Spectral method: Difference between revisions

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{{Short description|Class of methods used in numerical analysis and scientific computing to solve ODE/PDE}}

{{More footnotes|date=August 2013}}

'''Spectral methods''' are a class of techniques used in [[applied mathematics]] and [[scientific computing]] to numerically solve certain [[differential equation]]s, potentially involving the use of the [[fast Fourier transform]]. The idea is to write the solution of the differential equation as a sum of certain "[[basis function]]s" (for example, as a [[Fourier series]] which is a sum of [[Sine wave|sinusoid]]s) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.

Spectral methods and [[finite -element method]]s are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole ___domain, while finite element methods use basis functions that are nonzero only on small subdomains. In([[compact othersupport]]). wordsConsequently, spectral methods takeconnect on avariables ''global approachglobally'' while finite elementelements methodsdo use aso ''local approachlocally''. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is [[Smooth function|smooth]]. However, there are no known three-dimensional single -___domain spectral [[shock capturing]] results (shock waves are not smooth).<ref name="CHQZ">[https://books.google.com/books?id=7COgEw5_EBQC pp 235, Spectral Methods]: evolution to complex geometries and applications to fluid dynamics, By Canuto, Hussaini, Quarteroni and Zang, Springer, 2007.</ref> In the finite -element community, a method where the degree of the elements is very high or increases as the grid parameter ''h'' decreases to zeroincreases is sometimes called a [[spectral -element method]].

Spectral methods can be used to solve [[ordinary differential equations]] (PDEs, ODEs, eigenvalue, etc) <ref>{{cite journal |last1=Muradova |first1=Aliki D. |title=The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions |journal=Adv Comput Math |year=2008 |volume=29 |issue=2 |pages=179–206, [[partial2008 differential|doi=10.1007/s10444-007-9050-7|hdl=1885/56758 equations]]|s2cid=46564029 |hdl-access=free }}</ref> (PDEs) and [[eigenvalueoptimization problem]] problems involving differential equationss. When applying spectral methods to time-dependent PDEs, the solution is typically written as a sum of basis functions with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients which can be solved using any [[numerical methods for ordinary differential equations|numerical method for ODEs]].<ref>{{cite journal |last1=Muradova |first1=Aliki D. |title=A time spectral method for solving the nonlinear dynamic equations of a rectangular elastic plate |journal=Journal of Engineering Mathematics |year=2015 |volume=92 |pages=83–101, 2015 |doi=10.1007/s10665-014-9752-z }}</ref> Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems {{Citation needed|date=August 2013}}.

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. The implementation of the spectral method is normally accomplished either with [[collocation method|collocation]] or a [[Galerkin method|Galerkin]] or a [[Tau method|Tau]] approach . For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations.

Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple domains with smooth solutions. However, because of their global nature, the matrices associated with step computation are dense and computational efficiency will quickly suffer when there are many degrees of freedom (with some exceptions, for example if matrix applications can be written as [[Fourier transform]]s). For larger problems and nonsmooth solutions, finite elements will generally work better due to sparse matrices and better modelling of discontinuities and sharp bends.

Spectral methods are computationally less expensive than finite element methods, but become less accurate for problems with complex geometries and discontinuous coefficients. This increase in error is a consequence of the [[Gibbs phenomenon]].

==Examples of spectral methods==

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===A concrete, linear example===

Here we presume an understanding of basic multivariate [[calculus]] and [[Fourier series]]. If <math>g(x,y)</math> is a known, complex-valued function of two real variables, and g is periodic in x and y (that is, <math>g(x,y)=g(x+2\pi,y)=g(x,y+2\pi)</math>) then we are interested in finding a function ''f''(''x'',''y'') so that

:<math>\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)f(x,y)=g(x,y)\quad \text{for all } x,y</math>

<!--math>f_{xx}(x,y)+f_{yy}(x,y)=g(x,y)\quad \text{for all} x,y</math-->

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, or a problem in potential theory, among other possibilities.

If we write ''f'' and ''g'' in Fourier series:

:<math>f=:\sum a_begin{j,kalign}e^{i(jx+ky)}</math>

:<math>gf&=:\sum b_a_{j,k}e^{i(jx+ky)}</math>, \\[5mu]

g&=:\sum b_{j,k}e^{i(jx+ky)},

\end{align}</math>

and substitute into the differential equation, we obtain this equation:

:<math>\sum -a_{j,k}(j^2+k^2)e^{i(jx+ky)}=\sum b_{j,k}e^{i(jx+ky)}.</math>

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

{{NumBlk|:(*)| <math>a_{j,k}=-\frac{b_{j,k}}{j^2+k^2}</math>|{{EquationRef|<nowiki>*</nowiki>}}}}

which is an explicit formula for the Fourier coefficients ''a''_''j'',''k''.

With periodic boundary conditions, the [[Poisson equation]] possesses a solution only if ''b''_''0'',''0'' = 0. Therefore, we can freely choose ''0a''_0,0 which will be equal to the mean of the resolution. Therefore,This corresponds to choosing the integration constant.

we can freely choose ''a''_''0'',''0'' which 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 <math>h^n</math>, where <math>h := 1/n</math> and <math>n</math> is the highest frequency treated.

==== Algorithm ====

# Compute the Fourier transform (''b_j,k'''') of ''g''.

# Compute the Fourier transform (''a_j,k'') of ''f'' via the formula ({{EquationNote|*}}).

# Compute ''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 {{nowrap|time ''O''(''n'' log ''n'').}}

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:<math>\partial_{t} u + u \partial_{x} u = \rho \partial_{xx} u + f \quad \forall x\in\left[0,2\pi\right), \forall t>0</math>

where &rho; is the [[viscosity]] coefficient. In weak conservative form this becomes

:<math>\left\langle \partial_{t} u , v \right\rangle = \leftBigl\langle \partial_x \left(-\frac{1}{2}tfrac12 u^2 + \rho \partial_{x} u\right) , v \rightBigr\rangle + \left\langle f, v \right\rangle \quad \forall v\in \mathcal{V}, \forall t>0</math>

where \overline{g(x)}\,dx</math> following [[inner product space|inner product]] notation. [[integration by parts|Integrating by parts]] and using periodicity grants

where <math>\langle f, g \rangle := \int_{0}^{2\pi} f(x)

:<math>\langle \partial_{t} u , v \rangle = \left\langle \frac{1}{2}tfrac12 u^2 - \rho \partial_{x} u , \partial_x v\right\rangle+\left\langle f, v \right\rangle \quad \forall v\in \mathcal{V}, \forall t>0.</math>

\overline{g(x)}\,dx</math> following [[inner product space|inner product]] notation. [[integration by parts|Integrating by parts]] and using periodicity grants

:<math>\langle \partial_{t} u , v \rangle = \left\langle \frac{1}{2} u^2 - \rho \partial_{x} u , \partial_x v\right\rangle+\left\langle f, v \right\rangle \quad \forall v\in \mathcal{V}, \forall t>0.</math>

To apply the Fourier-Fourier–[[Galerkin method]], choose both

:<math>\mathcal{U}^N := \leftbiggl\{ u : u(x,t)=\sum_{k=-N/2}^{N/2-1} \hat{u}_{k}(t) e^{i k x}\rightbiggr\}</math>

and

:<math>\mathcal{V}^N :=\operatorname{span}\left\{ e^{i k x} : k\in -\tfrac12 N/2,\dots,N/2\tfrac12N - 1\right\}</math>

where <math>\hat{u}_k(t):=\frac{1}{2\pi}\langle u(x,t), e^{i k x} \rangle</math>. This reduces the problem to finding <math>u\in\mathcal{U}^N</math> such that

:<math>\langle \partial_{t} u , e^{i k x} \rangle = \left\langle \frac{1}{2}tfrac12 u^2 - \rho \partial_{x} u , \partial_x e^{i k x} \right\rangle + \left\langle f, e^{i k x} \right\rangle \quad \forall k\in \left\{ -N/2\tfrac12N,\dots,N/2\tfrac12N-1 \right\}, \forall t>0.</math>

Using the [[orthogonality]] relation <math>\langle e^{i l x}, e^{i k x} \rangle = 2 \pi \delta_{lk}</math> where <math>\delta_{lk}</math> is the [[Kronecker delta]], we simplify the above three terms for each <math>k</math> to see

:<math>

\begin{align}

\left\langle \partial_{t} u , e^{i k x}\right\rangle &= \leftbiggl\langle \partial_{t} \sum_{l} \hat{u}_{l} e^{i l x} , e^{i k x} \rightbiggr\rangle = \leftbiggl\langle \sum_{l} \partial_{t} \hat{u}_{l} e^{i l x} , e^{i k x} \rightbiggr\rangle = 2 \pi \partial_t \hat{u}_k,

\\

\left\langle f , e^{i k x} \right\rangle &= \leftbiggl\langle \sum_{l} \hat{f}_{l} e^{i l x} , e^{i k x}\rightbiggr\rangle= 2 \pi \hat{f}_k, \text{ and}

\\

\left\langle

\frac{1}{2}tfrac12 u^2 - \rho \partial_{x} u

,

\partial_x e^{i k x}

\right\rangle

&=

\leftbiggl\langle

\frac{1}{2}tfrac12

\leftBigl(\sum_{p} \hat{u}_p e^{i p x}\rightBigr)

\leftBigl(\sum_{q} \hat{u}_q e^{i q x}\rightBigr)

- \rho \partial_x \sum_{l} \hat{u}_l e^{i l x}

,

\partial_x e^{i k x}

\rightbiggr\rangle

\\

&=

\leftbiggl\langle

\frac{1}{2}tfrac12

\sum_{p} \sum_{q} \hat{u}_p \hat{u}_q e^{i \left(p+q\right) x}

,

i k e^{i k x}

\rightbiggr\rangle

-

\leftbiggl\langle

\rho i \sum_{l} l \hat{u}_l e^{i l x}

,

i k e^{i k x}

\rightbiggr\rangle

\\

&=

-\frac{tfrac12 i k}{2}

\leftbiggl\langle

\sum_{p} \sum_{q} \hat{u}_p \hat{u}_q e^{i \left(p+q\right) x}

,

e^{i k x}

\rightbiggr\rangle

- \rho k

\leftbiggl\langle

\sum_{l} l \hat{u}_l e^{i l x}

,

e^{i k x}

\rightbiggr\rangle

\\

&=

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- 2\pi\rho{}k^2\hat{u}_k

+ 2 \pi \hat{f}_k

\quad k\in\left\{ -N/2\tfrac12N,\dots,N/2\tfrac12N-1 \right\}, \forall t>0.

</math>

Dividing through by <math>2\pi</math>, we finally arrive at

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- \rho{}k^2\hat{u}_k

+ \hat{f}_k

\quad k\in\left\{ -N/2\tfrac12N,\dots,N/2\tfrac12N-1 \right\}, \forall t>0.

</math>

With Fourier transformed initial conditions <math>\hat{u}_{k}(0)</math> and forcing <math>\hat{f}_{k}(t)</math>, this coupled system of ordinary differential equations may be integrated in time (using, e.g., a [[Runge Kutta]] technique) to find a solution. The nonlinear term is a [[convolution]], and there are several transform-based techniques for evaluating it efficiently. See the references by Boyd and Canuto et al. for more details.

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* [http://www-personal.umich.edu/~jpboyd/BOOK_Spectral2000.html Chebyshev and Fourier Spectral Methods] by John P. Boyd.

* Canuto C., [[M. Yousuff Hussaini|Hussaini M. Y.]], Quarteroni A., and Zang T.A. (2006) ''Spectral Methods. Fundamentals in Single Domains.'' Springer-Verlag, Berlin Heidelberg

* Javier de Frutos, Julia Novo (2000): [httphttps://epubs.siam.org/sam-bindoi/dbq/article10.1137/35198S0036142999351984 A Spectral Element Method for the Navier–Stokes Equations with Improved Accuracy]

* [http://cdm.unimo.it/home/matematica/funaro.daniele/bube.htm Polynomial Approximation of Differential Equations], by Daniele Funaro, Lecture Notes in Physics, Volume 8, Springer-Verlag, Heidelberg 1992

* D. Gottlieb and S. Orzag (1977) "Numerical Analysis of Spectral Methods : Theory and Applications", SIAM, Philadelphia, PA

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* Jie Shen, Tao Tang and Li-Lian Wang (2011) "Spectral Methods: Algorithms, Analysis and Applications" (Springer Series in Computational Mathematics, V. 41, Springer), {{ISBN|354071040X}}

* Lloyd N. Trefethen (2000) ''Spectral Methods in MATLAB.'' SIAM, Philadelphia, PA

* Muradova A. D. (2008) "The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions", Advances in Computational Mathematics, 29, pp. 179–206, https://doi.org/10.1007/s10444-007-9050-7.

* Muradova A. D. (2015) "A time spectral method for solving the nonlinear dynamic equations of a rectangular elastic plate", Journal of Engineering Mathematics, 92, pp. 83–101, https://doi.org/10.1007/s10665-014-9752-z.

{{Numerical PDE}}