Splitting and composition methods in the numerical integration of differential equations

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table

Parallel Computation of functions of matrices and their action on vectors

We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of several matrices) and with a proper linear combination of the results, allows us to obtain new high order approximations to the desired functions of matrices. An error analysis to obtain forward and backward error bounds is presented. The coefficients of each method, which depends on the number of processors, can be adjusted to improve the accuracy, the stability or to reduce round off errors of the methods. We illustrate this procedure by explicitly constructing some methods which are then tested on several numerical examples

Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schr\"odinger equations

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using Fast Fourier Transforms. To split the system into the quantum harmonic oscillator problem and the remaining part allows to get higher accuracies in many cases, but it requires to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build new methods which combine the advantages of using Fourier methods while solving the timedependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition).Comment: 12 pages of RevTex4-1, 8 figures; substantially revised and extended versio

Continuous numerical solutions of coupled mixed partial differential systems using Fer's factorization

AbstractIn this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate time-dependent systems of the type ut − A(t)uxx − B(t)u = 0, 0 < x < p, t > 0, u(0,t) = u(p,t) = 0, u(x,0) = f(x), 0⩽x⩽p. After truncation of an exact series solution, the numerical solution is constructed using Fer's factorization. Given ε > 0 and t0,t1, with 0< t0 < t1 and D(t0,t1) = {s(x,t); 0⩽x⩽p, t0⩽t⩽t1} the error of the approximated solution with respect to the exact series solution is less than ε uniformly in D(t0,t1). An algorithm is also included

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

Exponential integrators for coupled self-adjoint non-autonomous partial differential systems

We consider the numerical integration of coupled self-adjoint non-autonomous partial differential systems. Under convergence conditions, the solution can be written as a series expansion where each of its terms correspond to solutions of linear time dependent matrix differential equations with oscillatory solutions that must be solved numerically. In this work, we analyze second order of Magnus integrators whose numerical error grows with the number of terms considered in the truncated series, n, at a rate that still allows us to guarantee convergence of the numerical series. In addition, the integrator can be implemented with a recursive algorithm such that the computational cost of the method grows only linearly with the number of terms of the series. Higher order Magnus integrators are also analyzed. Commutator-free Magnus integrators can be used with a similar recursive algorithm and can provide highly accurate results, but they show a faster error growth with n, and some caution must be taken if these methods are used. Numerical experiments confirm the performance of the proposed algorithm. (C) 2014 Elsevier Inc. All rights reserved.The work of S. Blanes has been partially supported by Ministerio de Ciencia e Innovacion (Spain) under project MTM2010-18246-C03. The work of E. Ponsoda has also been partially supported by the Universitat Politecnica de Valencia under project 2087.Ponsoda Miralles, E.; Blanes Zamora, S. (2014). Exponential integrators for coupled self-adjoint non-autonomous partial differential systems. Applied Mathematics and Computation. 243:1-11. https://doi.org/10.1016/j.amc.2014.05.050S11124

On the Linear Stability of Splitting Methods

A comprehensive linear stability analysis of splitting methods is carried out by means of a 2 × 2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by two). An algorithm is provided for determining the coefficients of all possible time- reversible splitting schemes for a prescribed stability polynomial. It is shown that p(x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By selecting conveniently the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available