Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
74 Discretization and numerical simulations
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.2 0.4 0.6 0.8 1
0.12
0.1
0.08
0.06
0.04
0.02
(a) (b)
0
0 0.2 0.4 0.6 0.8 1
Figure 3.3: Evolution of the velocity field for initial condition (3.2.1) and 15 layers, at
times t = 1 (a) and t = 10 (b)
completed with the non penetration condition at the bottom and the divergence free condition
to reconstruct the vertical velocity:
w(t,x,0) = 0, ∀(t,x) ∈ R + ×R, ∂xu+∂zw = 0.
In order to design our numerical scheme for this system, we introduce a tracer parameter
0 a 1 of particules of fluids in the vertical direction [38]. Thus the vertical position
of one particle of fluid at time t and horizontal position x reads:
z = Z(t,x,a), 0 a 1.
Then, we have Z(t,x,a = 0) = 0 for particles at the bottom, and Z(t,x,a = 1) = H(t,x)
at the free surface. Moreover we ask the advection equation at the free surface to be
satisfied for any a, namely:
∂tZ(t,x,a)+u(t,x,Z(t,x,a)) ∂xZ(t,x,a) = w(t,x,Z(t,x,a)) . (3.2.2)
Hence, noting c = ∂aZ and injecting these new variables in the primitive equations, we
get an hyperbolic system posed in a fixed domain Ω = R + ×R×[0,1]. It reads, for all
(t,x,a) ∈ Ω:
∂tũ(t,x,a)+∂x
ũ 2
2
(t,x,a) = −g∂xZ(t,x,1), (3.2.3)
∂tc(t,x,a)+∂x(cũ)(t,x,a) = 0, (3.2.4)