On Spectral Method for Volterra Functional Integro-Differential Equations of Neutral Type

The main purpose of this work is to provide a numerical method for the solution of Volterra functional integro-differential equations of neutral type based on a spectral approach. We analyze the convergence properties of the spectral method to approximate smooth solutions of Volterra functional integro-differential equations of neutral type. It is shown that for the neutral integro-differential equations, the spectral methods yield an exponential order of convergence.     

Extreme eigenvalues of real symmetric Toeplitz matrices

We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, along with an efficient stopping rule, an error analysis, and extensive numerical results.

     

非线性弱奇性Volterra积分方程的谱配置法

基于已有文献的研究成果及前期工作,我们考察了非线性弱奇性Volterra积分方程（VIE）的谱配置法,并对该方法进行了收敛性分析.得到的结论是数值误差呈谱收敛.误差收敛阶与配置点个数及方程解的正则性相关.数值实验也证实了这一结论.本文的方法解决了已有文献中类似数值方法（Allaei（2016）,Sohrabi（2017））存在的问题.     

Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper, we consider the numerical approximation of stochastic partial differential equations with nonlinear multiplicative trace class noise. Discretization is obtained by spectral collocation method in space, and semi‐implicit Euler method is used for the temporal approximation. Our purpose is to investigate the convergence of the proposed method. The rate of convergence is obtained, and some numerical examples are included to illustrate the estimated convergence rate.     

线性变系数中立型变延迟微分方程谱方法的收敛性

本文主要研究了应用谱方法求解线性变系数中立型变延迟微分方程,构造了相应的基于Chebyshev和Legendre正交多项式的数值方法, 证明了其收敛性,最后给出了数值算例. 这些结果表明应用谱方法求解延迟微分方程可以获得谱收敛与谱精度的计算效果.     

一类偏微分积分方程的Legendre-Galerkin谱方法空间半离散的全局性

本文借助拉普拉斯变换,运用Legendre-Galerkin谱方法来研究一类线性偏积分微分方程的半离散问题,这类问题就出现在比如粘弹性模型中,我们得到了空间半离散的稳定性和收敛性结果。     

发展型Navier-Stokes方程基于压力稳定化方法的谱逼近格式

本文采用压力稳定化方法近似模拟不可压缩条件,进而构造了发展型非周期NavierStokes方程的全离散Legendre谱逼近计算格式,严格分析了格式的广义稳定性与收敛性．本文建立的逼近结果也适用其它非周期问题．     

求解Fisher型方程的混合Legendre-球面调和谱方法

主要研究两同心球所界球形区域上偏微分方程的谱方法,建立了与区域形状相适应的混合Legendre-球面调和正交逼近的部分结果,在此基础上提出了数值求解两同心球所界球形区域上Fisher型方程的混合Legendre-球面调和谱格式,并分别给出了格式的收敛性及相关的数值结果.     

Couette-Taylor流的谱Galerkin逼近

利用谱方法对轴对称的旋转圆柱间的Couette-Taulor流进行数值模拟．首先给出Navier-Stokes方程流函数形式,利用Couette流把边界条件齐次化．其次给出Stokes算子的特征函数的解析表达式,证明其正交性,并对特征值进行估计．最后利用Stokes算子的特征函数作为逼近子空间的基函数,给出谱Galerkin逼近方程的表达式．证明了Navier-Stokes方程非奇异解的谱Galerkin逼近的存在性、唯一性和收敛性,给出了解谱Galerkin逼近的误差估计,并展示了数值计算结果．     

非线性玻尔兹曼方程的傅里叶谱方法

玻尔兹曼方程作为空气动理学中最基本的方程之一,是连接微观牛顿力学和宏观连续介质力学的重要桥梁.该方程描述了一个由大量粒子组成的复杂系统的非平衡态时间演化:除了基本的输运项,其最重要的特性是粒子间的相互碰撞由一个高维,非局部且非线性的积分算子来描述,从而给玻尔兹曼方程的数值求解带来非常大的挑战.在过去的二十年间,基于傅里叶级数的谱方法成为了数值求解玻尔兹曼方程的一种很受欢迎且有效的确定性算法.这主要归功于谱方法的高精度及它可以被快速傅里叶变换加速的特质.本文将回顾玻尔兹曼方程的傅里叶谱方法,具体包括方法的导出,稳定性和收敛性分析,快速算法,以及在一大类基于碰撞的空气动理学方程中的推广.     

三阶微分方程的多区域Legendre-Petrov-Galerkin谱方法

提出三阶微分方程初边值问题的多区域Legendre-Petrov-Galerkin谱方法.对于三阶线性微分方程,证明该方法全离散格式的稳定性,并给出L~2-误差估计.进而将该方法和Legendre配置方法相结合,应用于某些非线性问题.数值算例对单区域和多区域方法的结果进行比较.     

A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in two‐dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi‐implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element‐wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third‐order in time. Copyright © 2014 John Wiley & Sons, Ltd.     

一类带弱奇异核偏积分微分方程空间谱配置方法的全局性

借助拉普拉斯变换,运用谱配置方法研究一类线性偏积分微分方程的半离散问题,这类问题出现在粘弹性模型中.它是一种基于Gauss-Lobatto求积节点的配置方法.我们得到了空间半离散解的稳定性和收敛性结果.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

On the best constant in the inf-sup-condition for elongated rectangular domains

The Ladyzhenskaya-Babushka-Brezzi inequality (inf-sup-condition) is often used in the analysis of convergence of approximate solutions of hydrodynamic equations to exact ones. The constant involved in it depends on the shape of the domain and determines the efficiency of different algorithms. In this paper, its asymptotics and two-sided estimates for rectangular domains are obtained. To this end, a new method for estimating eigenvalues of a certain spectral problem with Stokes’ saddle operator is used. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 387–396, March, 2000.     

Klein-Gordon-Zakharov方程初边值问题的Legendre谱方法

研究Klein-Gordon-Zakharov方程初边值问题的Legendre谱方法.在先验估计的基础上,证明了该格式的稳定性和收敛性,并得到最优阶误差估计.另外,还设计了一个半隐格式,并给出数值例子.在文章的后面给出了多区域谱格式,数值结果表明精度要高于单区域.     

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

The convergence of a class of combined spectral-finite difference methods using Hermite basis, applied to the Fokker-Planck equation, is studied. It is shown that the Hermite based spectral methods are convergent with spectral accuracy in weighted Sobolev space. Numerical results indicating the spectral convergence rate are presented. A velocity scaling factor is used in the Hermite basis and is shown to improve the accuracy and effectiveness of the Hermite spectral approximation, with no increase in workload. Some basic analysis for the selection of the scaling factors is also presented.

     

广义Benjamin-Bona-Mahony方程Fourier谱方法的大时间误差估计

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＢｅｎｊａｍｉｎ Ｂｏｎａ Ｍａｈｏｎｙｅｑｕａｔｉｏｎｕｔ+ｕｘ+ｕｕｘ -ｕｘｘ-ｕｘｘｔ =0 ( 1 .1 )ｉｎｃｏｒｐｏｒａｔｅｓｎｏｎｌｉｎｅａｒｄｉｓｐｅｒｓｉｖｅａｎｄｄｉｓｓｉｐａｔｉｖｅｅｆｆｅｃｔｓ ,ａｎｄｈａｓｂｅｅｎｐｒｏｐｏｓｅｄａｓａｍｏｄｅｌｆｏｒｂｏｔｈｔｈｅｂｏｒｅｐｒｏｐａｇａｔｉｏｎａｎｄｔｈｅｗａｔｅｒｗａｖｅｓ[1,2 ] .Ｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｕｎｉｑｕｅｎｅｓｓｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｉｓｅｑｕａｔｉｏｎｈａｖｅｂｅｅ…     

An adaptive spectral least-squares scheme for the Burgers equation

A least-squares spectral collocation method for the one-dimensional inviscid Burgers equation is proposed. This model problem shows the stability and high accuracy of these schemes for nonlinear hyperbolic scalar equations. Here we make use of a least-squares spectral approach which was already used in an earlier paper for discontinuous and singular perturbation problems (Heinrichs, J. Comput. Appl. Math. 157:329–345, 2003). The domain is decomposed in subintervals where continuity is enforced at the interfaces. Equal order polynomials are used on all subdomains. For the spectral collocation scheme Chebyshev polynomials are employed which allow the efficient implementation with Fast Fourier Transforms (FFTs). The collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The scheme exhibits exponential convergence where the exact solution is smooth. In parts of the domain where the solution contains discontinuities (shocks) the spectral solution displays a Gibbs-like behavior. Here this is overcome by some suitable exponential filtering at each time level. Here we observe that by over-collocation the results remain stable also for increasing filter parameters and also without filtering. Furthermore by an adaptive grid refinement we were able to locate the precise position of the discontinuity. Numerical simulations confirm the high accuracy of our spectral least-squares scheme.      

Continuous collocation approximations to solutions of first kind Volterra equations

In this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind Volterra integral equations. The results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations. The convergence analysis is based on a Runge-Kutta or ODE approach.