2D Simulation of Discontinuous Shallow Flows - Experimental Methods in Hydraulic Research - page 145

T . A i is the cell area and the

E operator represents

spatial average. Integrating along the n i edges of the i cell and performing a first-
order time integration, ( 5 ) becomes:

n ¼ Fi þ Gj with n ¼ ðÞ i ; j hi

X

n i

A i D V hi

Dt þ

L k D ik E h n i ¼ A i Hhi (6)

k ¼ 1

where L k is the length of the k edge.
The fluxes through the k edge of cell i represent, in the simplest interpretation,
the differences between the values of the independent variables on the adjacent
cells i and j , separated by edge k . D ik represents that difference, through edge k from
cell i to j . The flux variations can be expressed as a function of the independent
conservative variables using an approximate Jacobian matrix, J
n;ik

, orthogonal to the

edge in question:

J D ik E h n i ¼

n;ik D ik V

hi

(7)

Such approximation arises from the fact that, for the shallow-water-type equa-
tions, even if the problem is hyperbolic, the flux vectors are not homogeneous
functions of the dependent variables F

. Therefore, it is not possible to use
the natural Jacobian of the system, and an approximated matrix, built on the basis
of a local linearization, must be used (Roe 1981 ).
The eigenvalues and the eigenvectors of the approximate Jacobian matrix are
given by:

ð 6¼ JU ðÞ Þ

l ð 1 Þ l ð 2 Þ l ð 3 Þ

ik ¼ð~

u n þ ~ c Þ ik ;

ik ¼ð~

u n Þ ik ;

ik ¼ð~

u n ~ c Þ ik (8)

2 3 2 3 2 3

1 1

~

1

e ð 1 Þ

4 5 ik ;

e ð 2 Þ

4 5 ik ;

e ð 3 Þ

4 5 ik

~

ik ¼

u ~ þ ~ ci jj ~

ik ¼

cj jj ~

ik ¼

~ u ~ ci jj (9)

u ~ þ ~ cj jj ~ ci jj u ~ ~ cj jj

The approximate variables are given by simple arithmetic means (Roe 1981 ).

r

g h i þ

u i

p

u j

p

v i

p

v j

p

h i þ h j h i þ h j h l

~ u ik ¼ p

h i

p

h j

v ik ¼ ~ p

h i

p

h j

~ c ik ¼ (10)

2

þ þ

The dependant variables variations are projected on a new base, formed by the
eigenvectors of the system:

X

3

a ðnÞ

ik e ðnÞ

D hi¼ (11)

ik

n¼ 1