Spectral Methods · 2020. 11. 12. · •Legendre spectral method: •Laguerre spectral method: ... The truncated Fourier series can be expressed in the convolution form as ... Spectral

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.


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.


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.


Sinc Interpolation

Sinc interpolation is accurate only for smooth function.

The Gibbs phenomenon can be observed.


Sinc Interpolation

Given a periodic grid such that

For simplicity, let N is even. So the grid spacing is

Periodic Grids


Fourier analysis:

Fourier synthesis:

Discrete Fourier Transforms

In this case, and we obtain the interpolant

Impulse Response


Differentiating the interpolant

yields the differentiation matrix



Spectral differentiation of rough and smooth functions




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


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?


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


Program 20

Linear Wave Propagation


Program 27: Solitary waves from KdV equation

Nonlinear Wave Propagation


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, L. N., 2000, Spectral Methods in MATLAB,

SIAM.

Reference

Spectral Methods · 2020. 11. 12. · •Legendre spectral method: •Laguerre spectral method: ... The truncated Fourier series can be expressed in the convolution form as ... Spectral

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