spectral methods · 2020. 11. 12. · •legendre spectral method: •laguerre spectral method: ......
TRANSCRIPT
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Chaiwoot Boonyasiriwat
April 10, 2019
Spectral Methods
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▪ 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)
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▪ 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)
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▪ “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)
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▪ 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)
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▪ 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)
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▪ 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)
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“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)
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“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)
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▪ 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
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▪ 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
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▪ 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
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▪ 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
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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
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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
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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)
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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
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Given the Kronecker delta function
It can be shown that for
and the corresponding interpolant is
which is called the sinc function.
Sinc Interpolation
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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
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Sinc interpolation is accurate only for smooth function.
The Gibbs phenomenon can be observed.
Trefethen (2000, p. 14)
Sinc Interpolation
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Given a periodic grid such that
For simplicity, let N is even. So the grid spacing is
Periodic Grids
Trefethen (2000, p. 18)
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Fourier analysis:
Fourier synthesis:
Discrete Fourier Transforms
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In this case, and we obtain the interpolant
Impulse Response
Trefethen (2000, p. 21)
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Differentiating the interpolant
yields the differentiation matrix
Trefethen (2000, p. 5)
Spectral Differentiation
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Spectral differentiation of rough and smooth functions
Trefethen (2000, p. 22)
Spectral Differentiation
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Trefethen (2000, p. 26)
Wave Propagation
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Chebyshev Spectral
Method
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▪ 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
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▪ 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)
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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.
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Chebyshev nodes are extreme points of Chebyshev
polynomial.
Chebyshev Nodes
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“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
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Comparing the monic polynomials of uniform and
Chebyshev nodes shows large errors near boundaries
for uniform nodes.
Why Chebyshev Nodes?
Trefethen (2000, p. 47)
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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
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Chebyshev Differentiation Matrix
Trefethen (2000, p. 53)
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Program 20
Linear Wave Propagation
Trefethen (2000, p. 84)
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Program 27: Solitary waves from KdV equation
Nonlinear Wave Propagation
Trefethen (2000, p. 112)
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Radial : Chebyshev
Angular: Fourier
Chebyshev-Fourier Spectral Method
Trefethen (2000, p. 116, 123)
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Program 37: Fourier in x, Chebyshev in y
Chebyshev-Fourier Spectral Method
Trefethen (2000, p. 144)
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▪ Trefethen, L. N., 2000, Spectral Methods in MATLAB,
SIAM.
Reference