Geoscience Reference
In-Depth Information
the bottom elevations, which are used to construct a background mesh for feedback
to the mesh generator.
3.2 Merging of Altimetric Information with 2D Mesh
Since the 2D meshing procedures described are valid for the plane alone, the output
elements from Gmsh have that coordinate set to 0 by default. If provided a Digital
Terrain Model (DTM), which is a file with a set of points represented in 3D
resulting of a topographical survey, it is possible to superimpose the mesh over
the DTM and calculate altimetric coordinates for every point on the mesh. A simple
algorithm, using linear interpolation on the DTM, was developed to efficiently
estimate the elevation of the points in the mesh. Assuming that the DTM is regular
(uniform spacing between orthogonal measured points), it is possible to devise a
way to directly access the points in question in the altimetric matrix of the DTM
that surrounds a point belonging to the mesh and interpolate information.
4 Numerical Discretization
4.1 Finite-Volume Discretization
The finite-volume method (FVM) was developed to represent conservation PDEs in
the form of simple algebraic equations (reviews in LeVeque
1992
; Hirch
2001
;
Toro
1999
). In this method, volume integrals in a PDE that contains a divergence
term are converted in surface integrals using Gauss's theorem. These terms can then
be evaluated as boundary fluxes in each computational cell, and since the flux that
enters one cell is the one that leaves the adjacent one, these methods are inherently
conservative.
The hyperbolic, nonhomogeneous, first-order, quasi-linear system that expresses
the conservation laws (
1
-
3
) can be written in vector notation as:
@
t
U
ð
ð
V
Þ
Þ :
E
ð
U
Þ¼
H
ð
U
Þ,@
t
V
ð
ð
U
Þ
Þ @
x
F
ð
ð
U
Þ
Þ @
y
G
ð
ð
U
Þ
Þ¼
H
ð
U
Þ
(4)
3
where
U
:
R
0
; þ1
½!
R
is the independent conservatives variables vector;
3
3
3
3
and
G
3
3
are
V
:
R
!
R
is the primitive variables vector;
F
:
R
!
R
:
R
!
R
3
3
the flux vectors in
x
and
y
, respectively;
H
:
R
!
R
is the source terms vector;
and
x
,
y
, and
t
are space and time coordinates.
To obtain the discretization scheme, (
4
) is integrated in a computational cell,
i
, and Gauss's theorem is applied. This yields:
ð
O
i
r
@
t
A
i
Vhiþ
E
ð
U
Þ
dS
¼
A
i
Hhi
(5)
