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numerical tests have shown that it can provide satisfactory results for many practical
nonlinear problems.
The advantage of the operator splitting method is that each operator can be handled
with an appropriate method specific to that operator. However, boundary conditions
may be difficult to implement, and the overall accuracy is hard to judge even though
high-order schemes might be used for every operator.
4.2.3 Finite difference method for multidimensional
problems on curvilinear grids
River flow problems usually have irregular and even movable domains. When the
classic finite difference method on regular grids is used to solve these problems, diffi-
culties may arise near boundaries. However, boundary conditions are essential to the
properties of the solution. Therefore, the finite difference method on fixed and moving
curvilinear grids has been established in the past decades via coordinate transformation
and interpolation, as described below.
4.2.3.1 Governing equations in generalized coordinate
system
In general, the unsteady coordinate transformation from the Cartesian coordinate
system ( x i , t ) to a moving, curvilinear coordinate system (
ξ
m ,
τ
) can be written as
x i
=
x i
m ,
τ) (
i
=
1, 2, 3; m
=
1, 2, 3
)
(4.63)
t
= τ
where
ξ
1 ,
ξ
2 , and
ξ
( = ξ
,
η
,
ζ)
are the coordinates, and
τ
is the time in the curvilinear
3
system.
Coordinate transformation (4.63) includes time and hence can be applied to both
fixed and movable grids (Wu, 1996a; Shyy et al ., 1996). Its Jacobian matrix is
x 1
∂ξ
x 2
∂ξ
x 3
∂ξ
0
1
1
1
x 1
∂ξ
x 2
∂ξ
x 3
∂ξ
0
2
2
2
B
=
(4.64)
x 1
∂ξ
x 2
∂ξ
x 3
∂ξ
0
3
3
3
x 1
∂τ
x 2
∂τ
x 3
∂τ
1
and Jacobian determinant is J
. For a monotonic and reversible coordinate trans-
formation, the Jacobian determinant should be non-zero and have finite bounds, i.e.,
0
=|
B
|
<
J
< +∞
.
m
i
i
m
m
Denote
α
= ∂ξ
/∂
x i and
β
=
x i /∂ξ
m . Then
α
i and
∂ξ
/∂
t can be determined by
m
m
M i m
J
M m
J
∂ξ
m
m
i
α
=
,
=
(4.65)
t
 
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