(F.26)

= cos(mxj ) cos(mx)

N pm

m=0

where the 1/pm term means that the m = 0 and m = N terms in the sum should be divided

by (1/2).

Quadrature Rule:

If f (x) is a periodic but symmetric function with a period of 2π, then

N

π

f (x) dx ≈ (F.27)

wj f (xj )

0 j=1

where the quadrature weights are given by

π

wj ≡ [“Rectangle Rule”] (F.28)

N

independent of j. Although the “rectangle rule” is rather crude [O(1/N 2 )] accuracy for

integrating non-periodic integrals, it is has an error that decreases exponentially fast with N

when applied to an integrand which is (i) periodic and (ii) symmetric about the origin.

APPENDIX F. CARDINAL FUNCTIONS

568

F.6 Sine Cardinal Functions on the Interior (Rectangle Rule)

Grid

Previously unpublished.

Grid points:

(2i ’ 1)π

[N degrees of freedom] (F.29)

xi =

2N

Cj (x) ≡ sin(N xj ) cos(N x) sin(x)/(N [cos(x) ’ cos(xj )])

sin

(F.30a)

N

2 1

(F.30b)

= sin(mxj ) sin(mx)

N cm

m=1

where the 1/cm term means that the m = N term in the sum should be divided by (1/2).

Quadrature Rule:

If f (x) is a periodic but antisymmetric function with a period of 2π, then

N

π

f (x) dx ≈ (F.31)

wj f (xj )

0 j=1

where the quadrature weights are given by

N ’1

2 Nπ 4 1 mπ

wj ≡ 2 sin(N xj ) sin2 sin(mxj ) sin2 (F.32)

+

N 2 N m 2

m=1

F.7. SINC(X): WHITTAKER CARDINAL FUNCTION 569

F.7 Sinc(x): Whittaker cardinal function

These are employed to represent functions on the interval x ∈ [’∞, ∞]. It is necessary to

simultaneously (i) decrease the spacing h between neighboring interpolation points and (ii)

increase the number of sinc functions retained in the expansion to obtain higher accuracy

for a given f (x). The function being expanded must decay as |x| ’ ∞. The formulas are

taken from Stenger(1981).

Grid points:

i = 0, ±1, ±2, . . . (F.33)

xi = h i

Cardinal Functions:

x ’ jh

Cj (x) ≡ sinc (F.34)

h

where

sin(πx)

sinc(x) ≡ (F.35)

πx

1st Derivative:

0 n=0

sinc (F.36)

h δ1 =

(’1)n+1 /n n=0

i,i+n

2d Derivative:

’π 2 /3 n=0

δ sinc

2

(F.37)

h =

2

’2(’1)n /n2 n=0

i,i+n

0 n=0

sinc

h 3 δ3 (F.38)

=

(’1)n+1 6/n3 ’ π 2 /n n=0

i,i+n

π 4 /5 n=0

δ sinc

4

(F.39)

h =

4

4(’1)n [π 2 /n2 ’ 6/n4 ] n=0

i,i+n

0 n=0

sinc

h 5 δ5 (F.40)

=

(’1)n+1 [π 4 /n ’ 20π 2 /n3 + 120/n5 ] n=0

i,i+n

’π 2 /7 n=0

δ sinc

6

(F.41)

h =

6

’6(’1)n [π 4 /n2 ’ 20π 2 /n4 + 120/n6 ] n=0

i,i+n

Fourier representation:

π

1

sinc(x) ≡ (F.42)

exp(±ikx) dk

2π ’π

APPENDIX F. CARDINAL FUNCTIONS

570

F.8 Chebyshev Polynomials: Extrema & Endpoints (“Gauss-

Lobatto”) Grid

Taken from Gottlieb, Hussaini, and Orszag(1984).

Grid points:

πi

(F.43)

xi = cos i = 0, . . . , N

N

Cardinal Funcs:

(1 ’ x2 ) dTN (x)

Cj (x) ≡ (’1)j+1 (F.44a)

2 (x ’ x )

cj N dx

j

N

2 1

(F.44b)

= Tm (xj ) Tm (x)

N pj pm

m=0

where pj = 2 if j = 0 or N and pj = 1 if j = 1, . . . , N ’ 1; cj = 1 if |j| < N ; c±N = 2.

1st Derivative:

±

(1 + 2N 2 )/6

i=j=0

’(1 + 2N 2 )/6 i=j=N

dCj

δ Cheb ≡ (F.45)

=

1

’xj /[2(1 ’ x2 )]

dx i = j; 0 < j < N

ij

x=xi

j

(’1)i+j pi /[pj (xi ’ xj )] i=j

where p0 = pN = 2, pj = 1 otherwise.

Higher Derivative:

k

Cheb = δ Cheb (F.46)

δk 1

where the exponent k denotes the usual matrix multiplication of k copies of the ¬rst deriva-

tive matrix.

Note: The ¬rst derivative matrix is not antisymmetric (with respect to the interchange of i

and j) and the second derivative matrix is not symmetric. This makes it dif¬cult to prove

stability theorems for time-integration methods (Gottlieb et al., 1984).

F.9. CHEBYSHEV POLYNOMIALS: INTERIOR OR “ROOTS” GRID 571

F.9 Chebyshev Polynomials: Interior or “Roots” Grid

Grid Points:

2i ’ 1

(F.47)

xi = cos π i = 1, . . . , N

2N

Cardinal Functions:

TN (x)

Cj (x) ≡ [x ’ xj ] (F.48)

TN (xj )

cos(N t) sin(tj )

≡ (F.49)

N sin(N tj )[cos(t) ’ cos(tj )]

where the TN is the x-derivative of TN (x) and where in the trigonometric form, t = arccos(x)

and tj = arccos(xj ) where the branch is chosen so that 0 ¤ t ¤ π.

1st Derivative:

±

0.5 xj /(1 ’ x2 )

i=j

j

dCj

≡ (δ1 )ij = (1 ’ x2 )/(1 ’ x2 ) (F.50)

j i

dx (’1)i+j

x=xi i=j

xi ’ xj

Second Derivative:

±

x2 N2 ’ 1

j

’

i=j

(1 ’ x2 )2 3(1 ’ x2 )

d2 Cj

≡ (δ2 )ij = j j

(F.51)

dx2 xi 2

x=xi