Novel symplectic integrators for the Klein-Gordon equation with space- and time-dependent mass

[EN] We consider the numerical time-integration of the non-stationary Klein-Gordon equation with position- and time-dependent mass. A novel class of time-averaged symplectic splitting methods involving double commutators is analyzed and 4th- and 6th-order integrators are obtained. In contrast with standard splitting methods (that contain negative coefficients if the order is higher than two), additional commutators are incorporated into the schemes considered here. As a result, we can circumvent this order barrier and construct high order integrators with positive coefficients and a much reduced number of stages, thus improving considerably their efficiency. The performance of the new schemes is tested on several examples.This work has been funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE). Kopylov has also been partly supported by grant GRISOLIA/2015/A/137 from the Generalitat Valenciana.Bader, P.; Blanes Zamora, S.; Casas, F.; Kopylov, N. (2019). Novel symplectic integrators for the Klein-Gordon equation with space- and time-dependent mass. Journal of Computational and Applied Mathematics. 350:130-138. https://doi.org/10.1016/j.cam.2018.10.011S13013835

Symplectic integrators for second-order linear non-autonomous equations

[EN] Two families of symplectic methods specially designed for second-order time-dependent linear systems are presented. Both are obtained from the Magnus expansion of the corresponding first-order equation, but otherwise they differ in significant aspects. The first family is addressed to problems with low to moderate dimension, whereas the second is more appropriate when the dimension is large, in particular when the system corresponds to a linear wave equation previously discretised in space. Several numerical experiments illustrate the main features of the new schemes. (C) 2017 Elsevier B.V. All rights reserved.Bader, Blanes, Casas and Kopylov acknowledge the Ministerio de Economia y Competitividad (Spain) for financial support through the coordinated project MTM2013-46553-C3-3-P. Additionally, Kopylov has been partly supported by fellowship GRISOLIA/2015/A/137 from the Generalitat Valenciana.Bader, P.; Blanes Zamora, S.; Casas, F.; Kopylov, N.; Ponsoda Miralles, E. (2018). Symplectic integrators for second-order linear non-autonomous equations. Journal of Computational and Applied Mathematics. 330:909-919. https://doi.org/10.1016/j.cam.2017.03.028S90991933

Splitting methods in the numerical integration of non-autonomous dynamical systems

[EN] We present a procedure leading to efficient splitting schemes for the time integration of explicitly time dependent partitioned linear differential equations arising when certain partial differential equations are previously discretized in space. In the first stage we analyze the order conditions of the corresponding autonomous problem and construct new 6th-order methods. In the second stage, by following a procedure previously designed by the authors, we generalize the methods to the time dependent case in such a way that no order reduction is present. The resulting schemes compare favorably with other integrators previously available.This work has been supported by Ministerio de Ciencia e Innovacion (Spain) under project MTM2007-61572(co-financed by the ERDF of the European Union). SB also acknowledges financial support from Generalitat Valenciana through project GV/2009/032.Blanes Zamora, S.; Casas Perez, F.; Murua, A. (2012). Splitting methods in the numerical integration of non-autonomous dynamical systems. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 106(1):49-66. https://doi.org/10.1007/s13398-011-0024-849661061Blanes S., Casas F.: Splitting methods for non-autonomous separable dynamical systems. J. Phys. A. Math. Gen. 39, 5405–5423 (2006)Blanes S., Casas F., Murua A.: Symplectic splitting operator methods tailored for the time-dependent Schrödinger equation. J. Chem. Phys. 124, 234105 (2006)Blanes S., Casas F., Murua A.: Splitting methods for non-autonomous linear systems. Int. J. Comput. Math. 84, 713–727 (2007)Blanes S., Casas F., Murua A.: On the linear stability of splitting methods. Found. Comp. Math. 8, 357–393 (2008)Blanes S., Casas F., Murua A.: Splitting and composition methods in the numerical integration of differential equations. Bol. Soc. Esp. Math. Apl. 45, 87–143 (2008)Blanes, S., Casas, F., Murua, A.: Error analysis of splitting methods for the time dependent Schrödinger equation. arXiv:1001.1549 (2011)Blanes S., Casas F., Oteo J.A., Ros J.: The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009)Blanes S., Casas F., Ros J.: Improved high order integrators based on Magnus expansion. BIT 40, 434–450 (2000)Blanes S., Diele F., Marangi C., Ragni S.: Splitting and composition methods for explicit time dependence in separable dynamical systems. J. Comput. Appl. Math. 235, 646–659 (2010)Blanes S., Moan P.C.: Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 142, 313–330 (2002)Gray S., Manolopoulos D.E.: Symplectic integrators tailored to the time-dependent Schrödinger equation. J. Chem. Phys. 104, 7099–7112 (1996)Gray S., Verosky J.M.: Classical Hamiltonian structures in wave packet dynamics. J. Chem. Phys. 100, 5011–5022 (1994)Hairer E., Lubich C., Wanner G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd ed. Springer, Berlin (2006)Iserles A., Munthe-Kaas H.Z., Nørsett S.P., Zanna A.: Lie group methods. Acta Numer. 9, 215–365 (2000)Leimkuhler B., Reich S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004)Magnus W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)McLachlan R.I, Quispel R.: Splitting methods. Acta Numer. 11, 341–434 (2002)McLachlan R.I, Quispel R.G.W.: Geometric integrators for ODEs. J. Phys. A. Math. Gen. 39, 5251–5285 (2006)Rieben R., White D., Rodrigue G.: High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations. IEEE Trans. Antennas Propag. 52, 2190–2195 (2004)Sanz-Serna J.M., Calvo M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)Sanz-Serna J.M., Portillo A.: Classical numerical integrators for wave-packet dynamics. J. Chem. Phys. 104, 2349–2355 (1996)Sofroniou M., Spaletta G.: Derivation of symmetric composition constants for symmetric integrators. Optim. Methods Softw. 20, 597–613 (2005)Walker R.B., Preston K.: Quantum versus classical dynamics in treatment of multiple photon excitation of anharmonic-oscillator. J. Chem. Phys. 67, 2017–2028 (1977

Symplectic time-average propagators for the Schrodinger equation with a time-dependent Hamiltonian

[EN] Several symplectic splitting methods of orders four and six are presented for the step-by-step time numerical integration of the Schrodinger equation when the Hamiltonian is a general explicitly time-dependent real operator. They involve linear combinations of the Hamiltonian evaluated at some intermediate points. We provide the algorithm and the coefficients of the methods, as well as some numerical examples showing their superior performance with respect to other available schemes. Published by AIP Publishing.The authors acknowledge Ministerio de Economia y Competitividad (Spain) for financial support through Project Nos. MTM2013-46553-C3 and MTM2016-77660-P (AEI/FEDER, UE). Additionally, A.M. has been partially supported by the Basque Government (Consolidated Research Group No. IT649-13).Blanes Zamora, S.; Casas, F.; Murua, A. (2017). Symplectic time-average propagators for the Schrodinger equation with a time-dependent Hamiltonian. The Journal of Chemical Physics. 146(11):1-10. https://doi.org/10.1063/1.4978410S11014611Castro, A., Marques, M. A. L., & Rubio, A. (2004). Propagators for the time-dependent Kohn–Sham equations. The Journal of Chemical Physics, 121(8), 3425-3433. doi:10.1063/1.1774980Kormann, K., Holmgren, S., & Karlsson, H. O. (2008). Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian. The Journal of Chemical Physics, 128(18), 184101. doi:10.1063/1.2916581Poulin, D., Qarry, A., Somma, R., & Verstraete, F. (2011). Quantum Simulation of Time-Dependent Hamiltonians and the Convenient Illusion of Hilbert Space. Physical Review Letters, 106(17). doi:10.1103/physrevlett.106.170501Lubich, C. (2008). From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. doi:10.4171/067Jahnke, T., & Lubich, C. (2000). Bit Numerical Mathematics, 40(4), 735-744. doi:10.1023/a:1022396519656Neuhauser, C., & Thalhammer, M. (2009). On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT Numerical Mathematics, 49(1), 199-215. doi:10.1007/s10543-009-0215-2Thalhammer, M. (2008). High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations. SIAM Journal on Numerical Analysis, 46(4), 2022-2038. doi:10.1137/060674636Thalhammer, M. (2012). Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations. SIAM Journal on Numerical Analysis, 50(6), 3231-3258. doi:10.1137/120866373Park, T. J., & Light, J. C. (1986). Unitary quantum time evolution by iterative Lanczos reduction. The Journal of Chemical Physics, 85(10), 5870-5876. doi:10.1063/1.451548Blanes, S., Casas, F., & Murua, A. (2015). An efficient algorithm based on splitting for the time integration of the Schrödinger equation. Journal of Computational Physics, 303, 396-412. doi:10.1016/j.jcp.2015.09.047Feit, M. ., Fleck, J. ., & Steiger, A. (1982). Solution of the Schrödinger equation by a spectral method. Journal of Computational Physics, 47(3), 412-433. doi:10.1016/0021-9991(82)90091-2Tremblay, J. C., & Carrington, T. (2004). Using preconditioned adaptive step size Runge-Kutta methods for solving the time-dependent Schrödinger equation. The Journal of Chemical Physics, 121(23), 11535-11541. doi:10.1063/1.1814103Sanz‐Serna, J. M., & Portillo, A. (1996). Classical numerical integrators for wave‐packet dynamics. The Journal of Chemical Physics, 104(6), 2349-2355. doi:10.1063/1.470930Peskin, U., Kosloff, R., & Moiseyev, N. (1994). The solution of the time dependent Schrödinger equation by the (t,t’) method: The use of global polynomial propagators for time dependent Hamiltonians. The Journal of Chemical Physics, 100(12), 8849-8855. doi:10.1063/1.466739Lauvergnat, D., Blasco, S., Chapuisat, X., & Nauts, A. (2007). A simple and efficient evolution operator for time-dependent Hamiltonians: the Taylor expansion. The Journal of Chemical Physics, 126(20), 204103. doi:10.1063/1.2735315Tal-Ezer, H., Kosloff, R., & Cerjan, C. (1992). Low-order polynomial approximation of propagators for the time-dependent Schrödinger equation. Journal of Computational Physics, 100(1), 179-187. doi:10.1016/0021-9991(92)90318-sNdong, M., Tal-Ezer, H., Kosloff, R., & Koch, C. P. (2010). A Chebychev propagator with iterative time ordering for explicitly time-dependent Hamiltonians. The Journal of Chemical Physics, 132(6), 064105. doi:10.1063/1.3312531Tal-Ezer, H., Kosloff, R., & Schaefer, I. (2012). New, Highly Accurate Propagator for the Linear and Nonlinear Schrödinger Equation. Journal of Scientific Computing, 53(1), 211-221. doi:10.1007/s10915-012-9583-xBlanes, S., Casas, F., & Murua, A. (2007). Splitting methods for non-autonomous linear systems. 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The Magnus expansion and some of its applications

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its applications to several physical problem

Magnus-based geometric integrators for dynamical systems with time-dependent potentials

[ES] Esta tesis trata sobre la integración numérica de sistemas hamiltonianos con potenciales explícitamente dependientes del tiempo. Los problemas de este tipo son comunes en la física matemática, porque provienen de la mecánica cuántica, clásica y celestial. La meta de la tesis es construir integradores para unos problemas relevantes no autónomos: la ecuación de Schrödinger, que es el fundamento de la mecánica cuántica; las ecuaciones de Hill y de onda, que describen sistemas oscilatorios; el problema de Kepler con la masa variante en el tiempo. El Capítulo 1 describe la motivación y los objetivos de la obra en el contexto histórico de la integración numérica. En el Capítulo 2 se introducen los conceptos esenciales y unas herramientas fundamentales utilizadas a lo largo de la tesis. El diseño de los integradores propuestos se basa en los métodos de composición y escisión y en el desarrollo de Magnus. En el Capítulo 3 se describe el primero. Su idea principal consta de una recombinación de unos integradores sencillos para obtener la solución del problema. El concepto importante de las condiciones de orden se describe en ese capítulo. En el Capítulo 4 se hace un resumen de las álgebras de Lie y del desarrollo de Magnus que son las herramientas algebraicas que permiten expresar la solución de ecuaciones diferenciales dependientes del tiempo. La ecuación lineal de Schrödinger con potencial dependiente del tiempo está examinada en el Capítulo 5. Dado su estructura particular, nuevos métodos casi sin conmutadores, basados en el desarrollo de Magnus, son construidos. Su eficiencia es demostrada en unos experimentos numéricos con el modelo de Walker-Preston de una molécula dentro de un campo electromagnético. En el Capítulo 6, se diseñan los métodos de Magnus-escisión para las ecuaciones de onda y de Hill. Su eficiencia está demostrada en los experimentos numéricos con varios sistemas oscilatorios: con la ecuación de Mathieu, la ec. de Hill matricial, las ecuaciones de onda y de Klein-Gordon-Fock. El Capítulo 7 explica cómo el enfoque algebraico y el desarrollo de Magnus pueden generalizarse a los problemas no lineales. El ejemplo utilizado es el problema de Kepler con masa decreciente. El Capítulo 8 concluye la tesis, reseña los resultados y traza las posibles direcciones de la investigación futura.[CA] Aquesta tesi tracta de la integració numèrica de sistemes hamiltonians amb potencials explícitament dependents del temps. Els problemes d'aquest tipus són comuns en la física matemàtica, perquè provenen de la mecànica quàntica, clàssica i celest. L'objectiu de la tesi és construir integradors per a uns problemes rellevants no autònoms: l'equació de Schrödinger, que és el fonament de la mecànica quàntica; les equacions de Hill i d'ona, que descriuen sistemes oscil·latoris; el problema de Kepler amb la massa variant en el temps. El Capítol 1 descriu la motivació i els objectius de l'obra en el context històric de la integració numèrica. En Capítol 2 s'introdueixen els conceptes essencials i unes ferramentes fonamentals utilitzades al llarg de la tesi. El disseny dels integradors proposats es basa en els mètodes de composició i escissió i en el desenvolupament de Magnus. En el Capítol 3, es descriu el primer. La seua idea principal consta d'una recombinació d'uns integradors senzills per a obtenir la solució del problema. El concepte important de les condicions d'orde es descriu en eixe capítol. El Capítol 4 fa un resum de les àlgebres de Lie i del desenvolupament de Magnus que són les ferramentes algebraiques que permeten expressar la solució d'equacions diferencials dependents del temps. L'equació lineal de Schrödinger amb potencial dependent del temps està examinada en el Capítol 5. Donat la seua estructura particular, nous mètodes quasi sense commutadors, basats en el desenvolupament de Magnus, són construïts. La seua eficiència és demostrada en uns experiments numèrics amb el model de Walker-Preston d'una molècula dins d'un camp electromagnètic. En el Capítol 6 es dissenyen els mètodes de Magnus-escissió per a les equacions d'onda i de Hill. El seu rendiment està demostrat en els experiments numèrics amb diversos sistemes oscil·latoris: amb l'equació de Mathieu, l'ec. de Hill matricial, les equacions d'onda i de Klein-Gordon-Fock. El Capítol 7 explica com l'enfocament algebraic i el desenvolupament de Magnus poden generalitzar-se als problemes no lineals. L'exemple utilitzat és el problema de Kepler amb massa decreixent. El Capítol 8 conclou la tesi, ressenya els resultats i traça les possibles direccions de la investigació futura.[EN] The present thesis addresses the numerical integration of Hamiltonian systems with explicitly time-dependent potentials. These problems are common in mathematical physics because they come from quantum, classical and celestial mechanics. The goal of the thesis is to construct integrators for several import ant non-autonomous problems: the Schrödinger equation, which is the cornerstone of quantum mechanics; the Hill and the wave equations, that describe oscillating systems; the Kepler problem with time-variant mass. Chapter 1 describes the motivation and the aims of the work in the historical context of numerical integration. In Chapter 2 essential concepts and some fundamental tools used throughout the thesis are introduced. The design of the proposed integrators is based on the composition and splitting methods and the Magnus expansion. In Chapter 3, the former is described. Their main idea is to recombine some simpler integrators to obtain the solution. The salient concept of order conditions is described in that chapter. Chapter 4 summarises Lie algebras and the Magnus expansion ¿ algebraic tools that help to express the solution of time-dependent differential equations. The linear Schrödinger equation with time-dependent potential is considered in Chapter 5. Given its particular structure, new, Magnus-based quasi-commutator-free integrators are build. Their efficiency is shown in numerical experiments with the Walker-Preston model of a molecule in an electromagnetic field. In Chapter 6, Magnus-splitting methods for the wave and the Hill equations are designed. Their performance is demonstrated in numerical experiments with various oscillatory systems: the Mathieu equation, the matrix Hill eq., the wave and the Klein-Gordon-Fock eq. Chapter 7 shows how the algebraic approach and the Magnus expansion can be generalised to non-linear problems. The example used is the Kepler problem with decreasing mass. The thesis is concluded by Chapter 8, in which the results are reviewed and possible directions of future work are outlined.Kopylov, N. (2019). Magnus-based geometric integrators for dynamical systems with time-dependent potentials [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/118798TESI