谱方法(spectrum method)，伪谱法（pseudoospectrum method）及谱元法（spectrum element method） 2012年04月11日星期三12:47
华夏艺术中心谱方法（spectrum method）
Spectral methods are a class of techniques to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential convergence" being the fastest possible. Spectral methods were developed in a long series of papers by Steven Orszag starting in 1969 including, but not limited to, Fourier series methods for periodic geometry problems, polynomial spectral methods for finite and unbounded geometry problems, pseudospectral methods for highly nonlinear problems, and spectral iteration methods for fast solution of steady state problems.
Partial differential equations (PDEs) describe a wide array of physical processes such as heat conduction, fluid flow, and sound propagation. In many such equations, there are underlying "basic waves" that can be used to give efficient algorithms for computing solutions to these PDEs. In a typical case, spectral methods take advantage of this fact by writing the solution as its Fourier series, substitut
ing this series into the PDE to get a system of ordinary differential equations (ODEs) in the time-dependent coefficients of the trigonometric terms in the series (written in complex exponential form), and using a time-stepping method to solve those ODEs.
从上面可以看到谱方法的思路：对PDE方程进行FFT变换，得到只对时间微分的常微分方程组。然后，using a time-stepping method来解ODE方程组。 孕妇死亡
The spectral method and the finite element method are closely related and built on the same ideas; the main difference between them is that the spectral method approximates the solution as linear combination of continuous functions that are generally nonzero over the domain of solution (usually sinusoids or Chebyshev polynomials), while the finite element method approximates the solution as a linear combination of piecewise functions that are nonzero on small subdomains.
谱方法和有限元法的思想很类似，差别在于：谱方法以一系列全局连续的函数（可以是三角函数，也可以是多项式）的叠加来近似真实解，而有限元法则是使用分片的简单函数的叠加来近似近似真实解。即，有限元的插值函数，只在该单元内作用。 Because of this, the spectral method takes on a global approach while the finite element method is a local approach. This is part of why the spectral method works best when the solution is smooth. In fa
ct there are no known three-dimensional single domain spectral shock capturing results.[1]
The implementation of the spectral method is normally accomplished either with collocation or a Galerkin or a Tau approach.
SM方法是解偏微分方程的一种数值方法。其要点是把解近似地展开成光滑函数(一般是正交多项式)的有限级数展开式，即所谓解的近似谱展开式，再根据此展开式和原方程，求出展开式系数的方程组。对于非定常问题，方程组还同时间有关谱方法实质上是标准的分离变量技术的一种推广。一般多取切比雪夫多项式和勒让德多项式作为近似展开式的基函数。对于周期性边界条件，用傅里叶级数和面调和级数比较方便。谱方法的精度，直接取决于级数展开式的项数。现以解简单一维热传导方程的初边值混合问题为例，说明这种方法的应用：
(1)
边界条件(0,)＝(,)＝0，(2)
初始条件(,0)＝()，(3)
式中为坐标；为时间；为大于零的常数根据周期性边界条件，可取近似谱展开式为：
情报科学
(4)
把式(4)代入式(1)得：
(5)
。(6)
利用快速傅里叶变换技术，可迅速完成求解过程，而且(4)至(6)式比任何有限阶的有限差分解，都更快地收敛到(1)至(3)的真解。一般说，谱方法远比普通一、二阶差分法准确。由于快速傅里叶变换之类的技术不断发展，谱方法的运算量越来越少，一般是很合算的。特别是对于二维以上的问题，用差分法计算必须设置足够多的网格点，造成计算量的增加，而用谱方法一般不需取太多的项就可得到较高精 度的解。因此谱方法在计算流体力学复杂流场的问题中有广泛应用。
下面是Wiki中的一个例子：
A concrete, linear example
Here we presume an understanding of basic multivariate calculus and Fourier series. If g(x,y) is a known, complex-valued function of two real variables, and g is periodic in x and y (that is, g(x,y)=g(x+2π,y)=g(x,y+2π)) then we are interested in finding a function f(x,y) so that
where the expression on the left denotes the second partial derivatives of f in x and y, respectively. This is the Poisson equation, and can be physically interpreted as some sort of heat conduction problem.
If we write f and g in Fourier series:
and substitute into the differential equation, we obtain this equation:
We have exchanged partial differentiation with an infinite sum, which is legitimate if we assume for instance that f has a continuous second derivative. By the uniqueness theorem for Fourier expansions, we must then equate the Fourier coefficients term by term, giving
(*)
which is an explicit formula for the Fourier coefficients a j,k.
With periodic boundary-conditions, the Poisson equation possesses a solution only if b0,0= 0. Therefore we can freely choose a0,0which will be equal to the mean of the resolution. This corresponds to choosing the integration constant.
To turn this into an algorithm, only finitely many frequencies are solved for. This introduces an
error which can be shown to be proportional to , where h=1/n and is the highest frequency treated.
从上面两个例子可以总结出来谱方法的算法步骤：
1Compute the Fourier transform (b j,k) of g.
2Compute the Fourier transform (a j,k) of f via the formula (*) and the Fourier transform of g.
3Compute f by taking an inverse Fourier transform of (a j,k).
Since we're only interested in a finite window of frequencies (of size n, say) this can be done using a Fast Fourier Transform algorithm. Therefore, globally the algorithm runs in time O(n log n).
伪谱法（pseudoospectrum method）
Pseudo-spectral methods [1] are a class of numerical methods for the solution of PDEs,such as the direct simulation of a particle with an arbitrary wavefunction interacting with an arbitrary potential. They are related to spectral methods and are used extensively in computational fluid dynamics and other areas, but are demonstrated below on an example from quantum physics.
Background
The Schrödinger wave equation,
can be written
which resembles the linearordinary differential equation
with solution
In fact, using the theory of linear operators, it can be shown that the general solution to the Schrödinger wave equation is
where exponentiation of operators is defined using power series. Now remember that
where the kinetic energy is given by
and the potential energy often depends only on position (i.e., ). We can write
It is tempting to write
so that we may treat each factor separately. However, this is only true if the operators and
commute, which is not true in general. Luckily, it turns out that
vdl
is a good approximation for small values of . This is known as the symmetric decomposition. The heart of the pseudo-spectral method is using this approximation iteratively to calculate the
wavefunction for arbitrary values of .
The method
For simplicity, we will consider the one-dimensional case. The method is readily extended to multiple dimensions.
Given , we wish to find where is small.
The first step i s to calculate an intermediate value by applying the rightmost operator in
the symmetric decomposition,
This requires only a pointwise multiplication.
The next step is to apply the middle operator,
This is an infeasible calculation to make in configuration space（就是以x,y为坐标的空间）. Fortunately, in momentum space（就是波数域）, the calculation is greatly simplified. If is
the momentum space representation of , then
which also requires only a pointwise multiplication. Numerically, is obtained from
using the Fast Fourier transform (FFT) and is obtained from using the
inverse FFT.
The final calculation is
This sequence can be summarized as
Analysis of algorithm
If the wavefunction is approximated by its value at distinct points, each iteration requires 3 pointwise multiplications, one FFT, and one inverse FFT. The pointwise multiplications each
require effort, and the FFT and inverse FFT each require effort. The total
computational effort is therefore determined largely by the FFT steps, so it is imperative to use an efficient (and accurate) implementation of the FFT. Fortunately, many are freely available.原道n11
谱元法（spectrum element method）
谱元法SEM(spectral element method)基于弹性力学方程弱形式基础之上，最初是Patera在1984年计算流体力学中提出. 它在有限元上进行谱展开，所以具有有限元方法和伪谱法的思想，同时兼备有限元可以模拟任何复杂介质模型的韧性和伪谱法的精度，谱元法又称为谱方法的域分解或高阶有限元法。
秩和比传统的方法以有限元通用性最好，但是有限元法中分析波的传播时需要使单元大小与波的波长相当，且时间分辨率也非常小，使得计算效率较低。需要一些新的方法来提高计算效率。谱元法有时域的和频域的两种，而这两种方法其实没有什么关系。
时域谱元法和传统的有限元法区别较小，应该说是一种高阶的有限元法，其为了达到精度，细分网格是通过切比雪夫多项式或者勒让德多项式等正交多项式的根来定网格节点。前人研究证明其在分析波的传播方面可以提高计算效率，减少存储空间。
频域谱元法也是在分析波的传播方面的一种有限元方法，其是在频域内，使位移函数采用波动方程的一般解，得到与频率相关的动刚度矩阵，利用快速傅里叶变换实现时域和频域的转换。以一根梁为例，分析波的传播时，有限元方法网格大小跟波长有关，可能需要成千上万的单元节点，而频域谱元法则只需一个单元两个节点。显然可以大幅度提高计算效率。然而这种方法对非均质或者二维三维结构的模拟能力较弱，通用性不好。
In mathematics, the spectral element method is a high order finite element method.（这句话，真是，直接说出来了谱元法的本质：高阶有限单元法）
Introduced in a 1984 paper[1] by A. T. Patera, the abstract begins: "A spectral element method that combines the generality of the finite element method with the accuracy of " The spectral element method is an elegant formulation of the finite element method with a high
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