 223–230.10.1023/A:1008346516048CrossRefGoogle Scholar
Varadarajan, V. S. (1984), Lie Groups, Lie Algebras, and Their Representations, Springer.10.1007/978-1-4612-1126-6CrossRefGoogle Scholar
Verlet, L. (1967), Computer ‘experiments’ on classical fluids I: Thermodynamical properties of Lennard-Jones molecules, Phys. Rev. 159, 98–103.10.1103/PhysRev.159.98CrossRefGoogle Scholar
Vlasov, A. A. (1961), Many-Particle Theory and its Application to Plasma, Gordon & Breach.Google Scholar
Wisdom, J. and Holman, M. (1991), Symplectic maps for the n-body problem, Astron. J. 102, 1528–1538.10.1086/115978CrossRefGoogle Scholar
Wisdom, J., Holman, M. and Touma, J. (1996), Symplectic correctors, in Integration Algorithms and Classical Mechanics (Marsden, J. E., Patrick, G. W. and Shadwick, W. F., eds), Vol. 10 of Fields Institute Communications, American Mathematical Society, pp. 217–244.Google Scholar
Yanenko, N. N. (1971), The Method of Fractional Steps, Springer.10.1007/978-3-642-65108-3CrossRefGoogle Scholar
Yoshida, H. (1990), Construction of higher order symplectic integrators, Phys. Lett. A 150, 262–268.10.1016/0375-9601(90)90092-3CrossRefGoogle Scholar
Yosida, K. (1971), Functional Analysis, third edition, Springer.10.1007/978-3-662-00781-5CrossRefGoogle Scholar
Zhang, R., Liu, J., Qin, H., Wang, Y., He, Y. and Sun, Y. (2015), Volume-preserving algorithm for secular relativistic dynamics of charged particles, Phys. Plasmas 22, art. 044501.10.1063/1.4916570CrossRefGoogle Scholar
Zhu, W., Zhao, X. and Tang, Y. (1996), Numerical methods with a high order of accuracy applied in the quantum system, J. Chem. Phys. 104, 2275–2286.10.1063/1.470923CrossRefGoogle Scholar