Geoscience Reference
In-Depth Information
Figure 4.11
Stretching coordinate transformation.
Stretching coordinate transformation (4.91) has the following analytic relations:
∂
x
∂ξ
=
∂
x
∂ζ
=
∂
x
∂τ
=
1,
0,
0,
∂
∂ξ
=
∂
z
∂ξ
+
H
∂
z
b
h
∂ξ
z
∂ζ
=
∂
h
H
,
∂
∂τ
=
∂
z
∂τ
+
H
∂
z
b
h
∂τ
,
,
(4.92)
∂
∂
∂
t
∂ξ
=
t
∂ζ
=
t
∂τ
=
0,
0,
1,
and
∂ξ
∂
∂ξ
∂
∂ξ
∂
x
=
1,
=
0,
=
0,
z
t
∂ζ
∂
H
h
∂
z
b
x
−
ζ
h
∂
h
∂ζ
∂
H
h
,
∂ζ
∂
H
h
∂
z
b
∂
−
ζ
h
∂
h
x
=−
x
,
=
=−
t
,
(4.93)
∂
∂
z
t
t
∂
∂τ
∂
∂τ
∂
∂τ
∂
x
=
0,
=
0,
=
1.
z
t
The 2-D stretching coordinate transformation (4.91) can be easily extended to the
3-D case by adding one stretching function in the third direction. Because analytic
transformation relations exist in the entire domain, it is very convenient to solve
the transformed governing equations in the fixed, regular logical domain. However,
this stretching coordinate transformation is inconvenient for the complex boundary
problems that do not have analytic transformation relations.
4.2.3.3 Discretization of the transformed equations
As mentioned above, the irregular (and/or moving) physical domain or element is
converted to a regular logical domain or element under coordinate transformations
(4.74), (4.83), (4.87), (4.89), and (4.91). Therefore, many classic finite difference
schemes based on regular grids can be used to solve the transformed equation on the
regular logical domain or element. For example, the convection-diffusion equation
(4.50) in the (
x
,
y
) coordinate system can be converted to the following form in the
