解高维广义BBM方程的谱方法和拟谱方法

在非线性色散介质的长波研究中,Benjanin,Bona和Mahony等人提出并讨论了BBM方程。这类方程在许多数学物理问题中出现,如热力学中的双温热传导问题、在岩石裂缝中的渗流问题等,因而引起了人们的重视。之后,Goldstein,Avrin,郭柏灵等进一步研究了高维广义BBM方程。这类方程的数值分析很多,但主要是差分法和有限元法,如[9-10],[11]在一维情形下用谱方法和拟谱方法作了研究。本文讨论高维     

半直线上BBM方程初边值问题的修正有理谱耦合方法

本文考虑使用修正的有理谱方法处理半直线上的BBM方程初边值问题.对非线性项使用Chebyshev有理插值显式处理,而线性项使用修正Legendre有理谱方法隐式处理.这种处理既可以节约运算又可以保持良好的稳定性.数值例子表明了算法的有效性     

Glowinski区域分解算法的收敛性方程：Stokes方程

Ｇｌｏｗｉｎｓｋｉ区域分解算法的收敛性方程──Ｓｔｏｋｅｓ方程储德林，胡显承（清华大学应用数学系，北京１０００８４）ＴＨＥＣＯＮＶＥＲＧＥＮＣＥＯＦＧＬＯＷＩＮＳＫＩ＇ＳＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＡＬＧＯＲＩＴＨＭ──ＳＴＯＫＥＳＥＱＵＡ...     

求解Fisher型方程的混合Jacobi-球面调和谱方法

本文提出了两种数值求解单位球内Fisher型方程的混合Jacobi-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。     

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.     

KdV方程的时间谱离散方法

本文提出了解KdV方程周期边值问题的安全港离散方法:在时间方向上采用Chebyshev拟谱逼近,在空间方向上采用Fourier Galerkin逼近。谱展开的系数由目标泛函的极小值来确定。同时证明了该方法的收敛性。     

一类Schrodinger方程的周期解和谱方法

本文利用半群理论讨论了一类非线性非自共轭Ｓｃｈｒｏｄｉｎｇｅｒ方程周期初边值解的存在唯一性，以及它的Ｆｏｕｒｉｅｒ谱方法的可解性，稳定性和收敛性     

输运方程谱间断流线扩散耦合方法

本文讨论了谱有限元方法,构造了求解Ｂｏｌｔｚｍ ａｎｎ 方程球谐函数谱展开和间断流线扩散有限元耦合格式．建立了这种耦合方法的稳定性及最优阶收敛性误差估计．得到了比标准有限元更高的精度．     

杂交有限元的区域分解法

近几年来,由于并行计算机的迅速发展,求解椭圆型微分方程的区城分解法又引起了人们的重视.这一方法的基本思想是把求解区域分解成许多子区域,每个子区域上用一台计算机求解.这种方法适应并行机的需要,是用并行机解大型椭圆型偏微分方程的一     

Spectral Deferred Correction Methods for Ordinary Differential Equations

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).Deferred correction methods based on the Picard equation appear to be promising candidates for further investigation.     

Spectral Element Viscosity Methods for Nonlinear Conservation Laws on the Semi-Infinite Interval

In this paper we propose a spectral element: vanishing viscosity (SEW) method for the conservation laws on the semi-infinite interval. By using a suitable mapping, the problem is first transformed into a modified conservation law in a bounded interval, then the well-known spectral vanishing viscosity technique is generalized to the multi-domain case in order to approximate this trarsformed equation more efficiently. The construction details and convergence analysis are presented. Under a usual assumption of boundedness of the approximation solutions, it is proven that the solution of the SEW approximation converges to the uniciue entropy solution of the conservation laws. A number of numerical tests is carried out to confirm the theoretical results.     

Stokes方程的投影／方向分裂谱方法

我们提出和分析了一种求解Stokes方程的数值方法．新方法基于空间上的Legendre谱离散,时间上则采用投影／方向分裂格式．更确切地说,时间离散的出发点是旋度形式的压力校正投影法,在此基础上进一步应用方向分裂法,把速度和压力方程分裂为一系列一维的椭圆型子问题．然后生成的这些一维子问题用Legendre谱方法进行空间离散．另外,我们证明了全离散格式的稳定性．一些数值实验验证了收敛性和方法的有效性．     

Multiscale Domain Decomposition Methods for Elliptic Problems with High Aspect Ratios

Abstract In this paper we study some nonoverlapping domain decomposition methods for solving a classof elliptic problems arising from composite materials and flows in porous media which contain many spatialscales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarsesolver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domaindecomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate inthe presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent ofthe aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework iscarried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numericalexperiments which include problems with multipl     

Well-Conditioned Spectral Collocation Methods for Problems in Unbounded Domains

Based on the generalized Laguerre and Hermite functions, we construct two types of Birkhoff-type interpolation basis functions. The explicit expressions are derived, and fast and stable algorithms are provided for computing these basis functions. As applications, some well-conditioned collocation methods are proposed for solving various second-order differential equations in unbounded domains. Numerical experiments illustrate that our collocation methods are more efficient than the standard Laguerre/Hermite collocation approaches.     

Spectral domain embedding for elliptic PDEs in complex domains

Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such as rectangles or disks. Domain decomposition methods/spectral element methods extend the applicability of spectral methods to more complex geometries. An alternative is to embed the irregular domain into a regular one. This paper uses the spectral method with domain embedding to solve PDEs on complex geometry. The running time of the new algorithm has the same order as that for the usual spectral collocation method for PDEs on regular geometry. The algorithm is extremely simple and can handle Dirichlet, Neumann boundary conditions as well as nonlinear equations.     

An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations

The time-harmonic Maxwell equations are considered in the low-frequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields.

     

The Spectral Decomposition of Covariance Matrices for the Variance Components Models

The aim of this paper is to propose a simple method to determine the number of distinct eigenvalues and the spectral decomposition of covariance matrix for a variance components model. The method introduced in this paper is based on a partial ordering of symmetric matrix and relation matrix. A method is also given for checking straightforwardly whether these distinct eigenvalues are linear dependent as functions of variance components. Some examples and applications to illustrate the results are presented.     

Spectral discretization of a model for organic pollution in waters

We are interested in a mixed reaction diffusion system describing the organic pollution in stream‐waters. In this work, we propose a mixed‐variational formulation and recall its well‐posedness. Next, we consider a spectral discretization of this problem and establish nearly optimal error estimates. Numerical experiments confirm the interest of this approach. Copyright © 2016 John Wiley & Sons, Ltd.     

A Posteriori Error Estimation of Spectral and Spectral Element Methods for the Stokes/Darcy Coupled Problem

In this paper, we carry out an a posteriori error analysis of Legendre spectral approximations to the Stokes/Darcy coupled equations. The spectral approximations are based on a weak formulation of the coupled equations by using the Beavers-Joseph-Saffman interface condition. The main contribution of the paper consists of deriving a number of posteriori error indicators and their upper and lower bounds for the single domain case. An extension of the upper bounds to the multi-domain case in the spectral element framework is also given.