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 (δ1 )ij i=j
1 ’ x2 xi ’ xj
i
APPENDIX F. CARDINAL FUNCTIONS
572

F.10 Legendre Polynomials: Extrema & Endpoints Grid (“Gauss-
Lobatto”)
These data are from Gottlieb, Hussaini, & Orszag(1984).

Grid points:

x0 = ’1 the (N -1) roots of dPN /dx (F.52)
& xN = 1 &

Cardinal Functions:
’(1 ’ x2 ) dPN (x)
Cj (x) ≡ (F.53)
N (N + 1)PN (xj )(x ’ xj ) dx

Quadrature weights:

2
wj ≡ (F.54)
2
N (N + 1) {PN (xj )}

First Derivative Matrix:
±
(1/4)N (N + 1) i=j=0




 ’(1/4)N (N + 1) i=j=N
Leg
(F.55)
δ1 =

 0 i=j & 0<j<N

ij


PN (xi )/[PN (xj )(xi ’ xj )] i=j




Four-Point Interpolation


3 5 dP3 3 15
P3 = ’ x + x3 , = ’ + x2 (F.56)
2 2 dx 2 2




Table F.1: Grid points and weights for 4-point Legendre-Lobatto

xj wj
±1 - 1/6 0.166666666666667
±0.447213595499958 ± 1/5 5/6 0.833333333333333
F.10. LEGENDRE POLYNOMIALS: GAUSS-LOBATTO GRID 573

Five-Point Interpolation
3 15 35 dP4 15 35
P4 = ’ x2 + x4 , = ’ x + x3 (F.57)
8 4 8 dx 2 2

Table F.2: Grid points and weights for 5-point Legendre-Lobatto

xj wj
±1 - 1/10 0.1
±0.654653670707977 ± 3/7 49/90 0.544444444444444
-
0 32/45 0.711111111111111



Six-Point Interpolation
15 35 63 dP5 15 105 2 315 4
x ’ x3 + x5 ; ’ (F.58)
P5 = = x+ x
8 4 8 dx 8 4 8

Table F.3: Grid points and weights for 6-point Legendre-Lobatto
√ √
Notes: Variants for weights: w(±0.765) = 9/(20 + 10/ 7) and w(±0.285) = 9/(20 ’ 10/ 7).

xj wj
±1 - 1/15 0.066666666666667

±0.765055323929464 ± 1
(7 + 2 7)/21 0.378474956297847
15{P5 (xj )}
2

±0.285231516480645 ± (7 ’ 2 7)/21 1
0.554858377035486
15{P5 (xj )}
2




Seven-Point Interpolation
5 105 2 315 4 231 6 dP6 105 315 3 693 5
P6 = ’ + x’ x’ (F.59)
x+ x, = x+ x
16 16 16 16 dx 8 4 8

Table F.4: Grid points and weights for 7-point Legendre-Lobatto
√ 2
Notes: Variants for weights: w(±0.8302) = 43923/(175 3 + 7 15 ) and
√ 2
w(±0.468) = 43923/(175 ’3 + 7 15 ).

xj wj
±1 - 1/21 0.047619047619047

±0.830223389627857 ± 1
(15 + 2 15)/33 0.276826047361566
21{P6 (xj )}
2

±0.468848793470714 ± (15 ’ 2 15)/33 1
0.431745381209862
21{P6 (xj )}
2
256
-
0 0.487619047619048
525
APPENDIX F. CARDINAL FUNCTIONS
574

Eight-Point Interpolation
35 315 3 693 5 429 7
=’ x’ (F.60)
P7 x+ x+ x
16 16 16 16
dP7 35 945 2 3465 4 3003 6
=’ + x’ (F.61)
x+ x
dx 16 16 16 16


Table F.5: Grid points and weights for 8-point Legendre-Lobatto
√ √
Notes: r = 320 55/265837 and φ = arccos( 55/30) . xj in the expression for the weights denotes the corre-
sponding grid point.

xj wj
±1 - 1/28 0.035714285714285
φ
±0.871740148509606 ± 5 1
2r1/3 cos + 0.210704227143506
28{P7 (xj )}
2
3 13

φ
±0.591700181433142 ± 5 4π 1
2r1/3 cos + + 0.341122692483504
28{P7 (xj )}
2
3 13 3

φ
±0.209299217902479 ± 5 2π 1
2r1/3 cos + + 0.412458794658704
28{P7 (xj )}
2
3 13 3




Nine-Point Interpolation
35 315 2 3465 4 3003 6 6435 8
’ x’ (F.62)
P8 = x+ x+ x
128 32 64 32 128
dP8 315 3465 3 9009 5 6435 7
=’ x’ (F.63)
x+ x+ x
dx 16 16 16 16


Table F.6: Grid points and weights for 9-point Legendre-Lobatto
√ √
Notes: r = 448 91/570375 and φ = arccos( 91/154) . xj in the expression for the weights denotes the corre-
sponding grid point.

xj wj
±1 - 1/36 0.027777777777777
φ
±0.899757995411460 ± 7 1
2r1/3 cos + 0.165495361560805
36{P8 (xj )}
2
3 15

φ
±0.677186279510737 ± 7 4π 1
2r1/3 cos + + 0.274538712500161
36{P8 (xj )}
2
3 15 3

φ
±0.363117463826178 ± 7 2π 1
2r1/3 cos + + 0.346428510973046
36{P8 (xj )}
2
3 15 3
4096
-
0 0.371519274376417
11025



Canuto, Hussaini, Quarteroni and Zang (1988), Appendix C, give a short FORTRAN
program to compute the Legendre-Lobatto points for general degree.
Appendix G

Transformation of Derivative
Boundary Conditions

“Pseudospectral algorithms are simply N-th order ¬nite differences in disguise.”
” J. P. Boyd


One modest complication in applying boundary conditions imposed on the derivatives
of the solution is that the mapping functions which relate the derivatives in the original co-
ordinate y to those of the new coordinate x are usually singular at the endpoints. However, it
is straightforward to obtain the correct conditions via l™Hopital™s Rule, which is equivalent
to making power series expansions about the endpoint and then taking x ’ endpoint.
To illustrate, consider mapping that converts Chebyshev polynomials in y into cosines
in x.
y ∈ [’1, 1] x ∈ [0, π] (G.1)
y = cos(x) &
The method, however, can be applied to any of the mappings discussed above.
Suppose that the boundary conditions are
(G.2)
uy (1) = ± & uyy (1) = β
Now from Table E.2,
ux
uy = ’ (G.3)
sin(x)
The complication is that as y ’ 1, x ’ 0, and the denominator of (G.3) is 0.
However, u(y) is mapped into u(cos[x]) which is always symmetric about both x = 0
and x = π. This in turn implies that all the odd derivatives of u(y[x]) with respect to x must
vanish at both endpoints. Thus, the R. H. S. of (G.3) is ¬nite because both its numerator and
denominator vanish. To calculate the limit at the endpoint, write the Taylor expansions
ux (x) ≈ ux (0) + x uxx (0) + O(x2 ) |x| (G.4)
1
≈ x uxx (0) + O(x3 ) (G.5)
since all the odd derivatives vanish. Since
x3
sin(x) ≈ x ’ + O(x5 ) (G.6)
6

575
576 APPENDIX G. TRANSFORMATION OF DERIVATIVE BOUNDARY CONDITIONS

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