Page 154 - Basic Structured Grid Generation

Differential models for grid generation 143

grid-cell area. We start with a single boundary curve on which η is taken to be zero.
The two first-order partial differential equations for the cartesian co-ordinates x, y as
functions of ξ, η are:

g 12 = g 1 · g 2 = x ξ x η + y ξ y η = 0,
|g 1 × g 2 |= x ξ y η − x η y ξ = V, (5.111)
where the first equation imposes orthogonality, and the second implies, by eqn (1.43),
that V is a measure of cell-area (assuming that δξδη is the same for each grid cell).
In general we can let V be a function of ξ, η, so that, for example, we could increase
grid-density near the boundary η = 0byensuringthat V is small there.
It follows from eqns (5.111) that

V V
x η =− y ξ , y η = x ξ , (5.112)
g 11 g 11
2
2
where g 11 = (x ξ ) + (y ξ ) .
The subdirectory Book/hyper.gds on the accompanying disk contains a program for
solving these equations numerically using a marching procedure. The cell-area V is
prescribed in the form
√ −λ(1−η)
V = K g 11 e , (5.113)
where K and λ are constants, and the equations are discretized with first-order accuracy
in the η-direction and second-order accuracy in the ξ-direction as

(x i,j+1 − x i,j ) V y i+1,j − y i−1,j
=− ,
η g 11 2 ξ
i,j

(y i,j+1 − y i,j ) V x i+1,j − x i−1,j
= , (5.114)
η g 11 2 ξ
i,j
where
2 2
x i+1,j − x i−1,j y i+1,j − y i−1,j
(g 11 ) i,j = + .
2 ξ 2 ξ
These equations determine explicitly the values x i,j+1 and y i,j+1 in terms of val-
ues x i−1,j ,x i,j ,x i+1,j ,y i−1,j ,y i,j ,and y i+1,j . Hence the method itself may be called
explicit. Only data on values of x and y on the initial boundary η = 0 are required
to start the process. The numerical solution starts from this boundary and marches
outward in the direction of increasing η, eventually constructing new co-ordinate lines
ξ = 0, 1and η = 1.

5.11 Solving the hosted equations

5.11.1 An example

In this section we illustrate the task of solving a partial differential equation, given
that the solution domain has been discretized by the generation of a suitable grid. We