Page 149 - Basic Structured Grid Generation

138 Basic Structured Grid Generation

where the effect will be to double the spacing between grid points in one part of the
physical domain.

Exercise 5. With ϕ(ξ) given by eqn (5.87), show by integrating eqn (5.82) in the two
intervals 0 ξ< ξ 0 and ξ 0 <ξ 1 and eliminating K that

(b − a)

 a + ξ, 0 ξ< ξ 0

(2 − ξ 0 )

x(ξ) = . (5.88)
 2a − bξ 0 (b − a)
+ 2ξ ,ξ 0 <ξ 1


2 − ξ 0 2 − ξ 0
In this example the grid spacing in the physical plane changes at x = x 0 ,where
x 0 = a + (b − a)ξ 0 /(2 − ξ 0 ).Thisgives
2(x 0 − a)
ξ 0 = .
b + x 0 − 2a
For more complicated weight functions, it may be difficult to ensure that changes in
grid spacing in the physical plane occur precisely where they are wanted. It may be
more convenient to use weight functions which are functions of physical co-ordinates,
so that in the one-dimensional case we have ϕ(x), with

x i+1 + x i
x i+1 − x i = Kϕ (ξ i+1 − ξ i ) (5.89)
2
instead of eqn (5.81), with the limiting forms

dx
= Kϕ(x), (5.90)
dξ
d 1 dx
= 0, (5.91)
dξ ϕ(x) dξ
in place of eqns (5.82) and (5.83), giving

2
d x 1 dϕ dx 2
= . (5.92)
dξ 2 ϕ dx dξ
This is a non-linear second-order differential equation (with the same end-conditions
x(0) = a, x(1) = b), with no guaranteed solution in general. Integration of eqn (5.90)
yields
b 1
K = dx. (5.93)
a ϕ(x)
Equation (5.92) is equivalent to eqn (5.80) if we express P as a function of x and put
1 dϕ
P(x) =− . (5.94)
2
Kϕ dx