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谱方法解微分方程谱方法（Spectral methods）是一种用离散傅里叶变换（Discrete Fourier Transform）或者离散余弦变换（Discrete Cosine Transform）等频域方法来求解微分方程的一类数值方法。
它在数学与工程领域被广泛应用，特别是用于解决具有周期性和高度光滑解的微分方程。
谱方法的优点包括高精度、快速收敛和适用于多维问题。
它可以在整个定义域内提供高度准确的解，而其他传统的常用差分法和有限元法则只在特定位置或单个点上提供近似解。
这使得谱方法具有广泛的应用领域，例如流体力学、量子力学、天体物理学等领域中的一维、二维和三维研究等。
谱方法主要包含三个主要步骤：离散化、求解以及逆变换。
首先，对微分方程进行空间离散化，通常使用Chebyshev多项式或者傅里叶基函数等等。
采用Chebyshev多项式进行离散化时，可以使用Chebyshev-Gauss-Lobatto（CGL）或Chebyshev-Gauss（CG）点进行节点选择。
对于一维问题，可以使用一维的Chebyshev系数，而对于二维和三维问题，需要扩展为二维或三维的Chebyshev系数。
其次，利用傅里叶变换或者离散余弦变换将微分方程转化为频域的代数方程。
通过数值求解这个代数方程，可以得到频域上的解。
最后，采用逆变换将频域的解转化为时域上的解。
这个逆变换可以是傅里叶逆变换或者离散余弦逆变换等。
谱方法的收敛性和精度主要依赖于离散化的方式以及选择的基函数。
在实践中，经验表明使用Chebyshev基函数的谱方法在解决光滑和非光滑问题时都能提供很高的精度和收敛性。
然而，谱方法的缺点也不能被忽视。
首先，谱方法对边界条件的处理相对复杂。
在实际应用中，可以通过使用特殊的基函数来处理这个问题。
其次，谱方法随着问题的维度增加，计算量会成指数级增加。
因此，尽管谱方法在一维和二维问题上表现出色，但在三维问题上的应用相对有限。
一维不可压缩navier stokes方程理论说明1. 引言1.1 概述本文将讨论一维不可压缩Navier-Stokes方程的理论说明。
Navier-Stokes方程是描述流体运动的基本方程之一，其在各个领域都具有重要应用价值。
本文将从介绍Navier-Stokes方程的基本概念开始，逐步展开对一维流动特征和不可压缩流体模型假设的理论说明。
1.2 文章结构文章分为五个主要部分：引言、一维不可压缩Navier-Stokes方程理论说明、理论推导和分析、数值方法和模拟研究以及结论与展望。
其中，引言部分将概述文章的目标和结构，提供读者对整篇文章内容的预览。
1.3 目的本文旨在深入探讨一维不可压缩Navier-Stokes方程，并通过理论推导和数值模拟研究解析该方程对流体运动行为的描述能力。
通过阐明不同数值方法在求解此类方程时的差异和优劣，我们可以更好地了解该方程在实践中的应用，并为进一步研究提供展望。
以上是关于引言部分的详细内容，请根据需要进行修改或补充。
2. 一维不可压缩Navier Stokes方程理论说明2.1 Navier Stokes方程简介Navier Stokes方程是描述流体运动的基本方程之一，它由质量守恒和动量守恒两个方程组成。
同时考虑流体的黏性和压力力作用，Navier Stokes方程能够准确描述流体在各种复杂情况下的运动。
2.2 一维流动特征描述在一维流动中，流体只在一个空间方向上（通常为x轴）有速度分量变化，而在其余两个空间方向上（通常为y轴和z轴）没有速度分量变化。
这样简化后的一维问题可以更容易地推导出Navier Stokes方程的解析解，并且提供了更直观的物理图像。
2.3 不可压缩流体模型假设不可压缩流体是指在任何情况下密度保持不变，即密度是常数。
这个假设适用于许多情况下，例如液体的非常小压缩性以及稳态条件下气体的高马赫数等。
通过这个假设，我们可以将Navier Stokes方程进一步简化为不含密度项的形式，并且使问题更具可行性。
fluent 二维大涡模拟命令Fluent（通常称为ANSYS Fluent）是一种基于计算流体动力学（CFD）的软件，它使用数值方法解决流体力学和热力学方程。
Fluent支持多个求解器，包括稳态、非稳态、可压缩和不可压缩流体求解器。
其中，二维大涡模拟（Large Eddy Simulation，LES）是一种用于模拟湍流流动的CFD方法，通过分解流体的速度场为大尺度和小尺度来模拟湍流流动。
本文将介绍Fluent中二维大涡模拟的相关命令。
1. 设定模拟参数在开始二维大涡模拟前，需要设定一些模拟参数，包括流体属性和边界条件。
在Fluent中，通过以下命令可以设定流体属性和边界条件：（1）设定流体属性DEFINE > MODELS > VISCOSITY2. 定义二维网格在进行CFD模拟前，需要先定义计算网格，以便数值求解器能够在其上执行算法。
在Fluent中，通过以下命令定义二维网格：（1）导入二维网格FILE > IMPORT > MESH3. 指定求解器有关Fluent的求解器已经在第一段中提到。
在进行二维大涡模拟时，可以选择可压缩或不可压缩流体求解器作为替代。
（2）可压缩流体求解器SOLVE > COMPRESSIBLE FLOW/HEAT TRANSFER > STEADY模拟模型是模拟过程中使用的具体模型。
在Fluent中，用户可以选择不同的模拟模型。
（1）可分离流边界层（Detached Eddy Simulation，DES）MODEL > VISCOSITY > DES（2）壁面函数（Wall Function）MODEL > VISCOSITY > WALL FUNCTION在进行CFD模拟时，需要设定一些计算参数以控制模拟进程，以及获得所需的结果。
在Fluent中，用户可以使用以下命令设定计算参数：6. 运行模拟在完成所有设定后，可以通过以下命令在Fluent中运行二维大涡模拟：SOLVE > EXECUTE COMMAND FILE > RUN此时，Fluent将自动执行过程，直至收敛或达到设定的计算时间。
luent中一些问题----(目录)1 如何入门2 CFD计算中涉及到的流体及流动的基本概念和术语2.1 理想流体（Ideal Fluid）和粘性流体（Viscous Fluid）2.2 牛顿流体（Newtonian Fluid）和非牛顿流体（non-Newtonian Fluid）2.3 可压缩流体（Compressible Fluid）和不可压缩流体（Incompressible Fluid）2.4 层流（Laminar Flow）和湍流（Turbulent Flow）2.5 定常流动（Steady Flow）和非定常流动（Unsteady Flow）2.6 亚音速流动(Subsonic)与超音速流动（Supersonic）2.7 热传导（Heat Transfer）及扩散（Diffusion）3 在数值模拟过程中，离散化的目的是什么？如何对计算区域进行离散化？离散化时通常使用哪些网格？如何对控制方程进行离散？离散化常用的方法有哪些？它们有什么不同？3.1 离散化的目的3.2 计算区域的离散及通常使用的网格3.3 控制方程的离散及其方法3.4 各种离散化方法的区别4 常见离散格式的性能的对比（稳定性、精度和经济性）5 流场数值计算的目的是什么？主要方法有哪些？其基本思路是什么？各自的适用范围是什么？6 可压缩流动和不可压缩流动，在数值解法上各有何特点？为何不可压缩流动在求解时反而比可压缩流动有更多的困难？6.1 可压缩Euler及Navier-Stokes方程数值解6.2 不可压缩Navier-Stokes方程求解7 什么叫边界条件？有何物理意义？它与初始条件有什么关系？8 在数值计算中，偏微分方程的双曲型方程、椭圆型方程、抛物型方程有什么区别？9 在网格生成技术中，什么叫贴体坐标系？什么叫网格独立解？10 在GAMBIT中显示的“check”主要通过哪几种来判断其网格的质量？及其在做网格时大致注意到哪些细节？11 在两个面的交界线上如果出现网格间距不同的情况时，即两块网格不连续时，怎么样克服这种情况呢？12 在设置GAMBIT边界层类型时需要注意的几个问题：a、没有定义的边界线如何处理？b、计算域内的内部边界如何处理（2D）？13 为何在划分网格后，还要指定边界类型和区域类型？常用的边界类型和区域类型有哪些？14 20 何为流体区域（fluid zone）和固体区域（solid zone）？为什么要使用区域的概念？FLUENT是怎样使用区域的？15 21 如何监视FLUENT的计算结果？如何判断计算是否收敛？在FLUENT中收敛准则是如何定义的？分析计算收敛性的各控制参数，并说明如何选择和设置这些参数？解决不收敛问题通常的几个解决方法是什么？16 22 什么叫松弛因子？松弛因子对计算结果有什么样的影响？它对计算的收敛情况又有什么样的影响？17 23 在FLUENT运行过程中，经常会出现“turbulence viscous rate”超过了极限值，此时如何解决？而这里的极限值指的是什么值？修正后它对计算结果有何影响18 24 在FLUENT运行计算时，为什么有时候总是出现“reversed flow”？其具体意义是什么？有没有办法避免？如果一直这样显示，它对最终的计算结果有什么样的影响26 什么叫问题的初始化？在FLUENT中初始化的方法对计算结果有什么样的影响？初始化中的“patch”怎么理解？27 什么叫PDF方法？FLUENT中模拟煤粉燃烧的方法有哪些？30 FLUENT运行过程中，出现残差曲线震荡是怎么回事？如何解决残差震荡的问题？残差震荡对计算收敛性和计算结果有什么影响？31数值模拟过程中，什么情况下出现伪扩散的情况？以及对于伪扩散在数值模拟过程中如何避免？32 FLUENT轮廓（contour）显示过程中，有时候标准轮廓线显示通常不能精确地显示其细节，特别是对于封闭的3D物体（如柱体），其原因是什么？如何解决？33 如果采用非稳态计算完毕后，如何才能更形象地显示出动态的效果图？34 在FLUENT的学习过程中，通常会涉及几个压力的概念，比如压力是相对值还是绝对值？参考压力有何作用？如何设置和利用它？35 在FLUENT结果的后处理过程中，如何将美观漂亮的定性分析的效果图和定量分析示意图插入到论文中来说明问题？36 在DPM模型中，粒子轨迹能表示粒子在计算域内的行程，如何显示单一粒径粒子的轨道（如20微米的粒子）？37 在FLUENT定义速度入口时，速度入口的适用范围是什么？湍流参数的定义方法有哪些？各自有什么不同？38 在计算完成后，如何显示某一断面上的温度值？如何得到速度矢量图？如何得到流线？39 分离式求解器和耦合式求解器的适用场合是什么？分析两种求解器在计算效率与精度方面的区别43 FLUENT中常用的文件格式类型：dbs，msh，cas，dat，trn，jou，profile等有什么用处？44 在计算区域内的某一个面（2D）或一个体（3D）内定义体积热源或组分质量源。
纳维-斯托克斯方程（NS方程）是一组描述粘性不可压缩流体动量守恒的运动方程。
在谱形式下，NS方程通常采用傅里叶级数或类似的方法展开，将物理量表示为频率或波数的函数。
这种形式允许我们分析流体的频率和波数特性，从而更好地理解流体运动的本质。
在谱形式下，NS方程可以表示为：
1. 连续性方程：ρ(u·∇)u = 0
2. 动量方程：ρ(u·∇)u + ∇p = μ∇²u
其中，ρ是流体密度，u是速度矢量，p是压力，μ是动力粘度。
在谱形式下，这些方程的解可以通过傅里叶分析或类似的方法找到。
值得注意的是，NS方程的谱形式求解非常复杂，通常需要高性能计算资源和数值方法。
在实际应用中，通常采用离散化方法，如有限差分法、有限元法等，将连续的物理量离散化后进行求解。
这些离散化方法可以在计算机上实现高效的数值模拟和计算。
流函数的存在条件
一、介绍流函数的概念
流函数是描述流体运动的一种数学方法，它是一个标量函数，可以用
来表示在流体运动中各点速度分量之间的关系。
在二维不可压缩流体中，流函数是一个标量场，它的等值线与速度场的线无交点。
二、流函数的存在条件
在二维不可压缩流体中，如果速度场满足连续性方程和无旋条件，则
存在一个与速度场相对应的流函数。
1. 连续性方程
连续性方程是描述质量守恒定律的基本方程之一。
在不可压缩流体中，连续性方程可以表示为：
∂ρ/∂t + ∇·(ρv) = 0
其中，ρ为密度，t为时间，v为速度向量。
该方程表达了质量守恒定律：任意时刻内部质量不变。
2. 无旋条件
无旋条件是指速度场满足以下条件：
∇×v = 0
该条件表达了角动量守恒定律：任意时刻内部角动量不变。
三、推导流函数公式
在二维不可压缩流体中，设速度分别为u(x,y)和v(x,y)，则有：∂u/∂x + ∂v/∂y = 0
根据无旋条件，有：
∂u/∂y - ∂v/∂x = 0
将u和v表示为流函数ψ的偏导数形式，有：
u = ∂ψ/∂y, v = -∂ψ/∂x
将上式代入连续性方程中，得到：
∇^2ψ = -ρ
其中，∇^2表示拉普拉斯算子，ρ为密度。
该式即为流函数的泊松方程。
四、总结
流函数是描述流体运动的一种数学方法，它是一个标量函数，可以用来表示在流体运动中各点速度分量之间的关系。
在二维不可压缩流体中，如果速度场满足连续性方程和无旋条件，则存在一个与速度场相对应的流函数。
通过推导可得到流函数的泊松方程公式。
Jou rnal of M athem atical R esearch&Expo siti onV o l.19,N o.2,3752390,M ay1999Spectral M ethods for Two-D i m en sional I ncom pressible FlowΞGuo B eny u M a H ep ing(ShanghaiU n iversity,Shanghai201800)Abstract: W e take the tw o2di m en si onal vo rticity equati on s as models to describe spectral m eth2ods and their com b inati on s w ith fin ite difference m ethods o r fin ite elem en t m ethods,w h ich are ap2p licab le to o ther si m ilar non linear p rob lem s.Som e num erical resu lts and erro r esti m ates of thesem ethods are given.Keywords: vo rticity equati on,spectral m ethod,com b inati on m ethod.Cla ssif ica tion: AM S(1991)65N30,76D99 CL C O241.82D ocu m en t Code:A Article I D:10002341X(1999)022*******1.I n troduction T he basic idea of sp ectral m ethods fo r p artial differen tial equati on s is to app rox i m ate ex2 act so lu ti on s by u sing sp ectral functi on s as bases.T he m ethods stem from the classical R itz2 Galerk in m ethod,w h ich have h igh accu racy so called the convergence of“infin ite o rder”.B u t they w ere no t w idely u sed fo r a long ti m e becau se of the exp en sive co st of com p u tati onal ti m e.How ever,the discovery of the Fast Fou rier T ran sfo rm ati on(FFT)and the rap id de2 velopm en t of m odern com p u ters rem oved th is ob stacle.A lthough fin ite difference m ethods and fin ite elem en t m ethods are very successfu l in num erical so lu ti on s of p artial differen tial e2 quati on s,it is no doub t that fo r som e p rob lem s sp ectral m ethods are m o re favo rab le becau se of its h igh accu racy.Sp ectral m ethods have been app lied successfu lly to num erical si m u lati on s in science and engineering.Go ttlieb and O rszag[5]summ arized theo retical resu lts and exp erience of app lica2 ti on s to m any p ractical p rob lem s.T hey p rovided num erical analysis fo r linear p rob lem s. Since then,sp ectral m ethods fo r non linear p rob lem s have also advanced rap idly w ith their app licati on s to flu id dynam ics,w eather p redicti on,and o ther fields[34,1,35,21,30,28,29, 33,19,2,9].T he sp ectral m ethod,p seudo sp ectralm ethod,and T au m ethod,w h ich are differen t ver2 si on s of sp ectral m ethods,can be derived from the m ethod of w eigh ted residual.W e con sider here an in itial2boundary value p rob lem as fo llow sΞRece ived da te:1996205230L u (x ,t )=f (x ,t ), x ∈8, t 〉0,B u (x ,t )=0, x ∈58,t Ε0,u (x ,0)=u 0(x ), x ∈8ϖ,(111)w here 8is a dom ain w ith the boundary 58,L is a differen tial op erato r ,and B is a linear boundary op erato r .T he m ethod of w eigh ted residual is to find an app rox i m ati on so lu ti on to (1.1)of the fo rmu N (x ,t )=u B (x ,t )+∑Nn =1an (t )Υn (x ),(1.2)w here trial functi on s {Υn (x )}(1Φn ΦN )are linearly indep enden t ,u B (x ,t )is cho sen such that u N (x ,t )satisfies the boundary conditi on ,and a n (t )are determ ined by the fo llow ing e 2quati on s∫8[L u N (x ,t )-f (x ,t )]w -n (x )d x =0, t 〉0,n =1,2,…,N ,(113)w here the w eigh t functi on s w n (x )are linearly indep enden t ,and by a si m ilar treatm en t of the in itial conditi on .Spectra l m ethod (Ga lerk i n approx i m a tion ) A ssum e that Υn (x )satisfy the boundary condi 2ti on so that u B (x ,t )=0and w eigh t functi on s w n (x )=Υn (x ).T herefo re ,(1.3)leads to (L u N (t ),Υn )=(f (t ),Υn ), n =1,2,…,N(1.4)w ith the inner p roduct (u ,v )=∫8u (x )v -(x )d x .Som eti m es ,it is m o re conven ien t to de 2scribe the schem e (1.4)via a p ro jecti on op erato r P N .So w e define a fin ite di m en si onal linear sp aceV N =Span{Υn :n =1,2,…,N },(1.5)and define P N u ∈V N such that(P N u ,Υn )=(u ,Υn ), n =1,2,…,N .(1.6)It is easy to show that P N u is un iquely determ ined since {Υn (x )}are linearly indep enden t .T hu s w e know that (1.4)is equ ivalen t to the fo llow ing schem eP N L u N (x ,t )=P N f (x ,t ).(1.7)Pseudospectra l m ethod (colloca tion approx i m a tion ) In th is case ,Υn (x )are the sam e as in sp ectral m ethods.B u t the w eigh t functi on s are taken as D irac ∆functi on s :w n (x )=∆(x -x n ), n =1,2,…,N ,w here x n ∈8ϖcalled co llocati on po in ts such that det (Υm (x n ))N ×N ≠0.N ow ,(1.3)leads to L u N (x n ,t )=f (x n ,t ), n =1,2,…,N .(1.8)A lso ,the schem e (1.8)can be described via an in terpo lati on op erato r P C .To th is end ,w edefine P C u ∈V N such that P C u (x n )=u (x n ), n =1,2,…,N .It is easy to see that P C u isun iquely determ ined since det (Υm (x n ))N ×N ≠0.T hu s ,w e know that (1.8)is equ ivalen t to the fo llow ing schem eP C L u N (x ,t )=P C f (x ,t ).(1.9)Tau m ethod H ere w e assum e that Υn (x )are o rthogonal ,bu t need no t satisfy the boundary conditi on .T he u B (x ,t )in (1.2)is of the fo rmu B (x ,t )=∑N +mn =N +1a n (t )Υn (x ),w here m is the num ber of indep enden t boundary conditi on s.T he w eigh t functi on s are taken as w n (x )=Υn (x )(n =1,2,…,N ).In th is case ,the schem e (1.3)is read as (L u N (t ),Υn )=(f (t ),Υn ), n =1,2,…,N (1.10)w ith the m equati on s given by the boundary con strain ts .T he T au app rox i m ati on schem e (1.10)can also be described viaa p ro jecti on op erato r [5].In th is p ap er ,w e take the tw o 2di m en si onal evo lu ti onary vo rticity equati on s as m odels to in troduce som e num erical m ethods related to sp ectral m ethods :.Fou rier sp ectral (o r p seudo sp ectral )m ethods.Fou rier sp ectral (o r p seudo sp ectral )2difference m ethods.Fou rier sp ectral (o r p seudo sp ectral )2fin ite elem en t m ethods.Fou rier 2Chebyshev sp ectral (o r p seudo sp ectral )m ethodsT he first m ethod is fo r p rob lem s w ith p eri odic boundary conditi on s and the o thers fo rones w ith sem i 2p eri odic boundary conditi on s ,w h ich are app licab le to o ther si m ilar p rob lem s.2.Four ier Spectra l or Pseudospectra l M ethods for the Per iod ic Problem s Fo r p rob lem s w ith p eri odic boundary conditi on s ,Fou rier sp ectral m ethods are pow er 2fu l.2.1 A Four ier Spectra l M ethod T he vo rticity equati on s are the stream functi on 2vo rticity fo rm u lati on p resen tati on s of in 2com p ressib le N avier 2Stokes equati on s.L et Ν(x ,y ,t )and Ω(x ,y ,t )be the vo rticity and stream functi on resp ectively .Ν0(x ,y )and f l (x ,y ,t )(l =1,2)are given .A ll of them have the p eri od 2Πfo r the variab le x and y .L et 8={(x ,y ):-Π〈x ,y 〈Π}.W e con sider the fo llow ing p rob lem 5t Ν+J (Ν,Ω)-v 2Ν=f 1, in 8×(0,T ),- 2Ω=Ν+f 2, in 8×(0,T ),Ν(x ,y ,0)=Ν0(x ,y ), in 8,(2.1)w here Τis a nonnegative con stan t and J (Ν,Ω)=5y Ω5x Ν-5x Ω5y Ν.L et (u ,v )=14Π2κ8u (x ,y )v -(x ,y )d x d y , u 2=(u,u ).To fix Ω(x ,y ,t ),w e requ ire that (Ω(t ),1)=0(0Φt ΦT ).Con sider the w eak fo rm u lati on of (2.1)as fo llow s (5t Ν(t ),v )+(J (Ν(t ),Ω(t )),v )+Τ( Ν(t ), v )=(f 1(t ),v ), t ∈(0,T ),( Ω(t ), v )=(Ν(t ),v )+(f 2(t ),v ), t ∈(0,T ).(2.2)Kuo Pen 2yu [22]u sed the sp ectral m ethod fo r the p rob lem (2.1)and p roved strict erro r esti 2m ates of the schem e .Fo r any po sitive in teger N ,set V N =Span{ei (lx +m y ):l 2+m 2ΦN 2}.L et Σbe the m esh step of the variab le t .D eno te u k (x ,y )=u (x ,y ,k Σ)by u k .D efineu k t =1Σ(u k +12u k ), u δk =12(u k +1+u k ).A Fou rier sp ectral schem e fo r so lving (2.1)is to findΓk =∑l 2+m 2÷N 2Γk l ,m e i (lx +m y ), Υk =∑l 2+m 2÷N 2Υk l ,m e i (lx +m y ),(2.3)such that fo r any v ∈V N and k Ε0,(Γk t ,v )+(J (Γk +∆ΣΓk t ,Υk ),v )+Τ( (Γk +ΡΣΓk t ), v )=(f k 1,v ),( Υk , v )=(Γk +f k 2,v ),(Υk ,1)=0,(Γ0,v )=(Ν0,v ),(2.4)w here ∆,ΡΕ0are p aram eters.W e po in t ou t that if ∆=0the schem e (2.4)can be so lved ex 2p licitly such thatΓk +1l ,m =11+ΤΡΣ(l 2+m 2){[1-Τ(1-Ρ)Σ(l 2+m 2)]Γk l ,m +g k l ,m }(2.5)w here g k l ,m =(f k l -J (Γk ,Υk ),e i (l +m .))(2.6)It is easy to see that the fo llow ing con servati on law s ho ld .If f k 1=0,w e have(Γk ,1)=(Γ0,1), k Ε0,(2.7)and if ∆=Ρ=12in additi on ,w e have Γn 2+2ΤΣ∑n -1k =0Γδk 2= Γ0 2.(2.8)Kuo Pen -yu [22]po in t ou t that if w e u se the filtered sp herical m ean summ ati onΓk(x ,y )=∑l 2+m 2÷N 21-l 2+m 2N 2ΧΓk l ,m ,e i (lx +m y ), ΧΕ0.(2.9)then w e can get better resu lts.2.2 A Four ier Pseudospectra lM ethod W hen the sp ectral m ethod is u sed,w e have to deal w ith the in tegrati on such as(2.6). In o rder to avo id th is troub le,the p seudo sp ectral m ethod is develop ed,w h ich is easier to p erfo rm and saves the co st in com p u tati on,and is m o re favo rab le fo r non linear p rob lem s.B u t th is m ethod som eti m es has non linear in stab ility due to the aliasing.A cco rding to Kuo Pen2yu[22],the Bochner summ ati on(2.9)cou ld be u sed fo r eli m inating these p henom ena and raise the accu racy of app rox i m ate so lu ti on s.M a H e2p ing and Guo B en2yu[26]develop ed a Fou rier p seudo sp ectral m ethod by u sing the Bochner summ ati on fo r p rob lem(2.1).L et W N =Span{e i(lx+m y):-NΦl,mΦN}, 8N={(qh,j h):-NΦq,jΦN}w ith h=2Π (2N +1).L et P N:L2(8)→V N be the o rthogonal p ro jecti on op erato r defined by(P N u,v)=(u,v), Πv∈V N(2.10) and P C:C(8)→W N be the in terpo lati on op erato r byP C u(x,y)=u(x,y), Π(x,y)∈8N.(2.11) Fo rΧΕ1and u∈V N w ith the coefficien ts u l,m,w e definea restrain t op erato r R=R(Χ)byR u(x,y)=∑l2+m2ΦN21-l2+m2N2Χ2u l,m e i(lx+m y).(2.12)To app rox i m ate the non linear term J(u,v),w e define P C=P N P C andJ C(u,v)=12{P C(5x u5y v-5y u5x v)+5x P C(u5y v)-5y P C(u5x v)}.(2.13) T he Fou rier p seudo sp ectral schem e fo r so lving(2.1)is to findΓk,Υk as(2.3)such thatfo r kΕ0,Γk t+R J C(RΓk+∆ΣRΓk t,RΥk)-Τ 2(Γk+ΡΣΓk t)=P C f k1,- 2Υ=Γk+P C f k2,(Υ,1)=0,Γ0=P CΝ0,(2.14)w here∆,ΡΕ0are p aram eters.If∆=0,then the schem e(2.14)can be so lved exp licitly as in(2.5)bu t w ithg k l,m=(f k1-R J C(RΓk,RΥk),e i(l.+m.))N,(2115) in w h ich(u,v)N=1(2N+1)2∑(x,y)∈8Nu(x,y)v-(x,y)and w h ich can be com p u ted w ith FFT efficien tly.B y the fact that fo r u,v,w∈V N,(J C(u,v),w)+(J C(w,v),u)=0,(2.16)it is easy to see that the sam e con servati on law s as (2.7)and (2.8)ho ld fo r (2.14).T he num erical resu lts given in M a H ep ing and Guo B enyu [26]show that the restrain t op 2erato r R (Χ)i m p roves the stab ility of the p seudo sp ectral m ethod ,esp ecially in the case w hen the so lu ti on s of the PD E change m o re rap idly .T he value of Χm u st be cho sen su itab ly to get good resu lts .How to choo se the p aram eter Χsu itab ly is relative to the s m oo thness of the ex 2act so lu ti on .Generally sp eak ing ,if the exact so lu ti on changes rap idly ,w e shou ld take s m all Χ,and conversely ,take large Χ.3.Four ier Spectra l or Pseudospectra l -D ifference M ethods for the Se m i -Per iod ic Proble m s H ereafter w e con sider the tw o 2di m en si onal vo rticity equati on s w ith p eri odic and nonp eri 2odic boundary conditi on s ,and assum e that Τis po sitive .L et I ={x :0〈x 〈1},I υ={y :0〈y 〈2Π},and 8=I ×I υ.W e assum e that all functi on s have the p eri od 2Πfo r the variab le y and thatΝ(0,y ,t )=Ν(1,y ,t )=Ω(0,y ,t )=Ω(1,y ,t )=0, Πy ∈I υ, t Ε0.(3.1) A lthough Fou rier sp ectral and p seudo sp ectral m ethods are favo rab le fo r p eri odic p rob 2lem s ,they does no t w o rk fo r the p rob lem (2.1)w ith (3.1).To so lve it ,M u rdock [31,32]u sedChebyshev sp ectral m ethods ,and Guo B enyu and X i ong Yueshan [17]fo llow ed the idea of GuoB enyu [7]to con struct a class of sp ectral 2difference schem es.T he key po in t is the u se of a skew symm etric decom po siti on of the non linear convecti on term s.T hen ,if the p aram eters in the schem e are cho sen su itab ly ,the num erical so lu ti on satisfies sem i discrete con servati on law s and better erro r esti m ates are ob tained .3.1 A Four ier Spectra l -D ifference M ethod L et h be the m esh sp acing in the x 2directi on w ith M h =1,and letI h ={x =jh :1Φj ΦM 21}.(3.2)D efineD u (x ,y ,t )=1h (u (x +h ,y ,t )-u (x ,y ,t )), D u (x ,y ,t )=D u (x -h ,y ,t ), D δu (x ,y ,t )=12(D u (x ,y ,t )+D {u (x ,y ,t )), u (x ,y ,t )=DD {u (x ,y ,t )+52y u (x ,y ,t ).T he key p rob lem in the con structi on of a reasonab le schem e is to si m u late as m uch as po ssi 2b le the p rop erties of the so lu ti on of (2.1).Indeed ,if f 1=f 2=0,then∫∫8Ν(x ,y ,t )d x d y -Τ∫t 0∫I [5x Ν(1,y ,t )-5x Ν(0,y ,t )]d x d y =∫∫8Ν0(x ,y )d x d y (3.3)and∫∫8Ν2(x ,y ,t )d x d y +2Τ∫t 0∫∫8[(5x Ν(x ,y ,s ))2+(5y Ν(x ,y ,s ))2]d x d y d s =∫∫8Ν20(x ,y )d x d y .(3.4) T he schem e w as con structed such that its so lu ti on satisfies sem i discrete con servati onlaw s.N o te that 5y w 5x u -5x w 5y u =5x (5y w u )-5y (5x w u )=5y (w 5x u )-5x (w 5y u ).T herefo re ,defineJ 1(u ,v )=5y vD ^u -D ^v 5y u , J 2(u ,v )=D ^(5y vu )-5y (D ^vu ),J 3(u ,v )=5y (vD ^u )-D ^(v 5y u ), J (Α)(u ,v )=∑3l =1ΑlJ l (u ,v ),w here Α=(Α1,Α2,Α3),Αl Ε0,and Α1+Α2+Α3= 1.N ow setV N =Sp an{e ily : l ΦN },(3.5)and let P N :L 2(I υ)→V N be the o rthogonal p ro jecti on op erato r such that ,∫I υ(P N u -u )v -d y =0, Πv ∈V N .(3.6)L et Γk and Υk be the app rox i m ati on s to Νand Υresp ectively such thatΓk (x ,y )=∑ l ΦN Γk l (x )e ily , Υk (x ,y )=∑ l ΦN Υk l (x )e ily .T he sp ectral 2difference schem e fo r (2.1)and (3.1)is Γk t +P N J (Α)(Γk +∆ΣΓk t ,Υk )2Τ∃(Γk +ΡΣΓk t )=P N f k 1, in I h ×I υ,k Ε0,-∃Υk =Γk +P N f k 2, in I h ×I υ,k Ε0,Γk (0,y )=Γk (1,y )=Υk (0,y )=Υk (1,y )=0, Πy ∈I υ,k Ε0,Γ0=P N Ν0, in I h ×I υ,(3.7)w here ∆and Ρare p aram eters such that 0Φ∆,ΡΦ1.If ∆=Ρ=0,then (3.7)is an exp licitschem e .O therw ise ,iterati on is needed to get Γk fo r k Ε 1.A ssum e that f k 1=f k 2=0,w ehave the fo llow ing con servati on law s(Γn ,1)+Σ∑n -1k =0{(Α2+Α3)A (Γk +∆ΣΓk t ,Υk )+2ΤS (Γk +ΡΣΓk t ,1)}=(Γ0,1)(3.8)w here (u ,v )h =h ∑x ∈I h (u (x ),v (x ))I υ, (u (x ),v (x ))I υ=12Π∫I υu (x ,y )v -(x ,y )d y , u 2h =(u ,u )h , u 21,h =12 D u 2h +12D -u 2h + 5y u 2h ,A (u ,v )=12(u (1-h ),5y v (1-h ))I υ-12(u (h ),5y v (h ))I υ,S (u ,v )=12h (u (h ),v (h ))I υ+12h(u (1-h ),v (1-h ))I υ.M o reover ,if Α1=Α2and ∆=Ρ=12,then Γn 2h +2ΤΣ∑n -1k =0{ Γ^k 21,h +S (Γ^k ,Γ^k )}= Γ0 2h .(3.9)C learly ,(3.8)and (3.9)are reasonab le analogies of (3.3)and (3.4),resp ectively .T he num erical resu lts given in Guo B enyu and X i ong Yueshan [17]show that w hen Α1=Α2and the so lu ti on s of (3.7)satisfy the sem i discrete con servati on law s ,better resu lts are ob 2tained .It is also show n that good resu lts can be go t even fo r s m all N .B y a skew 2symm etric decom po siti on of the non linear convecti on term ,w e ob tain better num erical resu lts than by the m o re conven ti onal fo rm .B u t a little m o re w o rk is requ ired fo r calcu lating the Fou rier co 2efficien ts of the non linear term .3.2 A Four ier Pseudospectra l -D ifference M ethod T he calcu lati on of Fou rier coefficien ts fo r the non linear convecti on term takes qu ite a lo t of ti m e in sp ectral 2difference schem e .To rem edy th is deficiency ,Guo B enyu and X i ong Yue 2shan [18]p rovided a p seudo sp ectral 2difference m ethod fo llow ing the w o rk of Guo B enyu andZheng J iadong [20].W e shall u se the sam e no tati on s as in the above secti on .W e first in troduce the po in ts onI υ:y j =2Πj (2N +1)(0Φj Φ2N ).L et P C :C (I υ-)→V N be the in terpo lati on op erato r suchthatP C u (y j )=u (y j ), 0Φj Φ2N .(3.10)Fo r ΧΕ1,w e define a restrain t op erato r by R =R (Χ)based on the Bochner summ ati on such that fo r any u ∈V N w ith the coefficien ts u l ,R u (y )=∑ l ΦN 1-l N Χu l e ily .(3.11)To app rox i m ate the non linear term J (u ,v ),w e define the fo llow ing non linear op erato rs J C ,1(u ,v )=P C (5y vD ^u -D ^v 5y u ), J C ,2(u ,v )=D ^[P C (u 5y v )]-5y [P C (uD ^v )], J C ,3(u ,v )=5y [P C (vD ^u )]-D ^[P C (v 5y u )], J (Α)C (u ,v )=∑3l =1Αl J C ,l (u ,v ),w here Α=(Α1,Α2,Α3),Αl Ε0,and Α1+Α2+Α3=1.L et Γk and Υk be the app rox i m ati on s to Νand Ωresp ectively ,w here Γk (x ,y ),Υk (x ,y )∈V N fo r all x ∈I h and k Ε0.T he p seudo sp ectral 2difference schem e fo r (2.1)and (3.1)is Γk t +R J (Α)C (R Γk +∆ΣR Γk t ,R Υk )-Τ∃(Γk +ΡΣΓk t )=P C f k 1,-∃Υk =Γk +P C f k 2,Γk (0,y )=Γk (1,y )=Υk (0,y )=Υk (1,y )=0,Γ0=P C Ν0.(3.12)If f k 1=f k 2=0,then (Γn,1)+Σ∑n -1k =0{(Α2+Α3)A (R Γk +∆k ΣR Γk t ,R Υk )+2ΤS (Γk +ΡΣΓk t ,1)}=(Γ0,1).(3.13)If in additi on Α1=Α2and ∆=Ρ=1 2,then (3.9)ho lds also .T he num erical resu lts in Guo B enyu and X i ong Yueshan [18]show the sam e advan tages of (3.12)as tho se of (3.7).In p articu lar ,even w e u se the skew symm etric decom po siti on of the non linear convecti on term ,the com p u tati onal ti m e is nearly the sam e as by the m o re con 2ven ti onal fo rm .4.Four ier Spectra l or Pseudospectra l -F i n ite Elem en t M ethods for the Sem i -Pe -r iod ic Problem s It m ay be hard to generalize the com b ined sp ectral 2difference m ethod to p rob lem s on non 2rectangu lar dom ain .T he fin ite elem en t m ethod can be successfu lly app lied to such p rob 2lem s .B u t the convergence rate is li m ited by the degree of in terpo lati on .O n the o ther hand ,the sp ectral and p seudo sp ectral m ethods have “infin ite ”o rder accu racy if the so lu ti on s to be app rox i m ated are infin itely differen tiab le .B u t it is very difficu lt to u se them to so lve p rob 2lem s on non 2rectangu lar dom ain s .In p articu lar ,Fou rier sp ectral o r p seudo sp ectral m ethodsare app licab le on ly to p eri odic p rob lem s.T hey cou ld no t be u sed directly to so lve (2.1)and (3.1).Canu to ,M aday ,and Q uarteron i [3]p ropo sed a com b ined p seudo sp ectral and fin ite ele 2m en t m ethod w ith app licati on to the steady p rob lem of N avier 2Stokes equati on s.Guo B enyu and Cao W ei m ing [10]con structed a sp ectral 2fin ite elem en t schem e fo r so lving (2.1)and(3.1).4.1 A Four ier Spectra l -F i n ite Ele m en tM ethod L et T h be a class of regu lar decom po siti on s of the in terval I and satisfy the inverse as 2sum p ti on [4].L et 0=x 0〈x 1〈…〈x M =1are the grid po in ts and I l =(x l -1,x l ).D efine h =m ax 1Φl ΦM x l -x l -1 ,h θ=m in 1Φl ΦM x l -x l -1 ,and assum e that there is a con stan t Θinde 2p enden t h such that h ΦΘh θ.L et I Pm be the set of po lynom ials of o rder Φm and S 0m .h ={v :v I l ∈IP m fo r 1Φl ΦM ,v is con tinuou s and v (0)=v (1)=0}.D eno te by II m the p iecew ise L agrange in terpo lati on op erato r of o rder m on to S 0m ,h ,i.e .,II m u is the L agrange in terpo lati on of o rder m of u on each I l (1Φl ΦM )and con tinuou s on I .V N and P N are defined by (3.5)and (3.6),resp ectively .T he sp ectral 2fin ite elem en t m ethod fo r so lving (2.1)and (3.1)is to find Γk ,Υk ∈S 0m ,hV N such that fo r any v ∈S 0m ,h V N and k Ε0,(Γk t ,v )+(J (Γk +∆ΣΓk t ,Υk ),v )+Τ( (Γk +ΡΣΓk t ), v )=(II m f k 1,v ),( Υk , v )=(Γk +II m f k 2,v ),Γ0=II m P N Ν0,(4.1)w here ∆,ΡΕ0are p aram eters. T he num erical resu lts given in Guo B enyu and Cao W ei m ing[10]show that(i) W ith the sam e m esh sizes,the sp ectral2fin ite elem en t m ethods give better resu lts than the fu lly fin ite elem en t m ethods.(ii) O n ly a relatively s m all N is needed to reso lve the so lu ti on s in y2directi on w ith the sp ectral2fin ite elem en t m ethods.T h is m ean s a great cu t in com p u tati on w o rk.T h is m ethod can be easily generalized to th ree2di m en si onal sem i2p eri odic p rob lem s,even w hen the dom ain is no t rectangu lar.4.2 A Four ier Pseudospectra l-F i n ite Ele m en tM ethod In com p arison w ith sp ectral m ethods,p seudo sp ectral m ethods can be i m p lem en ted m o re efficien tly.B u t their stab ility m ay be poo r due to the aliasing.In som e p rob lem s,the op ti2 m al rate of convergence in the L22no rm can be ob tained fo r sp ectral m ethods,bu t no t fo r p seudo sp ectral m ethods[29,25].Guo B enyu and M a H ep ing[14]p resen ted a Fou rier p seudo sp ectral2fin ite elem en t schem e fo r p rob lem(2.1)and(3.1).A con tro l op erato r based on the Bochner summ ati on is u sed toi m p rove the stab ility.L et the op erato rs P C,R(Χ),and J C(u,v)be the sam e as(3.10),(3.11),and(3.12),resp ectively.If u,v,w∈S0m,h V N,then in tegrating by p arts,w e get (2.16).T he p seudo sp ectral2fin ite elem en t schem e fo r so lving(2.1)and(3.1)is to findΓk,Υk∈S0m,h V N such that fo r any v∈S0m,h V N and kΕ0,(Γk t,v)+(R J C(RΓk+∆ΣRΓk t,RΥk),v)+Τ( (Γk+ΡΣΓk t), v)=(P C f k1,v),( Υk, v)=(Γk+P C f k2,v),(4.2)Γ0=0m P CΝ0,w here p aram eters0Φ∆,ΡΦ1.If∆=Ρ=1 2and f k1=0,w e have(2.8)from(2.16).H ere the op erato r J C(Γk,Υk)is con structed so that the app rox i m ati on so lu ti on satisfies a con servati on si m ilar to w hat the so lu ti on of(2.1)satisfies.T hu s,the stab ility is i m p roved and the convergence o rder is heigh tened.In fact,the m ain of the non linear erro r van ishes, and w e therefo re get better erro r esti m ates.A lso the con tro l op erato r R is u sed,w h ich i m2 p roves the stab ility and cu rb s erro rs,and by w h ich the L22op ti m al erro r esti m ate is ob2 tained.5.Four ier-Chebyshev Spectra l or Pseudospectra lM ethods for the Sem i-Per iod ic Problem sT he accu racy of bo th sp ectral2difference m ethods and sp ectral2fin ite elem en t m ethods is still li m ited due to the app rox i m ati on s in non2p eri odic directi on s.Guo B enyu,M a H ep ing, Cao W ei m ing and H uang H u i[15]p ropo sed ano ther k ind of m ixed m ethod fo r so lving(2.1) and(3.1)by u sing Fou rier2sp ectral app rox i m ati on in the p eri odic directi on and Chebyshev2 sp ectral app rox i m ati on in the non2p eri odic directi on.T he m ethod keep s the advan tage of the convergence of“infin ite o rder”.5.1 A Four ier-Chebyshev Spectra lM ethod In th is secti on ,let I =(-1,1),I υ=(0,2Π),and 8=I ×I υ.W e assum e that all func 2ti on s in (2.1)have the p eri od 2Πfo r the variab le y andΝ(±1,y ,t )=Ω(±1,y ,t )=0, Πy ∈I υ, t Ε0.(5.1)L et M and N be po sitive in tegers.D efine V M (I )={v ∈IP M :v (-1)=v (1)=0}.(5.2)L et V N (I υ)be the set of all real trigonom etric po lynom ials of degree less o r equal to N w ith the p eri od 2Π.D efineV M ,N (8)=V M (I ) V N (I υ).(5.3)L et Ξ(x )=(1-x 2)-1 2and define the sp aceL 2Ξ(8)={v is m easu rab le :(v ,v )Ξ〈∞}(5.4)equ i pp ed w ith the inner p roduct and the no rm(u ,v )Ξ=14Π∫∫8u (x ,y )v (x ,y )Ξ(x )d x d y , u 2Ξ=(u ,u )Ξ.(5.5)L et P M ,N :L 2Ξ(8)→V M ,N (8)be the o rthogonal p ro jecti on such that(u -P M ,N u ,v )Ξ=0, Πv ∈V M ,N (8).(5.6)L et Σbe the step of the variab le t and define u k t δ=12Σ(u k +1-u k -1).T he fu lly discrete Fou rier 2Chebyshev sp ectral schem e fo r so lving (2.1)and (5.1)is to find Γk ,Υk ∈V M ,N (8),app rox i m ating to Νand Ωresp ectively ,such that fo r any v ∈V M ,N (8),(Γk t δ,v )Ξ+(J (Γk ,Υk ),v )Ξ+Τ2a Ξ(Γk +1+Γk -1,v )=(f k 1,v )Ξ, k Ε1,a Ξ(Υk ,v )=(Γk +f k 2,v )Ξ, k Ε0,Γ1=P M ,N (Ν0+Σ5Ν(0)), Γ0=P M ,N Ν0,(5.7)w here a Ξ(u ,v )=14Π∫∫8 u (x ,y ) [v (x ,y )Ξ(x )]d x d y ,5t Ν(0)=Τ 2Ν0-J (Ν0,Ω(0))+f 1(0).W e give tw o exam p les to show the num erical resu lts of the m ethod in troduced above .Exam ple 1 L et the exact so lu ti on s of (2.1)and (3.1)beΝ(x ,y ,t )=A exp {B sin (C x +y )+Ξt }, Ω(x ,y ,t )=A exp {Ξt }(C x +sin y ).Fo r describ ing the erro rs ,w e let I h be as (3.2),I υN ={y =2Πj N :0Φj ΦN -1},and de 2fineE ∞(t )=m ax (x ,y )∈I h ×I υN Ν(x ,y ,t )-Γ(x ,y ,t ) ,E 2(t )=hN ∑(x ,y )∈I h ×I υN Ν(x ,y ,t )-Γ(x ,y ,t ) 21 2,w here Γ(x ,y ,t )is the so lu ti on of Fou rier sp ectral 2difference (FSD )schem e (3.7)o r the so lu ti on of Fou rier 2Chebyshev sp ectral (FCS )schem e (5.7).T he erro rs of bo th the FSD and FCS schem es are show n in T ab le I fo r A =C =Ξ=0.1,B =0.01,and Σ=Τ=0.001.T ab le I.E rro rs fo r the FSD and FCS Schem es .t =1FSDFCS M =10,N =4M =4,N =4E 2(t )0.2217E 230.5435E 25E ∞(t )0.6949E 230.6497E 25Exam ple 2 L et the exact so lu ti on s of (2.1)and (5.1)beΝ(x ,y ,t )=0.4(x 221)(x 228)sin2y e t 2,2 2Ω(x ,y ,t )=Ν(x ,y ,t ).(5.8)D efine I h ={x =co s (Πj M ):0Φj ΦM },I υN ={y =2Πj N :0Φj ΦN 21},and E (u (t ),v (t ))=∑(x ,y )∈I h ×I υN u (x ,y ,t )2v (x ,y ,t ) 2∑(x ,y )∈I h ×I υN u (x ,y ,t ) 21 2,L et Γ(x ,y ,t )and Υ(x ,y ,t )are the so lu ti on s of Fou rier sp ectral 2fin ite (FSF )schem e (4.1)o r the so lu ti on s of Fou rier 2Chebyshev sp ectral (FCS )schem e (5.7).T he erro rs of bo th the FSF and FCS schem es are show n in T ab le II fo r Τ=0.001,Σ=0.01.T ab le II.E rro rs fo r the FSF and FCS Schem es .t =5FSFFCS M =4,N =4M =10,N =4M =4,N =4E (Ν(t ),Γ(t ))0.4436E 220.7188E 220.3027E 24E (Ω(t ),Υ(t ))0.1592E 210.1455E 220.1687E 24 It can be seen that the resu lts of the Fou rier 2Chebyshev sp ectral m ethod are m uch better than tho se of the Fou rier sp ectral 2difference m ethod o r the Fou rier sp ectral 2fin ite elem en t m ethod .V ery h igh accu racy so lu ti on s can be ob tained w ith the Fou rier 2Chebyshev m ethodby u sing on ly a s m all num ber of m odes.T he w eakness of th is m ethod is that it can no t be app lied directly to the th ree 2di m en si onal sem i 2p eri odic p rob lem s on non 2rectangu lar do 2m ain s.5.2 A Four ier -Chebyshev Pseudospectra l M ethod Fo r saving the w o rk as w ell as keep ing the convergence rate of “infin ite o rder ”,GuoB enyu and L i J ian [12]develop ed a Fou rier 2Chebyshev p seudo sp ectral m ethod .W e u se thesam e no tati on s as in Secti on 5.1.Fu rtherm o re ,let {x j }and {w j }be the nodes and w eigh ts of Gau ss 2L obatto in tegrati on ,nam elyx q =co s q ΠM, fo r 0Φq ΦM ,w 0=w M =Π2M , w q =ΠM, fo r1Φq ΦM -1.A lso ,p u t y j =2Πj(2N +1)and define 8M ,N ={(x q ,y j ):0Φq ΦM , 0Φj Φ2N }.(5.9)W e deno te by P C the in terpo lati on from C (8ϖ)to V M ,N (8),i.e .,P C u (x ,y )=u (x ,y ), on 8M ,N . R ecen tlyM a H ep ing and Guo B enyu [27]generalized the restrain t op erato r u sed in the p re 2vi ou s secti on s to the Chebyshev app rox i m ati on .Fo r th is m ixed m ethod ,let Χ1,Χ2Ε1and R =R (Χ1,Χ2)such that ifΤ=∑Mq =0∑ l ΦN uql T q (x )e ily ,(5.10)T q (x )being the Chebyshev po lynom ial of degree q ,thenR u =∑Mq =0∑ l ΦN u ql (1-q M Χ11-lN Χ2T q (x )e ily .(5.11)Fo r app rox i m ating the non linear convecti on term ,let J C (u ,v )=5x [P C (u 5v )]-5x [P C (u 5x v )].L et Γk and Υk be the app rox i m ati on s to Νand Ωas in (5.7).T he Fou rier 2Chebyshev p seudo sp ectral schem e fo r (2.1)and (5.1)is to find Γk ,Υk ∈V M ,N (8)such that Γk t δ+R J C (R Γk ,R Υk )-Τ2 2(Γk +1+Γk -1)=P C f k 1,- 2Υk =Γk +P C f k 2,Γ1=P M ,N (Ν0+Σ5t Ν(0)), Γ0=P M ,N Ν0,(5.12) T he num erical resu lts given in Guo B enyu and L i J ian [12]show the advan tages of th is ap 2p roach .It p rovides the num erical so lu ti on s w ith h igh accu racy ,bu t needs less w o rk than the Fou rier 2Chebyshev sp ectral m ethod .6.Error Esti m a tes R ecen tly sp ectral m ethods ,p seudo sp ectral m ethods ,and related m ixed m ethods are de 2velop ing successfu lly .M uch w o rk has been done on the num erical analysis of these m ethodssystem atically (See Canu to ,H u ssain i ,Q uarteron l ,and Zang [2],and Guo B enyu [9]).In1981,Guo B enyu adop ted a Fou rier sp ectral m ethod fo r so lving the K .D .V .2B u rgers’equa 2ti on and strictly p roved the convergence (See Kuo Penyu [23]and its foo t no te ).T h is is one ofthe earliest theo retical w o rk of sp ectral m ethods fo r non linear p rob lem s.L ater ,Guo B enyu et al.generalized th is techn ique to the R .L .W .equati on ,vo rticity equati on ,N avier 2Stokes equati on ,and the flow w ith low M ach num ber .In p articu lar ,Guo B enyu and M a H ep ing [13]u sed such a m ethod fo r the th ree 2di m en si onal com p ressib le flow w ith strict erro r esti m atesw h ich is a difficu lt j ob .T hese w o rk ex tended g 2stab ility (i.e .,generalized stab ility ,see Guo。
