Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
72 Discretization and numerical simulations
Remark 1.3. We can mention that an explicit scheme in time may be very restrictive
because of the viscous terms. Nevertheless, we consider a small viscosity coefficient, which
reduces the constraint on the time step. Moreover, the instantaneous regularizing effect of
the viscous terms makes the nonconservative product h∂xuN well defined at the first time
step, even if we start with a discontinuous initial water height as in Section A.2.
Remark 1.4. We can note that this numerical scheme preserves at the discret level the
conservation of the mass hdx and the constant steady states with periodic boundary
condition. We will also see in the numerical results that it captures non constant steady
states in the last test case performed (see Section 3.2.5).
3.2 Numerical experiments
In this section we present numerical simulations performed to validate numerically the
multilayer system and its discretization. We also show its possible dynamic behavior,
depending on the wanted accuracy in the vertical direction, which is given by the choice
of the amplitude h of the inside fixed layers. In all the tests performed, the CFL number
is equal to 0.95.
3.2.1 Test 1: perturbation of rest in height, comparison with classical
Shallow Water model, flat bottom
In this section, we perturb the lake at rest in height. We take gravity constant g = 1,
viscosity µ = 0.0001, no friction, computational domain [0,1] and periodic boundary conditions.
We compare first our model to existing models, and then investigate its dynamic
behavior. We consider the initial condition
⎧
⎨ η(t = 0,x) = 0.1+0.01 (1+0.8 cos(2πx/L)) ,
(3.2.1)
⎩
ui(t = 0,x) ≡ 0 for 1 i N ,
where the number of layers N will be either 5 (h = 0.02) or 15 (h = 0.006). Then we
plot the solution to the multilayer scheme for 5 layers (+) and 15 layers (•), as well as the
result obtained with a simple global Lax-Friedrichs scheme for the classical viscous shallow
water system (??) (lines), for different times. Figure 3.1 shows the evolution of the free
surface, and Figure 3.2 shows the evolution of the averaged horizontal velocity.
On the one hand, we observe that the results fit well with the classical one layer
model, whatever the number of layers. Hence our multilayer model is at least as good
as the classical Saint-Venant model. On the other hand, we do have an advantage when
considering the multilayer model: information on the vertical velocity. Indeed, we see in
Figure 3.3 the velocity field inside the water for 15 layers. This information can not be
recovered by the classical shallow water model.