Page 282 - Computational Fluid Dynamics for Engineers

272 9. Grid Generation

Grid points are defined by evaluating the interpolation formula at successive
integer values i. Interior points r n for n = 2,3,...,iV — 1 can be specified to
serve as additional parameters to control the distribution. In most cases, how-
ever, interpolations are made solely from the boundaries, eliminating the need
for additional interior information within the gridded region. The Lagrange in-
terpolation is then reduced to its simplest, linear form:

r(i) = h - L \ f 1 + (j)r 2

Therefore <pi(j) = 1 — j and ^2(7) = 7
Ii = I (i = 0) and r 2 = r (i = I)

Hermite Interpolation

Lagrange interpolations match only function values. It is possible to match both
the function r and the first derivative r' = r^ by using Hermite interpolation,
defined by:

f(i) = £ > „ (j) -Vn + E ^ ( ! ) • < (9-4-14)
71=1 V J n=l ^ '
These polynomials can be obtained from the Lagrange Polynomials by:

n I I
* . : ; Vn ( 'j .<pi[j) (9.4.15a)

*• G) - Irr) * (7) <»•"»»

In the usual case with N = 2, the function matches two boundary values r x and
and the first derivatives r^ and r^ at the two boundaries (Fig. 9.9). In this
r 2
case, we have:
2
* i ( j ) = (l + 2 j ) . ( l - $ )
^ 2 (}) = ( 3 - 2 i ) . ( } ) 2

2
(
^(i) = f-i).(i)