Page 102 - Basic Structured Grid Generation

Structured grid generation – algebraic methods 91

and the last row becomes
2 2
(t n−1 − t ) (t n−1 + t n )(2t n−1 + t n )
n
0 – – – 0 . (4.60)
t n−1 t n−1

Method 4 Here we prescribe the gradients y , y of the interpolating curve at the

n
0
end-points. If these gradients are known, we obtain the best cubic spline fit of all
these methods. However, we commonly have only estimates of their values at our
disposal.
Using eqn (4.51), we obtain
1 y 1 − y 0 1 1 1 y 1 − y 0

y = φ (x 0 ) =− y t 1 + − t 1 (y − y ) =− y t 1 − y t 1 + ,
0 1 0 1 0 0 1
2 t 1 6 3 6 t 1
and hence
1 y 1 − y 0

y t 1 =− y t 1 − 3y + 3 . (4.61)
0 1 0
2 t 1
Substituting from (4.61) into the first equation (i = 1) of eqn (4.52) gives

3 y 2 − y 1 y 1 − y 0

y t 1 + 2t 2 + y t 2 = 6 − 9 + 3y (4.62)
1 2 0
2 t 2 t 1
Similarly we can show that the last equation (i = n − 1) of eqn (4.52) becomes
3 y n − y n−1 y n−1 − y n−2

y n−2 n−2 + y 2t n−1 + t n = 9 − 6 − 3y . (4.63)
t
n
n−1
2 t n t n−1
The other equations of (4.52) are unchanged, and hence we obtain the matrix
equation
 
y 2 − y 1 y 1 − y 0
6 − 9 + 3y
0
t 2 t 1
 
y
   
1  y 3 − y 2 y 2 − y 1 
6 − 6
 
 y 2   t 3 t 2 




y 4 − y 3 y 3 − y 2
   
y 6 − 6
   
 3   
t 4 t 3
   
A  –  =   , (4.64)
   – 
   
–
   – 
   
– y n−1 − y n−2 y n−2 − y n−3
   
6 − 6
   
 
y  t n−1 t n−2 
n−1  
y n − y n−1 y n−1 − y n−2
 
9 − 6 − 3y n
t n t n−1
where again the matrix A is the same as in the above sections, except for the first row,
which becomes
3 00 – – 0 , (4.65)

2 t 1 + 2t 2 t 2
and the last row, which is
3
0 – – – 0 t n−1 2t n−1 + t n . (4.66)
2