Modélisation, analyse mathématique et simulations numériques de ...
Modélisation, analyse mathématique et simulations numériques de ...
Modélisation, analyse mathématique et simulations numériques de ...
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tel-00656013, version 1 - 3 Jan 2012<br />
Ö×ÙÖ<br />
54 A dynamic multilayer shallow water mo<strong>de</strong>l<br />
z<br />
z4+1/2 = η(t, x)<br />
z3+1/2(t, x) u4(t, x)<br />
z2+1/2(t, x)<br />
z1+1/2(t, x)<br />
z1/2 = zb(x)<br />
0<br />
u3(t, x)<br />
u2(t, x)<br />
u1(t, x)<br />
H(t, x)<br />
h4(t, x)<br />
h3(t, x)<br />
h2(t, x)<br />
ÓØØÓÑ<br />
h1(t, x)<br />
Figure 2.2: Classical multilayer approach.<br />
equations by the heights, we rewrite the system on the unknown (U,h) = (u1 ... uN,h) T<br />
as follows.<br />
⎧<br />
⎨<br />
where the source terms are <strong>de</strong>scribed below.<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
⎩<br />
∂tU−µ∂xxU = S,<br />
∂th+∂x(huN) = F ,<br />
S = Sb +Sl +Snl,<br />
N−1 <br />
F = wN−1/2 = −<br />
where Sb refers to the bottom source term<br />
i=1<br />
Sb = −g∂xzb(1,...,1) T ,<br />
∂x(hiui) ,<br />
x<br />
(2.3.1)
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