High order efficient splittings for the semiclassical time-dependent Schrodinger equation

[EN] Standard numerical schemes with time-step h deteriorate (e.g. like epsilon(-2)h(2)) in the presence of a small semiclassical parameters in the time-dependent Schrodinger equation. The recently introduced semiclassical splitting was shown to be of order O (epsilon h(2)). We present now an algorithm that is of order O (epsilon h(7)+epsilon(2)h(6)+epsilon(3)h(4)) at the expense of roughly three times the computational effort of the semiclassical splitting and another that is of order O (epsilon h(6)+epsilon(2)h(4)) at the same expense of the computational effort of the semiclassical splitting.The work of SB has been funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).Blanes Zamora, S.; Gradinaru, V. (2020). High order efficient splittings for the semiclassical time-dependent Schrodinger equation. Journal of Computational Physics. 405:1-13. https://doi.org/10.1016/j.jcp.2019.109157S113405Bao, W., Jin, S., & Markowich, P. A. (2002). On Time-Splitting Spectral Approximations for the Schrödinger Equation in the Semiclassical Regime. Journal of Computational Physics, 175(2), 487-524. doi:10.1006/jcph.2001.6956Balakrishnan, N., Kalyanaraman, C., & Sathyamurthy, N. (1997). Time-dependent quantum mechanical approach to reactive scattering and related processes. Physics Reports, 280(2), 79-144. doi:10.1016/s0370-1573(96)00025-7Descombes, S., & Thalhammer, M. (2010). An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT Numerical Mathematics, 50(4), 729-749. doi:10.1007/s10543-010-0282-4Bader, P., Iserles, A., Kropielnicka, K., & Singh, P. (2014). Effective Approximation for the Semiclassical Schrödinger Equation. Foundations of Computational Mathematics, 14(4), 689-720. doi:10.1007/s10208-013-9182-8Gradinaru, V., & Hagedorn, G. A. (2013). Convergence of a semiclassical wavepacket based time-splitting for the Schrödinger equation. Numerische Mathematik, 126(1), 53-73. doi:10.1007/s00211-013-0560-6Keller, J., & Lasser, C. (2013). Propagation of Quantum Expectations with Husimi Functions. SIAM Journal on Applied Mathematics, 73(4), 1557-1581. doi:10.1137/120889186Gradinaru, V., Hagedorn, G. A., & Joye, A. (2010). Tunneling dynamics and spawning with adaptive semiclassical wave packets. The Journal of Chemical Physics, 132(18), 184108. doi:10.1063/1.3429607Gradinaru, V., Hagedorn, G. A., & Joye, A. (2010). Exponentially accurate semiclassical tunneling wavefunctions in one dimension. Journal of Physics A: Mathematical and Theoretical, 43(47), 474026. doi:10.1088/1751-8113/43/47/474026Coronado, E. A., Batista, V. S., & Miller, W. H. (2000). Nonadiabatic photodissociation dynamics ofICNin the à continuum: A semiclassical initial value representation study. The Journal of Chemical Physics, 112(13), 5566-5575. doi:10.1063/1.481130Church, M. S., Hele, T. J. H., Ezra, G. S., & Ananth, N. (2018). Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation. The Journal of Chemical Physics, 148(10), 102326. doi:10.1063/1.5005557Hagedorn, G. A. (1998). Raising and Lowering Operators for Semiclassical Wave Packets. Annals of Physics, 269(1), 77-104. doi:10.1006/aphy.1998.5843Faou, E., Gradinaru, V., & Lubich, C. (2009). Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets. SIAM Journal on Scientific Computing, 31(4), 3027-3041. doi:10.1137/080729724McLachlan, R. I. (1995). Composition methods in the presence of small parameters. BIT Numerical Mathematics, 35(2), 258-268. doi:10.1007/bf01737165Blanes, S., Casas, F., & Ros, J. (1999). Symplectic Integration with Processing: A General Study. SIAM Journal on Scientific Computing, 21(2), 711-727. doi:10.1137/s1064827598332497Blanes, S., Casas, F., & Ros, J. (2000). Celestial Mechanics and Dynamical Astronomy, 77(1), 17-36. doi:10.1023/a:1008311025472Blanes, S., Diele, F., Marangi, C., & Ragni, S. (2010). Splitting and composition methods for explicit time dependence in separable dynamical systems. Journal of Computational and Applied Mathematics, 235(3), 646-659. doi:10.1016/j.cam.2010.06.018Stefanov, B., Iordanov, O., & Zarkova, L. (1982). Interaction potential in1Σg+Hg2: fit to the experimental data. Journal of Physics B: Atomic and Molecular Physics, 15(2), 239-247. doi:10.1088/0022-3700/15/2/01

Time-average on the numerical integration of nonautonomous differential equations

[EN] In this work we show how to numerically integrate nonautonomous differential equations by solving alternate time-averaged differential equations. Given a quadrature rule of order 2s or higher for s = 1, 2, . . . , we show how to build a differential equation with an averaged vector field that is a polynomial function of degree s - 1 in the independent variable, t, and whose solution after one time step agrees with the solution of the original differential equation up to order 2s. Then, any numerical scheme can be used to solve this alternate averaged equation where the vector field is always evaluated at the chosen quadrature rule. We show how to use the Magnus series expansion, adapted to nonlinear problems, to build the formal solution, and this result is valid for any choice of the quadrature rule. This formal solution can be used to build new schemes that must agree with it up to the desired order. For example, we show how to build commutator-free methods from previous results in the literature. All methods can also be written in terms of moment integrals, and each integral can be computed using different quadrature rules. This procedure allows us to build tailored methods for different classes of problems. We illustrate the time-averaged procedure and its efficiency in solving several problems.This work was funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).Blanes Zamora, S. (2018). Time-average on the numerical integration of nonautonomous differential equations. SIAM Journal on Numerical Analysis. 56(4):2513-2536. https://doi.org/10.1137/17M1156150S2513253656

Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs

We consider time-averaging methods based on the Magnus series expansion jointly with exponential integrators for the numerical integration of general linear non-homogeneous differential equations. The schemes can be considered as averaged methods which transform, for one time step, a non-autonomous problem into an autonomous one whose flows agree up to a given order of accuracy at the end of the time step. The problem is reformulated as a particular case of a matrix Riccati differential equation and the Möbius transformation is considered, leading to a homogeneous linear problem. The methods proposed can be used both for initial value problems (IVPs) as well as for two-point boundary value problems (BVPs). In addition, they allow to use different approximations for different parts of the equation, e.g. the homogeneous and non-homogeneous parts, or to use adaptive time steps. The particular case of separated boundary conditions using the imbedding formulation is also considered. This formulation allows us to transform a stiff and badly conditioned BVP into a set of well conditioned IVPs which can be integrated using some of the previous methods. The performance of the methods is illustrated on some numerical examples. © 2012 IMACS. Published by Elsevier B.V. All rights reserved.We would like to thank the referees for their suggestions and comments that helped us to improve the presentation of the work as well as to clarify the main results. The authors acknowledge the support of the Generalitat Valenciana through the project GV/2009/032. The work of S.B. has also been partially supported by Ministerio de Ciencia e Innovacion (Spain) under the coordinated project MTM2010-18246-C03 (co-financed by the ERDF of the European Union) and the work of E.P. has also been partially supported by Ministerio de Ciencia e Innvacion of Spain, by the project MTM2009-08587.Blanes Zamora, S.; Ponsoda Miralles, E. (2012). Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs. Applied Numerical Mathematics. 62(8):875-894. doi:10.1016/j.apnum.2012.02.00187589462

Simulation of bimolecular reactions: Numerical challenges with the graph Laplacian

[EN] An important framework for modelling and simulation of chemical reactions is a Markov process sometimes known as a master equation. Explicit solutions of master equations are rare; in general the explicit solution of the governing master equation for a bimolecular reaction remains an open question. We show that a solution is possible in special cases. One method of solution is diagonalization. The crucial class of matrices that describe this family of models are non-symmetric graph Laplacians. We illustrate how standard numerical algorithms for finding eigenvalues fail for the non-symmetric graph Laplacians that arise in master equations for models of chemical kinetics. We propose a novel way to explore the pseudospectra of the non-symmetric graph Laplacians that arise in this class of applications, and illustrate our proposal by Monte Carlo. Finally, we apply the Magnus expansion, which provides a method of simulation when rates change in time. Again the graph Laplacian structure presents some unique issues: standard numerical methods of more than second-order fail to preserve positivity. We therefore propose a method that achieves fourth-order accuracy, and maintain positivity.We thank the organisers and delegates of the Canberra 2019 EMAC conference for helpful discussions about graph Laplacians. SM thanks the Australian Research Council Centre of Excellence fot Mathematical ans Statistical Frontiers (ACEMS). The work of SB was funded by Ministerio de Economía, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).Macnamara, S.; Blanes Zamora, S.; Iserles, A. (2020). Simulation of bimolecular reactions: Numerical challenges with the graph Laplacian. The ANZIAM Journal. 61:1-16. https://doi.org/10.21914/anziamj.v61i0.15169S1166

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. International Journal of Computer Mathematics, 84(6), 713-727. doi:10.1080/00207160701458567Blanes, S., Casas, F., & Murua, A. (2011). Splitting methods in the numerical integration of non-autonomous dynamical systems. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 106(1), 49-66. doi:10.1007/s13398-011-0024-8Gray, S. K., & Manolopoulos, D. E. (1996). Symplectic integrators tailored to the time‐dependent Schrödinger equation. The Journal of Chemical Physics, 104(18), 7099-7112. doi:10.1063/1.471428Gray, S. K., & Verosky, J. M. (1994). Classical Hamiltonian structures in wave packet dynamics. The Journal of Chemical Physics, 100(7), 5011-5022. doi:10.1063/1.467219Blanes, S., Casas, F., & Murua, A. (2006). Symplectic splitting operator methods for the time-dependent Schrödinger equation. The Journal of Chemical Physics, 124(23), 234105. doi:10.1063/1.2203609Blanes, S., Casas, F., & Murua, A. (2007). On the Linear Stability of Splitting Methods. Foundations of Computational Mathematics, 8(3), 357-393. doi:10.1007/s10208-007-9007-8Blanes, S., Casas, F., & Murua, A. (2011). Error Analysis of Splitting Methods for the Time Dependent Schrödinger Equation. SIAM Journal on Scientific Computing, 33(4), 1525-1548. doi:10.1137/100794535Sanz-Serna, J. M., & Calvo, M. P. (1994). Numerical Hamiltonian Problems. doi:10.1007/978-1-4899-3093-4Blanes, S., & Moan, P. C. (2002). Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. Journal of Computational and Applied Mathematics, 142(2), 313-330. doi:10.1016/s0377-0427(01)00492-7Rosen, N., & Zener, C. (1932). Double Stern-Gerlach Experiment and Related Collision Phenomena. Physical Review, 40(4), 502-507. doi:10.1103/physrev.40.502Kyoseva, E. S., Vitanov, N. V., & Shore, B. W. (2007). Physical realization of coupled Hilbert-space mirrors for quantum-state engineering. Journal of Modern Optics, 54(13-15), 2237-2257. doi:10.1080/09500340701352060Walker, R. B., & Preston, R. K. (1977). Quantum versus classical dynamics in the treatment of multiple photon excitation of the anharmonic oscillator. The Journal of Chemical Physics, 67(5), 2017. doi:10.1063/1.435085Li, X., Wang, W., Lu, M., Zhang, M., & Li, Y. (2012). Structure-preserving modelling of elastic waves: a symplectic discrete singular convolution differentiator method. Geophysical Journal International, 188(3), 1382-1392. doi:10.1111/j.1365-246x.2011.05344.

Positivity-preserving methods for ordinary differential equations

[EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.The authors thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Geometry, compatibility and structure preservation in computational differential equations" when work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1. S.B. has been supported by project PID2019-104927GB-C21 (AEI/FEDER, UE).Blanes Zamora, S.; Iserles, A.; Macnamara, S. (2022). Positivity-preserving methods for ordinary differential equations. ESAIM Mathematical Modelling and Numerical Analysis. 56(6):1843-1870. https://doi.org/10.1051/m2an/20220421843187056

The scaling, splitting, and squaring method for the exponential of perturbed matrices

[EN] We propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum A = D+epsilon B of a sparse and efficiently exponentiable matrix D with sparse exponential e(D) and a dense matrix epsilon B which is of small norm in comparison with D. The predominant algorithm is based on scaling the large matrix A by a small number 2(-s), which is then exponentiated by efficient Pade or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbed matrix B in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought.Bader, P.; Blanes Zamora, S.; Seydaoglu, M. (2015). The scaling, splitting, and squaring method for the exponential of perturbed matrices. SIAM Journal on Matrix Analysis and Applications. 36(2):594-614. doi:10.1137/14098003XS59461436

Propagators for Quantum-Classical Models: Commutator-Free Magnus Methods

[EN] We consider the numerical propagation of models that combine both quantum and classical degrees of freedom, usually, electrons and nuclei, respectively. We focus, in our computational examples, on the case in which the quantum electrons are modeled with time-dependent density-functional theory, although the methods discussed below can be used with any other level of theory. Often, for these so-called quantum classical molecular dynamics models, one uses some propagation technique to deal with the quantum part and a different one for the classical equations. While the resulting procedure may, in principle, be consistent, it can however spoil some of the properties of the methods, such as the accuracy order with respect to the time step or the preservation of the geometrical structure of the equations. Few methods have been developed specifically for hybrid quantum-classical models. We propose using the same method for both the quantum and classical particles, in particular, one family of techniques that proves to be very efficient for the propagation of Schrodinger-like equations: the (quasi)-commutator free Magnus expansions. These have been developed, however, for linear systems, yet our problem is nonlinear: formally, the full quantum-classical system can be rewritten as a nonlinear Schrodinger equation, i.e., one in which the Hamiltonian depends on the system itself. The Magnus expansion algorithms for linear systems require the application of the Hamiltonian at intermediate points in a given propagating interval. For nonlinear systems, this poses a problem as this Hamiltonian is unknown due to its dependence on the state. We approximate it by employing a higher order extrapolation using previous steps as input. The resulting technique can then be regarded as a multistep technique or, alternatively, as a predictor corrector formula.A.C. acknowledges support from the MINECO FIS2017-82426-P grant. S.B. acknowledges the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program "Geometry, compatibility and structure preservation in computational differential equations (2019)", EPSRC grant number EP/R014604/1. S.B. also acknowledges funding by the Ministerio de Economia y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE) and the Ministerio de Ciencia Innovacion y Universidades, through Programa de Estancias de profesores e investigadores senior en centros extranjeros, incluido el Programa "Salvador de Madariaga" 2019 (PRX19/00295).Gómez Pueyo, A.; Blanes Zamora, S.; Castro, A. (2020). Propagators for Quantum-Classical Models: Commutator-Free Magnus Methods. Journal of Chemical Theory and Computation. 16(3):1420-1430. https://doi.org/10.1021/acs.jctc.9b01031S14201430163Molner, S. P. (1999). The Art of Molecular Dynamics Simulation (Rapaport, D. C.). Journal of Chemical Education, 76(2), 171. doi:10.1021/ed076p171Fermi, E.; Pasta, J.; Ulam, S. Studies of Nonlinear Problems I, Los Alamos Report LA 1940; Los Alamos Scientific Laboratory, 1955; reproduced in Nonlinear Wave Motion. Providence, RI, 1974.Alder, B. J., & Wainwright, T. E. (1959). Studies in Molecular Dynamics. I. General Method. The Journal of Chemical Physics, 31(2), 459-466. doi:10.1063/1.1730376Bornemann, F. A., Nettesheim, P., & Schütte, C. (1996). Quantum‐classical molecular dynamics as an approximation to full quantum dynamics. The Journal of Chemical Physics, 105(3), 1074-1083. doi:10.1063/1.471952DOLTSINIS, N. L., & MARX, D. (2002). FIRST PRINCIPLES MOLECULAR DYNAMICS INVOLVING EXCITED STATES AND NONADIABATIC TRANSITIONS. Journal of Theoretical and Computational Chemistry, 01(02), 319-349. doi:10.1142/s0219633602000257Runge, E., & Gross, E. K. U. (1984). Density-Functional Theory for Time-Dependent Systems. Physical Review Letters, 52(12), 997-1000. doi:10.1103/physrevlett.52.997Marques, M. A. L., Maitra, N. T., Nogueira, F. M. S., Gross, E. K. U., & Rubio, A. (Eds.). (2012). Fundamentals of Time-Dependent Density Functional Theory. Lecture Notes in Physics. doi:10.1007/978-3-642-23518-4Verlet, L. (1967). Computer «Experiments» on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review, 159(1), 98-103. doi:10.1103/physrev.159.98Hairer, E., & Wanner, G. (1996). Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics. doi:10.1007/978-3-642-05221-7Castro, 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:10.1063/1.1774980Schleife, A., Draeger, E. W., Kanai, Y., & Correa, A. A. (2012). Plane-wave pseudopotential implementation of explicit integrators for time-dependent Kohn-Sham equations in large-scale simulations. The Journal of Chemical Physics, 137(22), 22A546. doi:10.1063/1.4758792Russakoff, A., Li, Y., He, S., & Varga, K. (2016). Accuracy and computational efficiency of real-time subspace propagation schemes for the time-dependent density functional theory. The Journal of Chemical Physics, 144(20), 204125. doi:10.1063/1.4952646Kidd, D., Covington, C., & Varga, K. (2017). Exponential integrators in time-dependent density-functional calculations. Physical Review E, 96(6). doi:10.1103/physreve.96.063307Dewhurst, J. K., Krieger, K., Sharma, S., & Gross, E. K. U. (2016). An efficient algorithm for time propagation as applied to linearized augmented plane wave method. Computer Physics Communications, 209, 92-95. doi:10.1016/j.cpc.2016.09.001Akama, T., Kobayashi, O., & Nanbu, S. (2015). Development of efficient time-evolution method based on three-term recurrence relation. The Journal of Chemical Physics, 142(20), 204104. doi:10.1063/1.4921465Kolesov, G., Grånäs, O., Hoyt, R., Vinichenko, D., & Kaxiras, E. (2015). Real-Time TD-DFT with Classical Ion Dynamics: Methodology and Applications. Journal of Chemical Theory and Computation, 12(2), 466-476. doi:10.1021/acs.jctc.5b00969Schaffhauser, P., & Kümmel, S. (2016). Using time-dependent density functional theory in real time for calculating electronic transport. Physical Review B, 93(3). doi:10.1103/physrevb.93.035115O’Rourke, C., & Bowler, D. R. (2015). Linear scaling density matrix real time TDDFT: Propagator unitarity and matrix truncation. The Journal of Chemical Physics, 143(10), 102801. doi:10.1063/1.4919128Oliveira, M. J. T., Mignolet, B., Kus, T., Papadopoulos, T. A., Remacle, F., & Verstraete, M. J. (2015). Computational Benchmarking for Ultrafast Electron Dynamics: Wave Function Methods vs Density Functional Theory. Journal of Chemical Theory and Computation, 11(5), 2221-2233. doi:10.1021/acs.jctc.5b00167Zhu, Y., & Herbert, J. M. (2018). Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation. The Journal of Chemical Physics, 148(4), 044117. doi:10.1063/1.5004675Rehn, D. A., Shen, Y., Buchholz, M. E., Dubey, M., Namburu, R., & Reed, E. J. (2019). ODE integration schemes for plane-wave real-time time-dependent density functional theory. The Journal of Chemical Physics, 150(1), 014101. doi:10.1063/1.5056258Gómez Pueyo, A., Marques, M. A. L., Rubio, A., & Castro, A. (2018). Propagators for the Time-Dependent Kohn–Sham Equations: Multistep, Runge–Kutta, Exponential Runge–Kutta, and Commutator Free Magnus Methods. Journal of Chemical Theory and Computation, 14(6), 3040-3052. doi:10.1021/acs.jctc.8b00197Stoer, J., & Bulirsch, R. (2002). Introduction to Numerical Analysis. doi:10.1007/978-0-387-21738-3Blanes, S., & Casas, F. (2017). A Concise Introduction to Geometric Numerical Integration. doi:10.1201/b21563Flocard, H., Koonin, S. E., & Weiss, M. S. (1978). Three-dimensional time-dependent Hartree-Fock calculations: Application toO16+O16collisions. Physical Review C, 17(5), 1682-1699. doi:10.1103/physrevc.17.1682Chen, R., & Guo, H. (1999). The Chebyshev propagator for quantum systems. Computer Physics Communications, 119(1), 19-31. doi:10.1016/s0010-4655(98)00179-9Hochbruck, M., & Lubich, C. (1997). On Krylov Subspace Approximations to the Matrix Exponential Operator. SIAM Journal on Numerical Analysis, 34(5), 1911-1925. doi:10.1137/s0036142995280572Frapiccini, A. L., Hamido, A., Schröter, S., Pyke, D., Mota-Furtado, F., O’Mahony, P. F., … Piraux, B. (2014). Explicit schemes for time propagating many-body wave functions. Physical Review A, 89(2). doi:10.1103/physreva.89.023418Caliari, M., Kandolf, P., Ostermann, A., & Rainer, S. (2016). The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential. SIAM Journal on Scientific Computing, 38(3), A1639-A1661. doi:10.1137/15m1027620Schaefer, I., Tal-Ezer, H., & Kosloff, R. (2017). Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems. Journal of Computational Physics, 343, 368-413. doi:10.1016/j.jcp.2017.04.017Trotter, H. F. (1959). On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4), 545-545. doi:10.1090/s0002-9939-1959-0108732-6Strang, G. (1968). On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis, 5(3), 506-517. doi:10.1137/0705041Feit, 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)90091-2Suzuki, M. (1990). Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Physics Letters A, 146(6), 319-323. doi:10.1016/0375-9601(90)90962-nSuzuki, M. (1992). General theory of higher-order decomposition of exponential operators and symplectic integrators. Physics Letters A, 165(5-6), 387-395. doi:10.1016/0375-9601(92)90335-jYoshida, H. (1990). Construction of higher order symplectic integrators. Physics Letters A, 150(5-7), 262-268. doi:10.1016/0375-9601(90)90092-3Sugino, O., & Miyamoto, Y. (1999). Density-functional approach to electron dynamics: Stable simulation under a self-consistent field. Physical Review B, 59(4), 2579-2586. doi:10.1103/physrevb.59.2579McLachlan, R. I., & Quispel, G. R. W. (2002). Splitting methods. Acta Numerica, 11, 341-434. doi:10.1017/s0962492902000053Ascher, U. M., Ruuth, S. J., & Wetton, B. T. R. (1995). Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. SIAM Journal on Numerical Analysis, 32(3), 797-823. doi:10.1137/0732037Cooper, G. J., & Sayfy, A. (1983). Additive Runge-Kutta methods for stiff ordinary differential equations. Mathematics of Computation, 40(161), 207-218. doi:10.1090/s0025-5718-1983-0679441-1Hochbruck, M., Lubich, C., & Selhofer, H. (1998). Exponential Integrators for Large Systems of Differential Equations. SIAM Journal on Scientific Computing, 19(5), 1552-1574. doi:10.1137/s1064827595295337Hochbruck, M., & Ostermann, A. (2005). Exponential Runge–Kutta methods for parabolic problems. Applied Numerical Mathematics, 53(2-4), 323-339. doi:10.1016/j.apnum.2004.08.005Hochbruck, M., & Ostermann, A. (2005). Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems. SIAM Journal on Numerical Analysis, 43(3), 1069-1090. doi:10.1137/040611434Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. doi:10.1017/s0962492910000048Houston, W. V. (1940). Acceleration of Electrons in a Crystal Lattice. Physical Review, 57(3), 184-186. doi:10.1103/physrev.57.184Chen, Z., & Polizzi, E. (2010). Spectral-based propagation schemes for time-dependent quantum systems with application to carbon nanotubes. Physical Review B, 82(20). doi:10.1103/physrevb.82.205410Sato, S. A., & Yabana, K. (2014). Efficient basis expansion for describing linear and nonlinear electron dynamics in crystalline solids. Physical Review B, 89(22). doi:10.1103/physrevb.89.224305Wang, Z., Li, S.-S., & Wang, L.-W. (2015). Efficient Real-Time Time-Dependent Density Functional Theory Method and its Application to a Collision of an Ion with a 2D Material. Physical Review Letters, 114(6). doi:10.1103/physrevlett.114.063004Magnus, W. (1954). On the exponential solution of differential equations for a linear operator. Communications on Pure and Applied Mathematics, 7(4), 649-673. doi:10.1002/cpa.3160070404Blanes, S., Casas, F., Oteo, J. A., & Ros, J. (2009). The Magnus expansion and some of its applications. Physics Reports, 470(5-6), 151-238. doi:10.1016/j.physrep.2008.11.001Nettesheim, P., Bornemann, F. A., Schmidt, B., & Schütte, C. (1996). An explicit and symplectic integrator for quantum-classical molecular dynamics. Chemical Physics Letters, 256(6), 581-588. doi:10.1016/0009-2614(96)00471-xHochbruck, M., & Lubich, C. (1999). A Bunch of Time Integrators for Quantum/Classical Molecular Dynamics. Lecture Notes in Computational Science and Engineering, 421-432. doi:10.1007/978-3-642-58360-5_24Hochbruck, M., & Lubich, C. (1999). Bit Numerical Mathematics, 39(4), 620-645. doi:10.1023/a:1022335122807Nettesheim, P., & Reich, S. (1999). Symplectic Multiple-Time-Stepping Integrators for Quantum-Classical Molecular Dynamics. Lecture Notes in Computational Science and Engineering, 412-420. doi:10.1007/978-3-642-58360-5_23Nettesheim, P., & Schütte, C. (1999). Numerical Integrators for Quantum-Classical Molecular Dynamics. Lecture Notes in Computational Science and Engineering, 396-411. doi:10.1007/978-3-642-58360-5_22Reich, S. (1999). Multiple Time Scales in Classical and Quantum–Classical Molecular Dynamics. Journal of Computational Physics, 151(1), 49-73. doi:10.1006/jcph.1998.6142Jiang, H., & Zhao, X. S. (2000). New propagators for quantum-classical molecular dynamics simulations. The Journal of Chemical Physics, 113(3), 930-935. doi:10.1063/1.481873Grochowski, P., & Lesyng, B. (2003). Extended Hellmann–Feynman forces, canonical representations, and exponential propagators in the mixed quantum-classical molecular dynamics. The Journal of Chemical Physics, 119(22), 11541-11555. doi:10.1063/1.1624062Blanes, S., & Moan, P. C. (2006). Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems. Applied Numerical Mathematics, 56(12), 1519-1537. doi:10.1016/j.apnum.2005.11.004Bader, P., Blanes, S., & Kopylov, N. (2018). Exponential propagators for the Schrödinger equation with a time-dependent potential. The Journal of Chemical Physics, 148(24), 244109. doi:10.1063/1.5036838Marques, M. (2003). octopus: a first-principles tool for excited electron–ion dynamics. Computer Physics Communications, 151(1), 60-78. doi:10.1016/s0010-4655(02)00686-0Castro, A., Appel, H., Oliveira, M., Rozzi, C. A., Andrade, X., Lorenzen, F., … Rubio, A. (2006). octopus: a tool for the application of time-dependent density functional theory. physica status solidi (b), 243(11), 2465-2488. doi:10.1002/pssb.200642067Schwerdtfeger, P. (2011). The Pseudopotential Approximation in Electronic Structure Theory. ChemPhysChem, 12(17), 3143-3155. doi:10.1002/cphc.201100387Alonso, J. L., Castro, A., Clemente-Gallardo, J., Cuchí, J. C., Echenique, P., & Falceto, F. (2011). Statistics and Nosé formalism for Ehrenfest dynamics. Journal of Physics A: Mathematical and Theoretical, 44(39), 395004. doi:10.1088/1751-8113/44/39/395004Iserles, A., & Nørsett, S. P. (1999). On the solution of linear differential equations in Lie groups. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357(1754), 983-1019. doi:10.1098/rsta.1999.0362Munthe–Kaas, H., & Owren, B. (1999). Computations in a free Lie algebra. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357(1754), 957-981. doi:10.1098/rsta.1999.0361Alvermann, A., & Fehske, H. (2011). High-order commutator-free exponential time-propagation of driven quantum systems. Journal of Computational Physics, 230(15), 5930-5956. doi:10.1016/j.jcp.2011.04.006Blanes, S., & Moan, P. C. (2000). Splitting methods for the time-dependent Schrödinger equation. Physics Letters A, 265(1-2), 35-42. doi:10.1016/s0375-9601(99)00866-xAuer, N., Einkemmer, L., Kandolf, P., & Ostermann, A. (2018). Magnus integrators on multicore CPUs and GPUs. Computer Physics Communications, 228, 115-122. doi:10.1016/j.cpc.2018.02.019Thalhammer, M. (2006). A fourth-order commutator-free exponential integrator for nonautonomous differential equations. SIAM Journal on Numerical Analysis, 44(2), 851-864. doi:10.1137/05063042Bader, P., Blanes, S., Ponsoda, E., & Seydaoğlu, M. (2017). Symplectic integrators for the matrix Hill equation. Journal of Computational and Applied Mathematics, 316, 47-59. doi:10.1016/j.cam.2016.09.04

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

Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrodinger equations

[EN] This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrodinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge-Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker-Campbell-Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrodinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-ofconstants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.Ministerio de Economia y Competitividad (Spain) (project MTM2016-77660-P (AEI/FEDER, UE) to S.B., F.C. and C.G.).Blanes Zamora, S.; Casas, F.; González, C.; Thalhammer, M. (2021). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrodinger equations. IMA Journal of Numerical Analysis. 41(1):594-617. https://doi.org/10.1093/imanum/drz058S59461741