Page 155 - Basic Structured Grid Generation

144 Basic Structured Grid Generation

take the two-dimensional form of Laplace’s Equation as an example. The subdirec-
tory Book/p.d.Equation on the accompanying disk, as listed in the last section of this
chapter, contains two files, the first of which, Conjugate.Laplace.f, simply shows how
to solve Laplace’s Equation
2
2
∂ ϕ ∂ ϕ
+ = 0
∂x 2 ∂y 2
in a square solution domain using the Conjugate Gradient method. Exact solutions,
x
such as u = e sin y, with appropriate boundary conditions, may be used to assess the
numerical accuracy of the method.
For non-rectangular domains we need to be able to transform the equation to gen-
2
eral curvilinear co-ordinates ξ, η. A convenient expression for ∇ ϕ was obtained in
eqn (1.173), with the addition of eqns (1.174) and (1.175). The file laplacian.general.f
contains a program which solves the discretized form of these equations using SOR.
First we set
√ √ √
g 11 / g = g oo , −2g 12 / g = g ot , g 22 / g = g tt ,
√ 2 √ 2
g∇ ξ = Delzi, g∇ η = Delet

so that eqns (1.174) and (1.175) can be written
(x η y ξξ − y η x ξξ ) (x η y ξη − y η x ξη ) (x η y ηη − y η x ηη )
Delzi = g tt √ + g ot √ + g oo √ ,
g g g
(y ξ x ξξ − x ξ y ξξ ) (y ξ x ξη − x ξ y ξη ) (y ξ x ηη − x ξ y ηη )
Delet = g tt √ + g ot √ + g oo √ .
g g g

Discretization of eqn (1.173) can now be represented by
$ %
(g tt ∗ ϕ) j+1,k − 2(g tt ∗ ϕ) j,k + (g tt ∗ ϕ) j−1,k
$ %
+ (g oo ∗ ϕ) j,k+1 − 2(g oo ∗ ϕ) j,k + (g oo ∗ ϕ) j,k−1
$
+ 0.25 (g ot ∗ ϕ) j+1,k+1 + (g ot ∗ ϕ) j−1,k−1
%
−(g ot ∗ ϕ) j+1,k−1 − (g ot ∗ ϕ) j−1,k+1
$ %
+ 0.5 (Delzi ∗ ϕ) j+1,k − (Delzi ∗ ϕ) j−1,k
$ %
+ 0.5 (Delet ∗ ϕ) j,k+1 − (Delet ∗ ϕ) j,k−1 = 0 (5.115)
in the transformed domain of curvilinear co-ordinates.
Taking terms containing ϕ j,k to one side of the equation, we can generate a ‘tem-
porary’ value
! "
t (g tt ∗ ϕ) j+1,k + (g tt ∗ ϕ) j−1,k + (g oo ∗ ϕ) j,k+1 + (g oo ∗ ϕ) j,k−1
ϕ j,k =
2[(g tt ) j,k + (g oo ) j,k ]
! "
(Delzi ∗ ϕ) j+1,k − (Delzi ∗ ϕ) j−1,k
+0.5
2[(g tt ) j,k + (g oo ) j,k ]