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     

Modulated Fourier Expansions of Highly Oscillatory Differential Equations

Modulated Fourier expansions are developed as a tool for gaining insight into the long-time behavior of Hamiltonian systems with highly oscillatory solutions. Particle systems of Fermi–Pasta–Ulam type with light and heavy masses are considered as an example. It is shown that the harmonic energy of the highly oscillatory part is nearly conserved over times that are exponentially long in the high frequency. Unlike previous approaches to such problems, the technique used here does not employ nonlinear coordinate transforms and can therefore be extended to the analysis of numerical discretizations.     

A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both types of integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for explicit integrators, a relationship between ERKN integrators and trigonometric integrators is established. For the long-term analysis of implicit integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion. By taking some adaptations of this technology for implicit methods, we derive the modulated Fourier expansion and show the near energy conservation.

     

基于修正的Magnus方法的高振荡动力系统的数值积分方法

基于建立于一般线性动力系统上的Magnus数值积分方法,针对随时间而高频率振荡的二阶动力系统,给出了有效的修正Magnus数值积分算法．首先,将二阶动力系统重新表示为一阶系统的形式,通过引进新变量进行参考坐标变换,使动力系统的高振荡性质保留在新形式内;进而基于局部线性化技术用修正的Magnus方法求解新形式下的系统方程;最后,通过一系列数值实验说明了文中方法的有效性．     

Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems

^** Email: David.Cohen{at}math.unige.ch. Present address: Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany (cohen{at}na.uni-tuebingen.de) Modulated Fourier expansion is used to show long-time near-conservationof the total and oscillatory energies of numerical methods forHamiltonian systems with highly oscillatory solutions. The numericalmethods considered are an extension of the trigonometric methods.A brief discussion of conservation properties in the continuousproblem and in the multi-frequency case is also given.     

A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.     

High Order Energy-Preserving Method of the "Good" Boussinesq Equation

The fourth order average vector field (AVF) method is applied to solve the "Good" Boussinesq equation. The semi-discrete system of the "good" Boussinesq equation obtained by the pseudo-spectral method in spatial variable, which is a classical finite dimensional Hamiltonian system, is discretized by the fourth order average vector field method. Thus, a new high order energy conservation scheme of the "good" Boussinesq equation is obtained. Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the "good" Boussinesq equation exactly and simulate evolution of different solitary waves well.     

High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.     

A semi-classical trace formula at a non-degenerate critical level

We study the semi-classical trace formula at a critical energy level for a h-pseudodifferential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.     

Exponential Integrators for Quantum-Classical Molecular Dynamics

We study time integration methods for equations of mixed quantum-classical molecular dynamics in which Newtonian equations of motion and Schrödinger equations are nonlinearly coupled. Such systems exhibit different time scales in the classical and the quantum evolution, and the solutions are typically highly oscillatory. The numerical methods use the exponential of the quantum Hamiltonian whose product with a state vector is approximated using Lanczos' method. This allows time steps that are much larger than the inverse of the highest frequencies.We describe various integration schemes and analyze their error behaviour, without assuming smoothness of the solution. As preparation and as a problem of independent interest, we study also integration methods for Schrödinger equations with time-dependent Hamiltonian.     

一类高振荡Hilbert变换的计算

Hilbert变换在信号处理与医学图像处理中都有着广泛的应用,但是对于一般的含有高振荡因子且在积分区阍中包含奇异值的‰变换往往处理起来较为困难,本文提出了一种基于等距节点插值的高效计算方法。     

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq"(t)+ Bq(t)=f(q(t)), where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q)=-▽_qV(q) for a real-valued function V(q). The energy-preserving formulae can exactly preserve the Hamiltonian H(q', q)=(1)/2q'^τ Aq' + (1)/2q^τ Bq + V(q). We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.     

Enhancing energy conserving methods

Recent observations [5] indicate that energy-momentum methods might be better suited for the numerical integration of highly oscillatory Hamiltonian systems than implicit symplectic methods. However, the popular energy-momentum method, suggested in [3], achieves conservation of energy by a global scaling of the force field. This leads to an undesirable coupling of all degrees of freedom that is not present in the original problem formulation. We suggest enhancing this energy-momentum method by splitting the force field and using separate adjustment factors for each force. In case that the potential energy function can be split into a strong and a weak part, we also show how to combine an energy conserving discretization of the strong forces with a symplectic discretization of the weak contributions. We demonstrate the numerical properties of our method by simulating particles that interact through Lennard-Jones potentials and by integrating the Sine-Gordon equation.This work was partly supported by NIH Grant P41RR05969, DOE/NSF Grant DE-FG02-91-ER25099/DMS-9304268, and NSF GCAG/HPCC ASC-9318159.     

非线性振动分析的均向量场法

通过构造向量形式的振动微分方程组，利用均向量场（AVF）法得到振动响应的向量差分迭代格式.该离散格式能够保能量，同时具有二阶精度的特征，从而给出非线性振动问题的均向量场法.介绍了均向量场法的基本步骤.在建立AVF格式时，对于微分方程中若干常见的项，直接给出相应的映射项.应用均向量场法研究了非线性单摆问题和Kepler(开普勒)问题，数值结果说明了该方法保能量和具有长时间求解能力的特性.     

Achieving Brouwer’s law with implicit Runge–Kutta methods

In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems. For implicit Runge–Kutta methods, a standard implementation shows an unexpected propagation. We propose a modification that reduces the effect of round-off and shows a qualitative and quantitative improvement for an accurate integration over long times. AMS subject classification (2000)  65L06, 65G50, 65P10     

Stroboscopic averaging of highly oscillatory nonlinear wave equations

In this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non‐oscillatory high‐order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz‐Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Weakly Nonlinear Oscillatory Waves with Multi-phases in Ideal Incompressible Fluid

In this paper, we apply the technique of weakly nonlinear geometric optics to study weakly nonlinear oscillatory waves with multi-phases in d-dimensional ideal incompressible fluid for d ≥ 2. Precisely, we give a rigorous asymptotic expansion for the solution of the oscillatory initial value problem to the ideal incompressible Euler equations. Generally, this problem is not well posed. However, the coherence assumption and small divisor property imposed on the phases functions lead to a compatibility condition for the solvability of profile equations from which we can determine every profile.     

Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems

We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) \(q^{\prime \prime }(t)+Mq(t)=f(q(t))\) with a principal frequency matrix \(M\in \mathbb {R}^{d\times d}\). If \(M\) is symmetric and positive semi-definite and \(f(q) = -\nabla U(q)\) for a smooth function \(U(q)\), then this is a multi-frequency oscillatory Hamiltonian system with the Hamiltonian \(H(q,p)=p^{T}p/2+q^{T}Mq/2+U(q),\) where \(p = q'\). The solution of this system is a nonlinear multi-frequency oscillator. The new trigonometric Fourier collocation method takes advantage of the special structure brought by the linear term \(Mq\), and its construction incorporates the idea of collocation methods, the variation-of-constants formula and the local Fourier expansion of the system. The properties of the new methods are analysed. The analysis in the paper demonstrates an important feature, namely that the trigonometric Fourier collocation methods can be of an arbitrary order and when \(M\rightarrow 0\), each trigonometric Fourier collocation method creates a particular Runge–Kutta–Nyström-type Fourier collocation method, which is symplectic under some conditions. This allows us to obtain arbitrary high-order symplectic methods to deal with a special and important class of systems of second-order ODEs in an efficient way. The results of numerical experiments are quite promising and show that the trigonometric Fourier collocation methods are significantly more efficient in comparison with alternative approaches that have previously appeared in the literature.     

MsFEM `a la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems

We introduce and analyze a multiscale finite element type method (MsFEM) in the vein of the classical Crouzeix-Raviart finite element method that is specifically adapted for highly oscillatory elliptic problems. We illustrate numerically the efficiency of the approach and compare it with several variants of MsFEM.     

多辛Dirac方程的高阶整体保能量格式

基于四阶平均向量场方法和拟谱方法构造了Dirac方程的高阶整体保能量格式，利用构造的高阶整体保能量格式数值模拟方程孤立波的演化行为.数值模拟结果表明构造的高阶整体保能量格式可以很好地模拟Dirac方程孤立波的演化行为，并且可以精确地保持方程的整体能量守恒特性.