Geoscience Reference
In-Depth Information
Table 4.1 Relation between k and ( i , j )
k
1
2
3
4
5
6
7
8
9
( i , j )
(1, 1)
(2, 1)
(3, 1)
(1, 2)
(2, 2)
(3, 2)
(1, 3)
(2, 3)
(3, 3)
4.2.4.3 Interpolation method on moving grids
For a moving grid, Eq. (4.96) can still be used to interpolate the function f at the
element shown in Fig. 4.10 at each time step. Consequently the spatial derivatives of
f are discretized by schemes (4.97) and (4.98), while the time derivative is discretized
as (Wu, 1996a)
5 =
f n + 1
5
9
f 5
f
e k f n + θ
k
+
(4.102)
t
τ
k
=
1
∂ξ + t ∂ϕ k
= ( t ∂ϕ k
where e k is the difference coefficient, defined as e k
∂η )
5 ;
τ
is the
time step length; and
θ
is an index:
=
0 for explicit schemes and
=
1 for implicit
schemes.
The second term on the right-hand side of scheme (4.102) appears due to grid
movement.
4.3 FINITE VOLUME METHOD
4.3.1 Finite volume method for 1-D problems
4.3.1.1 Discretization of 1-D steady problems
Consider the 1-D steady, homogeneous convection-diffusion equation, which is
written in conservative form as
d
dx
d
dx
d
dx
u
φ) =
(4.103)
where
φ
is the quantity to be determined, and
is the diffusion coefficient.
is
related to
ε
c in Eq. (4.15) by
= ρε
c . Note that the flow density
ρ
is included
in Eq.
(4.103) to consider its possible changes due to sediment,
temperature,
salinity, etc.
Fig. 4.14 shows the commonly used 1-D finite volume grid. For a grid point P , the
point on its west side or in the negative x direction is denoted as W , and the point on
its east side or in the positive x direction is denoted as E . The further west and east
points are WW and EE , respectively. The control volume (cell) for point P is embraced
by two faces w and e , which are located midway (not absolutely necessary) between
W and P and between P and E , respectively.
 
 
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