Numerical modeling of waves for a tsunami early warning system

and
Numerical modeling of waves for a tsunami early warning system
(ccg) x
ax =
(ccg) u
ux +(ccg) v
vx [auux + avvx] =
= u2 x (ccga) u +(ccg) u
uxvxav +(ccg) v
uxvxau + v2
x (ccga) v =
= au u2
x (ccg) u + uxvx (ccg) v + av v2
x (ccg) v + uxvx (ccg) u
(ccg) y ay =
[auuy + avvy] =
(ccg) u
uy +(ccg) v
vy
= u2 y (ccga) u +(ccg) u
uyvyav +(ccg) v
uyvyau + v2
y (ccga) v =
= au u2
y (ccg) u + uyvy (ccg) v + av v2
y (ccg) v + uyvy (ccg) u
Now we are ready to write equation (B.6)
ccg [auuu2 x + avvv2 x +2auvuxvx + au (ux) x + av
(vx) x ]+
+ccg auuu2 y + avvv2 y +2auvuyvy + au (uy) y + av (vy) y + k2a
+
+au u2
x (ccg) u + uxvx (ccg) v + av v2
x (ccg) v + uxvx (ccg) u +
+au u2
y (ccg) u + uyvy (ccg) v + av v2
y (ccg) v + uyvy (ccg) u =0
which can be rewritten as follows:
ccgauu u2 x + u2
y + ccgavv v2 x + v2
y
+ccg (au∇2u + av∇2v)+ccgk2
a+
+au (ccg) u u2 x + u2
y
+av (ccg) v
v 2 x + v 2 y
+ au (ccg) v (uxvx + uyvy)+
+2ccgauv (uxvx + uyvy)+
+ av (ccg) u (uxvx + uyvy) =0
(B.10)
(B.11)
(B.12)
(B.13)
the first and the sixth terms of equation (B.13) can be assembled as the
second and the seventh, yielding to
u2 x + u2
y (ccgau) u +
v2 x + v2
y
(uxvx + uyvy)
au (ccg) v + av (ccg) u +2ccgauv
(ccgav) v + ccgk 2 a+
+ ccg (au∇ 2 u + av∇ 2 v)=0
(B.14)
while the derivatives of a above are taken with respect to the mapped
coordinates (u, v), the coefficients of the derivatives still involve the
Università degli Studi di Roma Tre - DSIC 115