 Spectral method

Spectral methods are a class of techniques used in applied mathematics and scientific computing 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, substituting this series into the PDE to get a system of ordinary differential equations (ODEs) in the timedependent coefficients of the trigonometric terms in the series (written in complex exponential form), and using a timestepping method to solve those ODEs.
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. 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 fact there are no known threedimensional single domain spectral shock capturing results.^{[1]}
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 zero is sometimes called a spectral element method.
The implementation of the spectral method is normally accomplished either with collocation or a Galerkin or a Tau approach.
Contents
Examples of spectral methods
A concrete, linear example
Here we presume an understanding of basic multivariate calculus and Fourier series. If g(x,y) is a known, complexvalued 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 boundaryconditions, the Poisson equation possesses a solution only if b_{0,0} = 0. Therefore 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 h^{n}, where h = 1 / n and n 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 (*) and the Fourier transform of g.
 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 time O(n log n).
A concrete, nonlinear example
We wish to solve the forced, transient, nonlinear Burgers' equation using a spectral approach.
Given u(x,0) on the periodic domain , find such that
In weak, conservative form this becomes
where following inner product notation. Integrating by parts and using periodicity grants
To apply the Fourier−Galerkin method, choose both
and
where . This reduces the problem to finding such that
Using the orthogonality relation where δ_{lk} is the Kronecker delta, we simplify the above three terms for each k to see
Assemble the three terms for each k to obtain
Dividing through by 2π, we finally arrive at
With Fourier transformed initial conditions and forcing , 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 transformbased techniques for evaluating it efficiently. See the references by Boyd and Canuto et al. for more details.
A relationship with the spectral element method
One can show that if g is infinitely differentiable, then the numerical algorithm using Fast Fourier Transforms will converge faster than any polynomial in the grid size h. That is, for any n>0, there is a such that the error is less than Ch^{n} for all sufficiently small values of h. We say that the spectral method is of order n, for every n>0.
Because a spectral element method is a finite element method of very high order, there is a similarity in the convergence properties. However, whereas the spectral method is based on the eigendecomposition of the particular boundary value problem, the spectral element method does not use that information and works for arbitrary elliptic boundary value problems.
See also
 Discrete element method
 Gaussian grid
 Pseudospectral method
 Spectral element method
 Galerkin method
 Collocation method
Finite difference methods Heat Equation and related: FTCS scheme · Crank–Nicolson method Hyperbolic: Lax–Friedrichs method · Lax–Wendroff method · MacCormack method · Upwind scheme · Other: Alternating direction implicit method · Finitedifference timedomain methodFinite volume methods Finite element methods Other methods Spectral method · Pseudospectral method · Method of lines · Multigrid methods · Collocation method · Level set method · Boundary element method · Immersed boundary method · Analytic element method · Particleincell · Isogeometric analysisDomain decomposition methods Schur complement method · Fictitious domain method · Schwarz alternating method · Additive Schwarz method · Abstract additive Schwarz method · Neumann–Dirichlet method · Neumann–Neumann methods · Poincaré–Steklov operator · Balancing domain decomposition · BDDC · FETI · FETIDPReferences
 ^ pp 235, Spectral Methods: evolution to complex geometries and applications to fluid dynamics, By Canuto, Hussaini, Quarteroni and Zang, Springer, 2007. .
 Bengt Fornberg (1996) A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, UK
 Chebyshev and Fourier Spectral Methods by John P. Boyd.
 Canuto C., Hussaini M. Y., Quarteroni A., and Zang T.A. (2006) Spectral Methods. Fundamentals in Single Domains. SpringerVerlag, Berlin Heidelberg
 Javier de Frutos, Julia Novo: A Spectral Element Method for the NavierStokes Equations with Improved Accuracy
 Polynomial Approximation of Differential Equations, by Daniele Funaro, Lecture Notes in Physics, Volume 8, SpringerVerlag, Heidelberg 1992
 D. Gottlieb and S. Orzag (1977) "Numerical Analysis of Spectral Methods : Theory and Applications", SIAM, Philadelphia, PA
 J. Hesthaven, S. Gottlieb and D. Gottlieb (2007) "Spectral methods for timedependent problems", Cambridge UP, Cambridge, UK
 Steven A. Orszag (1969) Numerical Methods for the Simulation of Turbulence, Phys. Fluids Supp. II, 12, 250257
 Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 20.7. Spectral Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 9780521880688. http://apps.nrbook.com/empanel/index.html#pg=1083.
 Lloyd N. Trefethen (2000) Spectral Methods in MATLAB. SIAM, Philadelphia, PA
Categories: Numerical analysis
 Numerical differential equations
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