Geoscience Reference
In-Depth Information
Numerical methods
River engineering problems are usually governed by nonlinear differential equations
in irregular and movable domains, most of which have to be solved using numerical
methods. Introduced in this chapter are the discretization methods for 1-D, 2-D, and
3-D problems on fixed and moving grids, the solution strategies for the Navier-Stokes
equations, and the solution methods of algebraic equations. Some of these can be
found in Patankar (1980), Hirsch (1988), Fletcher (1991), Ferziger and Peric (1995),
Shyy
et al
. (1996), etc.
4.1 CONCEPTS OF NUMERICAL SOLUTION
4.1.1 General procedure of numerical solution
Consider the problem in a domain of
a
≤
x
≤
b
shown in Fig. 4.1, governed by a
differential equation
L
(
f
;
x
)
=
S
(4.1)
with boundary conditions
f
|
=
f
a
,
f
|
x
=
b
=
f
b
(4.2)
x
=
a
where
L
is the differential operator,
f
is the function to be determined,
x
is the spatial
coordinate, and
S
is the source term.
To acquire a numerical solution, the study domain is first represented by a finite
number of points, denoted as
x
1
,
x
2
,
...
, and
x
N
, which constitute the computational
grid (mesh). Here,
x
1
=
a
and
x
N
=
b
. The distance between two consecutive points,
x
, is the grid size or spacing.
Eq. (4.1) is discretized on the computational grid using a numerical method.
A discrete equation
L
d
is then established to approximate the differential equation
at each grid point:
L
d
(
f
;
x
i
)
=
S
i
(
i
=
2,
...
,
N
−
1
)
(4.3)
