Page 169 - Basic Structured Grid Generation

158 Basic Structured Grid Generation

giving finally
2
d x 1 dϕ dx 2
− = 0. (6.15)
dξ 2 ϕ(x) dx dξ
This is the same equation as (5.92), so the transformation x = x(ξ) which gives a
grid according to the weight function ϕ(x) makes I stationary (although minimizing
properties are now not so easy to prove).

Exercise 1. As an alternative derivation of the Euler-Lagrange equation, use the fact
that the integrand in eqn (6.14) is independent of ξ to show directly from eqn (6.4) that
2

1 dx

= const.
2
[ϕ(x)] dξ
(differentiation of this equation yielding eqn (6.15)), and hence dx/dξ is proportional
to ϕ(x).
Returning to eqn (6.13), the variational approach suggests a way of discretizing the
problem. Taking x 0 = a and x m = b as given end points and x 1 ,x 2 ,...,x m−1 as the
unknown interior points of the grid, a representation of the integral I, with a uniform
division of the range of integration into points ξ i = i/m, i = 0, 1, 2,. ..,m,is

m 2
I 1 x i − x i−1 ξ, (6.16)
2ϕ 1 ξ
i=1 i− 2
where ϕ(ξ) is evaluated at mid-points of the sub-intervals; ϕ 1 = ϕ(ξ i − 1/2m).
i− 2
Hence we have a problem of ordinary calculus in which, instead of minimizing a
functional, we want to minimize the function (since ξ is a constant)
m 2
(x i − x i−1 )

f(x 1 ,x 2 ,...,x m−1 ) =
2ϕ 1
i=1 i− 2
(x 1 − x 0 ) 2 (x 2 − x 1 ) 2 (x m − x m−1 ) 2
= + +· · · + . (6.17)
2ϕ 1 2ϕ 3 2ϕ 1
2 2 m− 2
Equating the partial derivatives ∂f/∂x j to zero gives the (m − 1) equations
(x j − x j−1 ) (x j+1 − x j )
− = 0, j = 1, 2,... ,(m − 1). (6.18)
ϕ 1 ϕ 1
j− j+
2 2
This is a system of (m−1) linear equations for (m−1) unknowns x 1 ,x 2 ,...,x m−1 ,
which we encountered in Chapter 5 at eqn (5.97). An equivalent system is clearly

(x 1 − x 0 ) (x 2 − x 1 ) (x m − x m−1 )

= =· · · = = K , (6.19)
ϕ 1 ϕ 3 ϕ m− 1
2 2 2

for some constant K . These equations are consistent with the condition (5.81) that the
distance between grid points is proportional to the value of the weight function at the
mid-point of the corresponding ξ-interval.