Numerical integrators for quantum dynamics close to the adiabatic limit

Summary.  We study the numerical solution of singularly perturbed Schr?-dinger equations with time-dependent Hamiltonian. Based on a reformulation of the equations, we derive time-reversible numerical integrators which can be used with step sizes that are substantially larger than with traditional integration schemes. A complete error analysis is given for the adiabatic case. To deal with avoided crossings of energy levels, which lead to non-adiabatic behaviour, we propose an adaptive extension of the methods which resolves the sharp transients that appear in non-adiabatic state transitions. Received November 12, 2001 / Revised version received May 8, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65L05, 65M15, 65M20, 65L70.     

Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22     

Geometric Integration Methods that Preserve Lyapunov Functions

We consider ordinary differential equations (ODEs) with a known Lyapunov function V. To ensure that a numerical integrator reflects the correct dynamical behaviour of the system, the numerical integrator should have V as a discrete Lyapunov function. Only second-order geometric integrators of this type are known for arbitrary Lyapunov functions. In this paper we describe projection-based methods of arbitrary order that preserve any given Lyapunov function. AMS subject classification (2000) 65L05, 65L06, 65L20, 65P40     

Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations

We study a class of numerical methods for a system of second-order SDE driven by a linear fast force generating high frequency oscillatory solutions. The proposed schemes permit the use of large step sizes, have uniform global error bounds in the position (i.e. independent of the large frequencies present in the SDE) and offer various additional properties. This new family of numerical integrators for SDE can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problems.     

Accurate dense output formula for exponential integrators using the scaling and squaring method

This paper proposes an accurate dense output formula for exponential integrators. The computation of matrix exponential function is a vital step in implementing exponential integrators. By scrutinizing the computational process of matrix exponentials using the scaling and squaring method, valuable intermediate results in this process are identified and then used to establish a dense output formula. Efficient computation of dense outputs by the proposed formula enables time integration methods to set their simulation step sizes more flexibly. The efficacy of the proposed formula is verified through numerical examples from the power engineering field.     

A Class of Explicit Exponential General Linear Methods

In this paper, we consider a class of explicit exponential integrators that includes as special cases the explicit exponential Runge–Kutta and exponential Adams–Bashforth methods. The additional freedom in the choice of the numerical schemes allows, in an easy manner, the construction of methods of arbitrarily high order with good stability properties. We provide a convergence analysis for abstract evolution equations in Banach spaces including semilinear parabolic initial-boundary value problems and spatial discretizations thereof. From this analysis, we deduce order conditions which in turn form the basis for the construction of new schemes. Our convergence results are illustrated by numerical examples. AMS subject classification (2000) 65L05, 65L06, 65M12, 65J10     

Energy Conservation with Non-Symplectic Methods: Examples and Counter-Examples

Energy conservation of numerical integrators is well understood for symplectic one-step methods. This article provides new insight into energy conservation with non-symplectic methods. Sufficient conditions and counter-examples are presented. AMS subject classification (2000) 65L06, 65P10, 37J99.Submitted June 2004. Accepted October 2004. Communicated by Syvert Nørsett.     

Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations

The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the St?rmer–Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included. AMS subject classification (2000) 65L05, 65P10     

Second-order accurate projective integrators for multiscale problems

We introduce new projective versions of second-order accurate Runge–Kutta and Adams–Bashforth methods, and demonstrate their use as outer integrators in solving stiff differential systems. An important outcome is that the new outer integrators, when combined with an inner telescopic projective integrator, can result in fully explicit methods with adaptive outer step size selection and solution accuracy comparable to those obtained by implicit integrators. If the stiff differential equations are not directly available, our formulations and stability analysis are general enough to allow the combined outer–inner projective integrators to be applied to legacy codes or perform a coarse-grained time integration of microscopic systems to evolve macroscopic behavior, for example.     

Raising the order of geometric numerical integrators by composition and extrapolation

We analyse composition and polynomial extrapolation as procedures to raise the order of a geometric integrator for solving numerically differential equations. Methods up to order sixteen are constructed starting with basic symmetric schemes of order six and eight. If these are geometric integrators, then the new methods obtained by extrapolation preserve the geometric properties up to a higher order than the order of the method itself. We show that, for a number of problems, this is a very efficient procedure to obtain high accuracy. The relative performance of the different algorithms is examined on several numerical experiments. AMS subject classification 17B66, 34A50, 65L05     

Variational multirate integration of constrained dynamics

Mechanical systems with dynamics on varying time scales, in particular those including highly oscillatory motion, impose challenging questions for numerical integration schemes. Tiny step sizes are required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a larger time step is accurate enough. Small time steps increase integration times unnecessarily, especially for costly function evaluations. For systems comprising fast and slow dynamics, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. Main challenges are the identification of fast and slow parts (e.g. by separating the energy or by distinguishing sets of variables), the synchronisation of their dynamics and in particular the treatment of mixed parts that often appear when fast and slow dynamics are coupled by constraints. In this contribution, a multirate integrator is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Variational integrators (based on a discrete version of Hamilton's principle) lead to symplectic and momentum preserving integration schemes that also exhibit good energy behavior. The resulting multirate variational integrator has the same preservation properties. An example demonstrates the performance of the multirate integrator for constrained multibody dynamics. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilization of the Lax-Wendroff method and a generalized one-step Runge-Kutta method for hyperbolic initial-value problems

In order ot integrate hyperbolic systems we distinguish explicit and implicit time integrators. Implicit methods allow large integration steps, but require more storage and are more difficult to implement than explicit methods. However explicit methods are subject to a restriction on the integration step. This restriction is a drawback if the variation of the solution in time is so small that accuracy considerations would allow a larger integration step. In this report we apply a smoothing technique in order to stabilize the Lax-Wendroff method and a generalized one-step Runge-Kutta method. Using this technique, the integration step is not limited by stability considerations.     

Numerical Evaluation of the Evans Function by Magnus Integration

We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and related partial differential equations. In a typical application scenario, we need to repeatedly sample the solution to a system of linear non-autonomous ordinary differential equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane. In this situation, a substantial portion of the computational effort—the numerical evaluation of the iterated integrals which appear in the Magnus series—can be performed independent of the parameters and hence needs to be done only once. More importantly, for any given tolerance Magnus integrators possess lower bounds on the step size which are uniform across large regions of parameter space and which can be estimated a priori. We demonstrate, analytically as well as through numerical experiment, that these features render Magnus integrators extremely robust and, depending on the regime of interest, efficient in comparison with standard ODE solvers. AMS subject classification (2000) 65F20     

A multirate time stepping strategy for stiff ordinary differential equations

To solve ODE systems with different time scales which are localized over the components, multirate time stepping is examined. In this paper we introduce a self-adjusting multirate time stepping strategy, in which the step size for a particular component is determined by its own local temporal variation, instead of using a single step size for the whole system. We primarily consider implicit time stepping methods, suitable for stiff or mildly stiff ODEs. Numerical results with our multirate strategy are presented for several test problems. Comparisons with the corresponding single-rate schemes show that substantial gains in computational work and CPU times can be obtained. AMS subject classification (2000)  65L05, 65L06, 65L50     

New open modified Newton Cotes type formulae as multilayer symplectic integrators

New modified open Newton Cotes integrators are introduced in this paper. For the new proposed integrators the connection between these new algorithms, differential methods and symplectic integrators is studied. Much research has been done on one step symplectic integrators and several of them have obtained based on symplectic geometry. However, the research on multistep symplectic integrators is very poor. Zhu et al. [1] studied the well known open Newton Cotes differential methods and they presented them as multilayer symplectic integrators. Chiou and Wu [2] studied the development of multistep symplectic integrators based on the open Newton Cotes integration methods. In this paper we introduce a new open modified numerical method of Newton Cotes type and we present it as symplectic multilayer structure. The new obtained symplectic schemes are applied for the solution of Hamilton’s equations of motion which are linear in position and momentum. An important remark is that the Hamiltonian energy of the system remains almost constant as integration proceeds. We have applied also efficiently the new proposed method to a nonlinear orbital problem and an almost periodic orbital problem.     

Numerical study of the asymptotics of the second Painlevé equation by a functional fitting method

The Painlevé equations arise as reductions of the soliton equations such as the Korteweg–de Vries equation, the nonlinear Schrödinger equation and so on. In this study, we are concerned with numerical approximation of the asymptotics of solutions of the second Painlevé equation on pole‐free intervals along the real axis. Classical integrators such as high order Runge–Kutta schemes might be expensive to simulate oscillation, decay and blow‐up behaviours depending on initial conditions. However, a lower order functional fitting method catches all kinds of solutions even for relatively large step sizes. Copyright © 2013 John Wiley & Sons, Ltd.     

飞机辅助动力装置引气特性计算方法

作为飞机环控系统与主发动机起动的气源,以目前广泛应用的带负载压气机结构APU(Auxiliary Power Unit)为研究对象,进行引气特性计算模型与计算方法研究。首先介绍了APU结构与引气工作特点,然后分析了建模时喘振控制阀SCV(Surge Control Valve)控制方法与APU共同工作机理,最后采用部件法建立了该类型APU引气计算数学模型。以某型APU为对象进行数值仿真并与实际试车数据比较,计算误差小于3%,表明所采用的建模方法是正确的,所建立的模型能够满足工程需求。      

Strong-stability-preserving 3-stage Hermite-Birkhoff time-discretization methods

Strong-stability-preserving (SSP) time-discretization methods have a nonlinear stability property that makes them particularly suitable for the integration of hyperbolic conservation laws. A collection of SSP explicit 3-stage Hermite-Birkhoff methods of orders 3 to 7 with nonnegative coefficients are constructed as k-step analogues of third-order Runge-Kutta methods, incorporating a function evaluation at two off-step points. Generally, these new methods have larger effective CFL coefficients than the hybrid methods of Huang with the same step number k. They have larger maximum scaled step sizes than hybrid methods on Burgers' equations.     

Positivity for explicit two-step methods in linear multistep and one-leg form

Positivity results are derived for explicit two-step methods in linear multistep form and in one-leg form. It turns out that, using the forward Euler starting procedure, the latter form allows a slightly larger step size with respect to positivity. AMS subject classification (2000) 65L06     

Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predatorprey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversified time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are di?cult to mimic using conventional time integrators that are used to solve systems of ordinary di?erential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow-fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which confirm that these multirate schemes outperform corresponding single-rate schemes substantially both in terms of computational work and CPU times.