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     

An infinite-dimensional Evans function theory for elliptic boundary value problems

An infinite-dimensional Evans function E(λ) and a stability index theorem are developed for the elliptic eigenvalue problem in a bounded domain Ω⊂Rm. The number of zero points of the Evans function in a bounded, simply connected complex domain D is shown to be equal to the number of eigenvalues of the corresponding elliptic operator in D. When the domain Ω is star-shaped, an associated unstable bundle E(D) based on D is constructed, and the first Chern number of E(D) also gives the number of eigenvalues of the elliptic operator inside D.     

Infinite-dimensional Evans function theory for elliptic eigenvalue problems in a channel

An infinite-dimensional Evans function theory is developed for the elliptic eigenvalue problem associated with the stability of travelling solitary waves in a channel. Also, a bundle is constructed over the complex domain, so that its first Chern number gives the number of eigenvalues inside the domain.     

Order reduction and determination of dominant state variables of nonlinear systems

The method presented can simplify nonlinear system models by reducing the number of state equations. Starting from a special state space representation, the main idea is to take over all nonlinear terms into the reduced system and to renew all couplings of state variables, input variables and nonlinear functions. The steady state performance can be influenced by additional measures which are discussed in detail and which are illustrated by a technical example. A dominance analysis is introduced which helps choosing the system order and the dominant state variables. All computations are based on proven algorithms and most of them are free of iterations.     

Order conditions for partitioned Runge-Kutta methods

We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations.     

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

A new strategy to avoid the order reduction of Runge-Kutta methods when integrating linear, autonomous, nonhomogeneous initial boundary value problems is presented. The solution is decomposed into two parts. One of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction. A numerical illustration is given. This idea applies to practical problems, where spatial discretization is also required, leading to the full order both in space and time.

     

代数体函数与其系数函数的增长级关系

应用代数体函数的Nevanlinna特征与Valiron特征,系统地研究了由v 1个整函数{Aj(z)}vj=0为系数所决定的代数体函数的级与Aj(z)(0≤j≤v)的级之间的关系;进而可以方便地求出代数体函数的级.     

Order Reduced Schemes for the Fourth Order Eigenvalue Problems on Multi-Connected Planar Domains

In this paper, we study the order reduced finite element method for the fourth order eigenvalue problems on multi-connected planar domains. Particularly, we take the biharmonic and the Helmholtz transmission eigenvalue problems as model problems, present for each an equivalent order reduced formulation and a corresponding stable discretization scheme, and present rigorous theoretical analysis. The schemes are readily fit for multilevel correction algorithms with optimal computational costs. Numerical experiments are given for verifications.     

化肥需要量灰色预测模型参数的四阶求法

针对文［１］化肥需要量灰色预测模型误差较大且预测结果与定性分析矛盾的问题，本文提出和应用了确定Ｖｅｒｈｕｌｓｔ模型参数的四阶方法，并重新合理配置建模数据而得一预测模型．计算表明新模型与实际值的拟合精度提高近十倍，而且预测结果与定性分析完全一致．     

On the Order Reduction Approach for Singularly Perturbed Optimal Control Systems

The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to be zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed problem with a small parameter. In this paper we establish that if relaxed controls are allowed then the answer to the question whether or not this method is valid depends essentially on one simple parameter: the dimension of the fast variable, denoted n. More specifically, if n=1 then the order reduction method is indeed applicable, while if n>1 then the set of singularly perturbed optimal control systems for which it is not applicable is dense (in the L norm).     

Banach空间中一阶积-微分方程的解

利用新的比较原理和半序方法讨论了Banach空间中的混合型一阶积-微分方程的非线性边值问题,并改进了已有的结果.     

The Dissipative Spectral Methods for the First Order Linear Hyperbolic Equations

In this paper, we introduce the dissipative spectral methods (DSM) for the first order linear hyperbolic equations in one dimension. Specifically, we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition. The error estimates of the methods are shown to be quasi-optimal for variable-coefficients equations. Numerical results are given to verify high accuracy of the DSM and to compare the proposed schemes with some high performance methods, showing some superiority in long-term integration for the periodic case and in dealing with limited smoothness near or at the boundary for the Dirichlet case.     

On the Mean Value of the Infinite Order Legendre Series

ＯｎｔｈｅＭｅａｎＶａｌｕｅｏｆｔｈｅＩｎｆｉｎｉｔｅＯｒｄｅｒＬｅｇｅｎｄｒｅＳｅｒｉｅｓＷａｎｇＡｎｂｉｎ（ＤｅｐｔｏｆＭａｔｈ．，ＹｕｅｙａｎｇＴｅａｃｈｅｒｓＣｏｌｌｅｇｅ．４１４０００）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｓｔｕｄ...     

具有特殊解结构的二阶线性微分方程的可积性判据

针对二阶线性微分方程y″+p(x)y′+q(x)y=0具有某种特殊解结构的情况下,进行可积性判据研究,利用降阶的思想,得到p(x),q(x)满足的关系式,找到了方程可积的充分条件.     

用AHP方法确定多目标的优先等级

本文给出了一种确定目标规划问题中多目标的优先等级的方法—— AHP方法 .     

基于时域误差限的大规模系统自适应模型降阶

为满足解大规模动态系统常微分方程组对精度和速度权衡的要求,提出了一种基于误差限的大规模系统自适应模型降阶方法,其中方法的误差分析基于时域最大误差限,降阶方法基于SVD-Krylov子空间的方法.方法既考虑了算法的复杂性,又保证了算法的精度.通过对典型实例分析,结果表明该方法在给定相对误差限10~(-4)下得出的降阶阶数在不同频率下都能给出很好的近似精度,低频1~10Hz平均相对误差为1.1812×10~(-5),高频1~10GHz平均相对误差为5.6408×10~(-5),即在很宽的频率范围内都能满足精度要求.     

Laguerre Spectral Method for High Order Problems

In this paper, we propose the Laguerre spectral method for high order problems with mixed inhomogeneous boundary conditions. It is also available for approximated solutions growing fast at infinity. The spectral accuracy is proved. Numerical results demonstrate its high effectiveness.     

A note: Every homogeneous difference equation of degree one admits a reduction in order

Every difference equation x _n+1 = f _n (x _n ,x _n ? 1,…,x _n ? k ) of order k+1 with each mapping f _n being homogeneous of degree 1 on a group G is shown to be equivalent to a system consisting of an equation of order k and a linear equation of order 1.     

基于序关系的竞赛评委评价方法

在特定的一种基于“平均分”的竞赛评阅方案下,对评委的阅卷水平、公正性进行分析,提出若干仅依赖于原始打分数据的量化指标,强调了序关系的重要性,可以较为准确的评价各个评委的水平和公正性.最后,对原评阅方案的优缺点进行定性讨论.     

Model order reduction of finite element models: improved component mode synthesis

The finite element (FE) approach constitutes an essential methodology when modelling the elastic properties of structures in various research disciplines such as structural mechanics, engine dynamics and so on. Because of increased accuracy requirements, the FE method results in discretized models, which are described by higher order ordinary differential equations, or, in FE terms, by a large number of degrees of freedom (DoF). In this regard, the application of an additional methodology, referred to as the model order reduction (MOR) or DoF condensation, is rather compulsory. Herein, a reduced dimension set of ordinary differential equations is generated, i.e. the initially large number of DoF is condensed, while aiming to keep the dynamics of the original model as intact as possible. In the commercially available FE software tools, the static and the component mode syntheses (CMS) are the only available integrated condensation methods. The latter represents the state of the art generating well-correlated reduced order models (ROMs), which can be further utilized for FE or multi-body systems simulations. Taking into consideration the information loss of the CMS, which is introduced by its part-static nature, the improved CMS (ICMS) method is proposed. Here the algorithmic scheme of the standard CMS is adopted, which is qualitatively improved by adequately considering the advantageous characteristics of another MOR approach, the so-called improved reduction system method. The ICMS results in better correlated reduced order models in comparison to all the aforementioned methods, while preserving the required structural properties of the original FE model.