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

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

[EN] The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution equations are formed by products of exponentials involving linear combinations of the defining operator evaluated at certain times. In comparison with other classes of time integration methods, such as Magnus integrators, an inherent advantage of CFQM exponential integrators is that structural properties of the operator are well preserved by the arising linear combinations. Employing the analytical framework of sectorial operators in Banach spaces, evolution equations of parabolic type and dissipative quantum systems are included in the scope of applications. In this context, however, numerical experiments show that CFQM exponential integrators of nonstiff order five or higher proposed in the literature suffer from poor stability properties. The given analysis delivers insight that CFQM exponential integrators are well defined and stable only if the coefficients occurring in the linear combinations satisfy a positivity condition and that an alternative approach for the design of stable high-order schemes relies on the consideration of complex coefficients. Together with suitable local error expansions, this implies that a high-order CFQM exponential integrator retains its nonstiff order of convergence under appropriate regularity and compatibility requirements on the exact solution. Numerical examples confirm the theoretical result and illustrate the favourable behaviour of novel schemes involving complex coefficients in stability and accuracy.Ministerio de Economia y Competitividad (Spain) through projects MTM2013-46553-C3 and MTM2016-77660-P (AEI/FEDER, UE) to S.B. and F.C.Blanes Zamora, S.; Casas, F.; Mechthild Thalhammer (2018). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type. IMA Journal of Numerical Analysis. 38(2):743-778. https://doi.org/10.1093/imanum/drx012S74377838

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

Fourier-Splitting methods for the dynamics of rotating Bose-Einstein condensates

We present a new method to propagate rotating Bose-Einstein condensates subject to explicitly time-dependent trapping potentials. Using algebraic techniques, we combine Magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the Hamiltonian that can be computed using only Fourier transforms at the cost of solving a small system of polynomial equations. The resulting scheme solves the challenging component of the (nonlinear) Hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating Bose-Einstein condensates. The method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates, e.g., for small nonlinearities, since it retains the near-integrable structure of the problem. For large nonlinearities, the method remains highly efficient if higher order p > 2 is sought. Furthermore, we show how it can adapted to the presence of dissipation terms. Numerical examples illustrate the performance of the scheme.Comment: 15 pages, 4 figures, as submitted to journa

Efficient time integration methods for Gross-Pitaevskii equations with rotation term

[EN] The objective of this work is the introduction and investigation of favourable time integration methods for the Gross-Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schrödinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.Part of this work was developed during a research stay at the Wolfgang Pauli Institute Vienna; the authors are grateful to the director Norbert Mauser and the staff members for their support and hospitality. Philipp Bader, Sergio Blanes, and Fernando Casas acknowledge funding by the Ministerio de Economía y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).Bader, P.; Blanes Zamora, S.; Casas, F.; Thalhammer, M. (2019). Efficient time integration methods for Gross-Pitaevskii equations with rotation term. Journal of Computational Dynamics (Online). 6(2):147-169. https://doi.org/10.3934/jcd.2019008S1471696

Geometric integration of non-autonomous Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.Comment: 20 pages, 3 figure

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

Strongly interacting electrons in solids are generically described by Hubbardtype models, and the impact of solar light can be modeled by an additional time-dependence. This yields a finite dimensional system of ordinary differential equations (ODE)s of Schr\"odinger type, which can be solved numerically by exponential time integrators of Magnus type. The efficiency may be enhanced by combining these with operator splittings. We will discuss several different approaches of employing exponential-based methods in conjunction with an adaptive Lanczos method for the evaluation of matrix exponentials and compare their accuracy and efficiency. For each integrator, we use defect-based local error estimators to enable adaptive time-stepping. This serves to reliably control the approximation error and reduce the computational effor

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