a reduced model for internal waves interacting with submarine ...
a reduced model for internal waves interacting with submarine ...
a reduced model for internal waves interacting with submarine ...
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V Flat bottom Rough bottom<br />
Linear<br />
Weakly<br />
Nonlinear<br />
(<br />
1−<br />
√ β<br />
ρ 2<br />
ρ 1<br />
T D ) u 1<br />
(<br />
1−<br />
√ β<br />
ρ 2<br />
ρ 1<br />
M −1 T D ) u 1<br />
(<br />
1−<br />
√ β<br />
ρ 2<br />
ρ 1<br />
T D ) u 1<br />
(<br />
1−<br />
√ β<br />
ρ 2<br />
ρ 1<br />
M −1 T D ) u 1<br />
Strongly<br />
Nonlinear<br />
(<br />
1−<br />
√ β<br />
ρ 2<br />
ρ 1<br />
T DA ) u 1<br />
(<br />
1−<br />
√ β<br />
ρ 2<br />
ρ 1<br />
M −1 T DA ) u 1<br />
Table 4.4: Discretizations <strong>for</strong> the relations in Table 4.2.<br />
The AB3 scheme used is:<br />
η n+1<br />
j<br />
V n+1<br />
j<br />
=η n j +∆t ( 23E<br />
n<br />
12 j − 16E n−1<br />
j<br />
= V n j+ ∆t ( 23F<br />
n<br />
12 j − 16F n−1<br />
j<br />
+ 5E n−2 )<br />
j ,<br />
+ 5F n−2 )<br />
j ,<br />
where E n j= E( j∆ξ, n∆t) and F n j= F( j∆ξ, n∆t). The AM3 scheme used is:<br />
η n+1<br />
j<br />
V n+1<br />
j<br />
The AB4 scheme used is:<br />
=η n j +∆t ( 9E<br />
n+1<br />
j + 19E n j<br />
24<br />
− 5En−1 j<br />
= V n j+ ∆t ( 9F<br />
n+1<br />
j + 19F n j<br />
24<br />
− 5Fn−1 j<br />
+ E n−2 )<br />
j ,<br />
+ F n−2 )<br />
j .<br />
η n+1<br />
j<br />
V n+1<br />
j<br />
=η n j +∆t ( 55E<br />
n<br />
24 j − 59E n−1<br />
j + 37E n−2<br />
j<br />
= V n j+ ∆t ( 55F<br />
n<br />
24 j − 59F n−1<br />
j + 37F n−2<br />
j<br />
− 9E n−3 )<br />
j ,<br />
− 9F n−3 )<br />
j ,<br />
65