Geoscience Reference
In-Depth Information
analytic solution of Eq. (4.15) in this segment:
f
−
f
i
−
1
−
(
x
−
x
i
−
1
)
S
i
/
u
exp
[
(
x
−
x
i
−
1
)
u
/ε
]−
1
c
u
=
(4.19)
f
i
+
1
−
f
i
−
1
−
(
x
i
+
1
−
x
i
−
1
)
S
i
/
exp
[
(
x
i
+
1
−
x
i
−
1
)
u
/ε
]−
1
c
Imposing
f
=
f
i
at
x
=
x
i
in Eq. (4.19) yields
f
i
−
f
i
−
1
−
S
i
x
/
u
exp
(
−
P
/
2
)
u
=
(4.20)
f
i
+
1
−
f
i
−
1
−
2
S
i
x
/
exp
(
P
/
2
)
+
exp
(
−
P
/
2
)
where
P
is the Peclet number, defined as
P
=
u
x
/ε
c
, which represents the relative
importance of convection and diffusion effects.
Eq. (4.20) can be rewritten as
a
P
f
i
=
a
W
f
i
−
1
+
a
E
f
i
+
1
+
S
i
(4.21)
u
u
where
a
W
=
x
exp
(
P
/
2
)/
sinh
(
P
/
2
)
,
a
E
=
x
exp
(
−
P
/
2
)/
sinh
(
P
/
2
)
, and
a
P
=
2
2
a
W
+
a
E
.
Eq. (4.21) is the exponential difference scheme. It was derived by Lu and Si (1990),
who called it a finite analytic scheme. It is similar to Spalding's (1972) exponential
scheme based on the finite volume approximation (see Section 4.3.1).
Scheme (4.21) is capable of automatically upwinding and has a diagonally dominant
coefficient matrix. It is very stable. It tends to the upwind difference scheme (4.17) for
a strong convection problem (large
P
) and to the central difference scheme (4.14) for
a strong diffusion problem (small
P
).
4.2.1.3 Discretization of 1-D unsteady problems
Time-marching schemes for 1-D unsteady problems include the Euler scheme, leapfrog
scheme, Lax scheme, Crank-Nicholson scheme, Preissmann scheme, characteristic
difference scheme, and Runge-Kutta method. The former five schemes are discussed
below, whereas the others can be found in Abbott (1966), Yeh
et al
. (1995), Fletcher
(1991), etc.
Euler scheme
Consider the 1-D unsteady convection equation:
∂
u
∂
f
f
+
x
=
S
(4.22)
∂
t
∂
The computational grid in the (
x
,
t
) plane for solving Eq. (4.22) is shown in Fig. 4.4.
The simplest scheme for the temporal term in Eq. (4.22) is the two-level difference
