Page 148 - Basic Structured Grid Generation

Differential models for grid generation 137

with end-conditions x(0) = a, x(1) = b. Although this is a non-linear differential
equation, and the existence of solutions (in particular, single-valued solutions for which
x increases monotonically with ξ) is not guaranteed, we may integrate once to give
−2
ξ
dx
= 2 P(t) dt
dξ c
for some constant c. This indicates that some restrictions on the possible values of P
are necessary for solutions to exist (and not to involve folding).
Another approach to grid generation in this situation may be derived by specify-
ing a weight function ϕ(ξ), which has the property that the grid spacing between
points x i ,x i+1 in the physical plane is proportional to the product of the grid spac-
ing between corresponding points ξ i ,ξ i+1 in the computational plane and the weight
function evaluated at the mid-point of ξ i and ξ i+1 . Thus

ξ i + ξ i+1
x i+1 − x i = Kϕ (ξ i+1 − ξ i ), (5.81)
2
where K is the constant of proportionality. The weight function is defined in the
computational domain 0 ξ 1 and takes only positive values.
Revising the definition a little so that it applies in a general way without having to
specify particular grid points, we proceed to the limit as the grid spacing tends to zero,
and re-write eqn (5.81) as
dx
= Kϕ(ξ). (5.82)
dξ
Eliminating K, we obtain the equation
d 1 dx
= 0, (5.83)
dξ ϕ (ξ) dξ
which is equivalent to
2
d x 1 dϕ dx
= . (5.84)
dξ 2 ϕ dξ dξ
This is the non-conservative version of eqn (5.83). Thus, given ϕ(ξ),we have a
differential model of grid generation, with the same boundary conditions x(0) = a,
x(1) = b.
Comparing eqn (5.84) with (5.80) shows that they are equivalent if we put
1 dϕ
P(ξ) =− . (5.85)
2 3
K ϕ dξ
Note that integration of eqn (5.82) with respect to ξ over the whole range from
ξ = 0 to1gives
1
K = (b − a) ϕ(ξ) dξ. (5.86)
0
As an example, we take

1, 0 ξ ξ 0
ϕ(ξ) = , (5.87)
2, ξ 0 <ξ 1