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Chaiwoot Boonyasiriwat
April 10, 2019
Spectral Methods
▪ Consider the problem
where L and is a spatial derivative operator.
▪ Approximate the solution by a finite sum
▪ Substitute the approximate solution in to the differential
equation yields the residual
▪ The weighted residual method forces the residual to be
orthogonal to the test functions k
Weighted Residual Methods
Shen et al. (2011, p.1-2)
▪ Spectral methods use globally smooth function (such as
trigonometric functions or orthogonal polynomials) as
the test functions while finite element methods use local
functions.
▪ Examples of spectral methods
• Fourier spectral method:
• Chebyshev spectral method:
• Legendre spectral method:
• Laguerre spectral method:
• Hermite spectral method:
where the polynomials are of degree k.
Spectral Methods
Shen et al. (2011, p.3)
▪ “The choice of test function distinguishes the following
formulations.”
• Bubnov-Galerkin: test functions are the same as the
basis functions
• Petrov-Galerkin: test functions are different from the
basis functions. The tau method is in this class.
• Collocation: test functions are the Lagrange basis
polynomial such that where xj are
collocation points.
Spectral Methods
Shen et al. (2011, p.3)
▪ Consider the problem
▪ Let xj, j = 0, 1, …, N be the collocation points.
▪ The spectral collocation method forces the residual to
vanish at the collocation points
▪ The spectral collocation method usually approximates
the solution as
where Lk are the Lagrange basis polynomials or nodal
basis functions with
Spectral Collocation Methods
Shen et al. (2011, p.4)
▪ Substituting into yields
▪ Assuming the Dirichlet boundary conditions
▪ We then obtain a linear system of N + 1 algebraic
equations in N + 1 unknowns.
Spectral Collocation Methods
Shen et al. (2011, p.4)
▪ The complex exponential are defined as
where
▪ The set forms a complete orthogonal
system in the complex Hilbert space L2(0,), equipped
with the inner product and norm
▪ The orthogonality of Ek is
Fourier Series
Shen et al. (2011, p.23)
“For any complex-valued function , its
Fourier series is defined by
where the Fourier coefficients are given by
“If u(x) is a real-valued function, its Fourier coefficients
satisfy
Fourier Series
Shen et al. (2011, p.23)
“For any complex-valued function , its
truncated
converges to u in the L2 sense, and there holds the
Parseval’s identity:
The truncated Fourier series can be expressed in the
convolution form as
where Dirichlet kernel is
Truncated Fourier Series
Shen et al. (2011, p.25)
▪ Finite difference (FD) coefficients can be obtained by
differentiating a polynomial interpolant passing through
points in the domain.
▪ When all domain points are used, FDM becomes a
spectral method called spectral collocation method.
▪ Spectral method has an exponential rate of convergence
or spectral convergence rate.
Spectral Method and FDM
▪ Spectral methods and finite element methods (FEM) are
closely related in that the solutions are written as a
linear combination of basis functions
▪ Spectral methods use global functions while FEM uses
local functions.
▪ A main drawback of spectral methods is that it is highly
accurate only when solutions are smooth.
Spectral Method and FEM
▪ Collocation method: solutions satisfy PDEs at a
number of points in the domain called collocation
points. The resulting method is also called
pseudospectral method.
▪ Galerkin method: solution satisfies
given
where is a set of linearly independent basis
functions.
▪ Tau method: similar to Galerkin except basis functions
are orthogonal polynomials.
Types of Spectral Methods
▪ Let p be a single function such that p( xj ) = uj for all j.
▪ Set wj = p'( xj )
▪ We are free to choose p to fit the problem.
▪ For a periodic domain, we use a trigonometric
polynomial on an equispaced grid resulting to the
Fourier spectral method.
▪ For nonperiodic domains, we use algebraic polynomials
on irregular grids such as Chebyshev grid leading to the
Chebyshev spectral method.
Spectral Collocation Methods
Fourier analysis:
The Fourier transform of a function u(x), x , is defined
by
Fourier synthesis:
The function u(x) can be reconstructed by
Fourier Transforms
Fourier analysis:
The semidiscrete Fourier transform of a function u(x),
x , is defined by
Fourier synthesis:
The function u(x) can be reconstructed by
Semidiscrete Fourier Transform
When , two complex exponentials
have the same values as long as
where m is an integer.
Example: sin(x) and sin(9x) on the discrete grid
Aliasing
Trefethen (2000, p. 11)
An interpolant can be obtained by
The Fourier transform is given by
Spectral differentiation can be performed by
differentiating the interpolant p(x) or
Spectral Differentiation
Given the Kronecker delta function
It can be shown that for
and the corresponding interpolant is
which is called the sinc function.
Sinc Interpolation
The band-limited interpolant of is
A discrete function can be written as
“So the band-limited interpolant of u is a linear
combination of translated sinc functions”
Differentiating this interpolant we obtain the
differentiation matrix.
Trefethen (2000, p. 13)
Sinc Interpolation
Sinc interpolation is accurate only for smooth function.
The Gibbs phenomenon can be observed.
Trefethen (2000, p. 14)
Sinc Interpolation
Given a periodic grid such that
For simplicity, let N is even. So the grid spacing is
Periodic Grids
Trefethen (2000, p. 18)
Fourier analysis:
Fourier synthesis:
Discrete Fourier Transforms
In this case, and we obtain the interpolant
Impulse Response
Trefethen (2000, p. 21)
Differentiating the interpolant
yields the differentiation matrix
Trefethen (2000, p. 5)
Spectral Differentiation
Spectral differentiation of rough and smooth functions
Trefethen (2000, p. 22)
Spectral Differentiation
Trefethen (2000, p. 26)
Wave Propagation
Chebyshev Spectral
Method
▪ When the boundary condition is non-periodic, algebraic
polynomial interpolation is used instead of Fourier
polynomials.
▪ Polynomial interpolation
• Given a set of points
• Find an interpolating polynomial of order n, given by
• This leads to a linear system of equations whose
solution is the polynomial coefficients {ai}.
Polynomial Interpolation
▪ When a uniform grid of points is used for higher-order
polynomial interpolation, large vibrations occur near the
boundaries.
▪ This is known as the Runge phenomenon.
Runge Phenomenon
Trefethen (2000, p. 44)
The Runge phenomenon can be avoided by using a
clustered grid, e.g., Chebyshev nodes defined by
Chebyshev Nodes
Trefethen (2000, p. 43-44)
Chebyshev nodes are projections of
equispaced points on a unit circle
onto x axis.
Chebyshev nodes are extreme points of Chebyshev
polynomial.
Chebyshev Nodes
“Given a function f on the interval [-1,1] and points
, there is a unique interpolation polynomial
of degree n with error
where .” So we want to minimize the infinity
norm of a monic polynomial g(x), i.e.
Polynomial Interpolation
http://en.wikipedia.org/wiki/Chebyshev_nodes
Comparing the monic polynomials of uniform and
Chebyshev nodes shows large errors near boundaries
for uniform nodes.
Why Chebyshev Nodes?
Trefethen (2000, p. 47)
Using the Chebyshev grid, we obtain an interpolant p(x)
whose derivatives are the approximation to the derivatives
of a given function u(x).
Chebyshev Spectral Differentiation
Image source: Trefethen (2000, p. 56)
Chebyshev differentiation of
Chebyshev Differentiation Matrix
Trefethen (2000, p. 53)
Program 20
Linear Wave Propagation
Trefethen (2000, p. 84)
Program 27: Solitary waves from KdV equation
Nonlinear Wave Propagation
Trefethen (2000, p. 112)
Radial : Chebyshev
Angular: Fourier
Chebyshev-Fourier Spectral Method
Trefethen (2000, p. 116, 123)
Program 37: Fourier in x, Chebyshev in y
Chebyshev-Fourier Spectral Method
Trefethen (2000, p. 144)
▪ Trefethen, L. N., 2000, Spectral Methods in MATLAB,
SIAM.
Reference