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Figure 26: Graphical description of the Fast Fourier Transform process which systematically continues to factor the problem in two. This process allows the FFT routine to drop to O(N log N ) operations.

There is no reason to stop the splitting process at this point. In fact, provided we choose the size of our domain and matrix FN so that N is a power of two, then we can continue to split the system until we have a simple algebraic, i.e.

a 1 × 1 system, solution to perform. The process is illustrated graphically in Fig. 26 where the switching and factorization are illustrated. Once the final level is reached, the algebraic expression is solved and the process is reversed.

This factorization process renders the FFT scheme O(N log N ).

4.2 Chebychev Polynomials and Transform

The Fast Fourier Transform is only one of many possible expansion bases, i.e.

there is nothing special about expanding in cosines and sines. Of course, the FFT expansion does have the unusual property of factorization which drops it

− −

to an O(N log N ) scheme. Regardless, there are a myriad of other expansion bases which can be considered. The primary motivation for considering other expansions is based upon the specifics of the given governing equations and its physical boundaries and constraints. Special functions are often prime candi- dates for use as expansion bases. The following are some important examples

• Bessel functions: radial, 2D problems

• Legendre polynomials: 3D Laplaces equation

• Hermite-Gauss polynomials: Schr¨odinger with harmonic potential

• Spherical harmonics: radial, 3D problems

• Chebychev polynomials: bounded 1D domains

The two key properties required to use any expansion basis successfully are its orthogonality properties and the calculation of the norm. Regardless, all the above expansions appear to require O(N ² ) calculations.

Chebychev Polynomials

Although not often encountered in mathematics courses, the Chebychev poly- nomial is an important special function from a computational viewpoint. The reason for its prominence in the computational setting will become apparent momentarily. For the present, we note that the Chebychev polynomials are solutions to the differential equation

j \

1 x² d dx

j

1 x² dT_n

dx + n²T_n = 0 x ∈ [−1, 1] . (4.2.1) This is a self-adjoint Sturm-Lioville problem. Thus the following properties are known

1. eigenvalues are real: λn = n² 2. eigenfunctions are real: T_n(x) 3. eigenfunctions are orthogonal:

1

(1 − x² )^−1/2Tn(x)Tm (x)dx =

−1

π

2 cnδnm (4.2.2) where c₀ = 2, c_n = 1(n > 0) and δnm is the Delta function

4. eigenfunctions form a complete basis

T T 1.0

T

0

T₁

0.5

2

0.0

T₃ T₄

−0.5

−1.0

−1 −0.5 0 0.5 1

x

Figure 27: The first five Chebychev polynomials over the the interval of defini- tion x ∈ [−1, 1].

Each Chebychev polynomial (of degree n) is defined by

Tn(cos θ) = cos nθ . (4.2.3)

Thus we find

T₀(x) = 1 (4.2.4a) T1(x) = x (4.2.4b) T₂(x) = 2x² − 1 (4.2.4c) T₃(x) = 4x³ − 3x (4.2.4d) T₄(x) = 8x⁴ − 8x² + 1 . (4.2.4e) The behavior of the first five Chebychev polynomials is illustrated in Fig. 27.

It is appropriate to ask why the Chebychev polynomials, of all the special functions listed, are of such computational interest. Especially given that the equation which the T_n(x) satisfy, and their functional form shown in Fig. 27, ap- pear to be no better than Bessel, Hermite-Gauss, or any other special function.

n

k=0

− ·

The distinction with the Chebychev polynomials is that you can transform them so that use can be made of the O(N log N ) discrete cosine transform. This effec- tively yields the Chebychev expansion scheme an O(N log N ) transformation.

Specifically, we transform from the interval x ∈ [−1, 1] by letting

x = cos θ θ ∈ [0, π] . (4.2.5) Thus when considering a function f (x), we have f (cos θ) = g(θ). Under differ- entiation we find

dg = f ^t sin θ (4.2.6)

dθ

Thus dg/dθ = 0 at θ = 0, π, i.e. no-flux boundary conditions are satisfied. This allows us to use the dct (discrete cosine transform) to solve a given problem in the new transformed variables.

The Chebychev expansion is thus given by

∞

f (x) = )

ak Tk (x) (4.2.7)

k=0

where the coefficients a_k are determined from orthogonality and inner products to be

ak =

1 1

√ f (x)T_k (x)dx . (4.2.8)

−1 1 − x²

It is these coefficients which are calculated in O(N log N ) time. Some of the properties of the Chebychev polynomials are as follows:

• Tn+1 = 2xT_n(x) − Tn−1(x)

• |Tn(x)| ≤ 1, |T ^t (x)| ≤ n²

• Tn(±1) = (±1)ⁿ

• d^p/dx^p (T_n(±1)) = (±1)^n+p Ti_p−1

(n² − k²)/(2k + 1)

• if n is even (odd), Tn(x) is even (odd)

There are a couple of critical practical issues which must be considered when using the Chebychev scheme. Specifically the grid generation and spatial reso- lution are a little more difficult to handle. In using the discrete cosine transform on the variable θ ∈ [0, π], we recall that our original variable is actually x = cos θ where x ∈ [−1, 1]. Thus the discretization of the θ variable leads to

(2m − 1)π \ xm = cos

2n m = 1, 2, ..., n . (4.2.9) Thus although the grid points are uniformly spaced in θ, the grid points are clustered in the original x variable. Specifically, there is a clustering of grid

n=0

p=n+1(p+nodd)

• Lf = f (x) : cnbn = 1

x_m=cos[(2m−1)π/2n] m=1,2,...,n 0

−1 −1 −0.5 0 0.5 1

x

Figure 28: Clustered grid generation for n = 30 points using the Chebychev polynomials. Note that although the points are uniformaly spaced in θ, they are clustered due to the fact that x_m = cos[(2m − 1)π/2n] where m = 1, 2, ..., n.

points at the boundaries. The Chebychev scheme then automatically has higher resolution at the boundaries of the computational domain. The clustering of the grid points at the boundary is illustrated in Fig. 28. So as the resolution is increased, it is important to be aware that the resolution increase is not uniform across the computational domain.

Solving Differential Equations

As with any other solution method, a solution scheme must have an efficient way of relating derivatives to the function itself. For the FFT method, there was a very convenient relationship between the transform of a function and the trans- form of its derivatives. Although not as transparent as the the FFT method, we can also relate the Chebychev transform derivatives to the Chebychev transform itself.

Defining L to be a linear operator so that

then with f (x) = L_∞

∞

Lf (x) = )

b_nT_n(x) , (4.2.10)

n=0

anTn(x) we find

• Lf = f ^t(x) : cn bn = 2 L

∞

tt L_∞

p=n+2(p+neven)

pap

• Lf = xf (x) : bn = (cn−1 an−1 + an+1)/2

• Lf = x² f (x) : bn = (cn−2 an−2 + (cn + cn−1)an + an+2)/4

where c₀ = 2, c_n = 0(n < 0), c_n = 1(n > 0), d_n = 1(n ≥ 0), and dn = 0(n < 0).