BAREMOS PARA LA EVALUACIÓN DE INVESTIGACIÓN:

Thus far, we have focused on the use of the FFT as the primary tool for spectral methods and their implementation. However, many problems do not in fact have periodic boundary conditions and the accuracy and speed of the FFT is rendered useless. The Chebychev polynomials are a set of mathematical functions which still allow for the construction of a spectral method which is both fast and accurate. The underlying concept is that Chebychev polynomials can be related to sines and cosines, and therefore they can be connected to the FFT routine.

Before constructing the details of the Chebychev method, we begin by con- sidering three methods for handling non-periodic boundary conditions.

Method 1: periodic extension with FFTs

Since the FFT only handles periodic boundary conditions, we can periodically extend a general function f (x) in order to make the function itself now peri- odic. The FFT routine can now be used. However, the periodic extension will in general generate discontinuities in the periodically extended function. The discontinuities give rise to Gibb’s phenomena: strong oscillations and errors are accumulated at the jump locations. This will greatly effect the accuracy of the scheme. So although spectral accuracy and speed is retained away from discon- tinuities, the errors and the jumps will begin to propagate out to the rest of the computational domain.

To see this phenomena, Fig. 29a considers a function which is periodically extended as shown in Fig. 29b. The FFT approximation to this function is shown in Fig. 29c where the Gibb’s oscillations are clearly seen at the jump lo- cations. The oscillations clearly impact the usefulness of this periodic extension technique.

yy y n = 16 Fourier modes. Note the Gibb’s oscillations.

Method 2: polynomial approximation with equi-spaced points

In moving away from an FFT basis expansion method, we can consider the most straight forward method available: polynomial approximation. Thus we simply discretize the given function f (x) with N + 1 equally spaced points and fit an N ^th degree polynomial through it. This amounts to letting

f (x) ≈ a0 + a1 x + a2 x² + a3x³ + · · · + anx^N (4.5.1) where the coefficients an are determined by an (N + 1) × (N + 1) system of equa- tions. This method easily satisfies the prescribed non-periodic boundary condi- tions. Further, differentiation of such an approximation is trivial. In this case however, Runge phenomena (polynomial oscillations) generally occurs. This is because a polynomial of degree n generally has N − 1 combined maximums and minimums.

The Runge phenomena can easily be illustrated with the simple example function f (x) = (1 + 16x²)⁻¹ . Figure 30(a)-(b) illustrates the large oscillations which develop near the boundaries due to Runge phenomena for N = 12 and N = 24. As with the Gibb’s oscillations generated by FFTs, the Runge oscil-

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

x x

yy

1.2 1 0.8 0.6 0.4 0.2 0

−0.2

1.2 1 0.8 0.6 0.4 0.2 0

−0.2

(a)

(c)

(b)

(d)

Figure 30: The function y(x) = (1 + 16x²)⁻¹ for x ∈ [−1, 1] (dotted line) with the bold points indicating the grid points for equi-spaced spaced points (a) n = 12 and (b) n = 24 and Chebychev clustered points (c) n = 12 and (d) n = 24. The solid lines are the polynomial approximations generated using the equi-spaced points (a)-(b) and clustered points (c)-(d) respectively.

lations render a simple polynomial approximation based upon equally spaced points useless.

Method 3: polynomial approximation with clustered points

There exists a modification to the straight forward polynomial approximation given above. Specifically, this modification constructs a polynomial approxima- tion on a clustered grid as opposed to the equal spacing which led to Runge phenomena. This polynomial approximation involves a clustered grid which is transformed to fit onto the unit circle, i.e. the Chebychev points. Thus we have the transformation

xn = cos(nπ/N ) (4.5.2)

2N ² +1

The clustered grid approximation results in a polynomial approximation shown in Fig. 30(c)-(d). There are reasons for this great improvement using a clustered grid [10]. However, we will not discuss them since they are beyond the scope of this course. This clustered grid suggests an accurate and easy way to represent a function which does not have periodic boundaries.

Clustered Points and Chebychev Differentiation

The clustered grid given in method 3 above is on the Chebychev points. The resulting algorithm for constructing the polynomial approximation and differ- entiating it is as follows.

1. Let p be a unique polynomial of degree ≤ N with p(xn ) = Vn, 0 ≤ n ≤ N where V (x) is the function we are approximating.

2. Differentiate by setting wn = p^t(xn ).

The second step in the process of calculating the derivative is essentially the matrix multiplication

w = D_N v (4.5.3)

where D_N represents the action of differentiation. By using interpolation of the Lagrange form [5], the matrix elements of p(x) can be constructed along with

Calculating the individual matrix elements results in the matrix D_N



top row and higher derivatives, simply raise the matrix to the appropriate power:

• D² the boundary conditions. Thus the general differentiation given by (4.5.3) must be modified to include the given boundary conditions. Consider, for example, the simple boundary conditions

v(−1) = v(1) = 0 . (4.5.6)

The given differentiation matrix is then written as

 w0  

Note that the new matrix created is the old D_N matrix with the top and bottom rows and the first and last columns removed.

j ) Connecting to the FFT

We have already discussed the connection of the Chebychev polynomial with the FFT algorithm. Thus we can connect the differentiation matrix with the FFT routine. After transforming via (4.5.2), then for real data the discrete Fourier transform can be used. For complex data, the regular FFT is used. Note that for the Chebychev polynomials

∂Tn(±1)

= 0 (4.5.9)

∂x

so that no-flux boundaries are already satisfied. To impose pinned boundary conditions v(±1) = 0, then the differentiation matrix must be imposed as shown above.