378
The time discretization of the momentum equation is carried out by using the characteristic method. In this equation, ." is treated implicitly. Making use of (36) a boundary
problem having only Qn+I as unknown is obtained. This problem is written in variational form and discretized using a finite element method. (Qn+I, ."n+l) is approached
by Raviart-Thomas mixed finite element (see [12]). The non-linear discretized problem
is solved using Bermudez-Moreno algorithm (see [2] and the references therein), that
takes advantage of the maximal monotonous character of the non-linear operator of."
appearing in (22).
This algorithm is unconditionally stable.
2.6 Initial and boundary conditions
The boundary conditions that can be considered depend on the variational formulation
chosen to discretize the equations. There are two kinds of boundary conditions: those
that must be imposed to test functions, called essential conditions, and those that are
obtained as a consequence of the variational formulation, called natural conditions.
With the variational formulation used to discretize the equations, the boundary
conditions that may be imposed can be classified into two different groups
(a) Normal component of the flux given:
Q. ii = j,
where ii is the outward unit vector. This condition is essential. It is useful on
the inflow/outflow boundaries when the flux is known, and on coastal boundaries
(f = ° in this case).
(b) Elevation given:
." = 'P.
This condition is natural. It is considered on the inflow/outflow boundaries, when
the elevation is known. For instance, this condition is useful to simulate the effects
due to tide fluctuations.
Finally, the following initial conditions are considered:
.,,(x,O) = ."o(x) in D,
u(x,O) = uo(x) in D.
The model used does not permit to impose no or more than one condition over
a piece of boundary. Therefore, this model can not efficiently deal with flows that
are supercritical near the inflow or the outflow boundary (see the remark concerning
boundary conditions in 2.4).
The time discretization of the momentum equation is carried out by using the characteristic method. In this equation, ." is treated implicitly. Making use of (36) a boundary
problem having only Qn+I as unknown is obtained. This problem is written in variational form and discretized using a finite element method. (Qn+I, ."n+l) is approached
by Raviart-Thomas mixed finite element (see [12]). The non-linear discretized problem
is solved using Bermudez-Moreno algorithm (see [2] and the references therein), that
takes advantage of the maximal monotonous character of the non-linear operator of."
appearing in (22).
This algorithm is unconditionally stable.
2.6 Initial and boundary conditions
The boundary conditions that can be considered depend on the variational formulation
chosen to discretize the equations. There are two kinds of boundary conditions: those
that must be imposed to test functions, called essential conditions, and those that are
obtained as a consequence of the variational formulation, called natural conditions.
With the variational formulation used to discretize the equations, the boundary
conditions that may be imposed can be classified into two different groups
(a) Normal component of the flux given:
Q. ii = j,
where ii is the outward unit vector. This condition is essential. It is useful on
the inflow/outflow boundaries when the flux is known, and on coastal boundaries
(f = ° in this case).
(b) Elevation given:
." = 'P.
This condition is natural. It is considered on the inflow/outflow boundaries, when
the elevation is known. For instance, this condition is useful to simulate the effects
due to tide fluctuations.
Finally, the following initial conditions are considered:
.,,(x,O) = ."o(x) in D,
u(x,O) = uo(x) in D.
The model used does not permit to impose no or more than one condition over
a piece of boundary. Therefore, this model can not efficiently deal with flows that
are supercritical near the inflow or the outflow boundary (see the remark concerning
boundary conditions in 2.4).