Spectral Methods Chaiwoot Boonyasiriwat April 10, 2019 Weighted - - PowerPoint PPT Presentation

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

• rthogonal 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

• r 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

• n 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(9x) 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

• nto 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

• f 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

• f 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