a reduced model for internal waves interacting with submarine ...

is to approximate theξ-derivative of a function f (ξ) by the fourth order, five point
formula
f ξ (ξ j )= 8( f j+1− f j−1 )+ f j−2 − f j+2
12∆ξ
+ O(∆ξ 4 ), (4.5)
where f j = f (ξ j ),ξ j = j∆ξ,∆ξ=2l/N, j=1,..., N. So the spatial discretization
for the LFM withβ=0 (which is just the wave equation) leads to the system of
Ordinary Differential Equations (ODEs)
⎧
⎪⎨ η t = Cu 1 ,
⎪⎩ u 1t = Cη,
whereη=[η 1 ,...,η N ] t , withη j =η( j∆ξ, t) and u 1 = [u 11 ,...,u 1N ] t , with u 1 j =
u 1 ( j∆ξ, t), j=1,..., N. C is the skew-symmetric, circulant matrix,
⎡
0 2/3 −1/12 0 ··· 0 1/12 −2/3
⎤
−2/3 0 2/3 −1/12
1/12 −2/3 0 2/3
... 1/12
... 0
C= 1 ∆ξ
0 1/12 −2/3 0
.
. . .
. . .
...
. . .
...
. . .
. . .
.
0
...
... 0 2/3 −1/12
−1/12
...
... −2/3 0 2/3
⎢⎣
2/3 −1/12 0 1/12 −2/3 0
⎥⎦
Since it is a skew-symmetric matrix, it has imaginary eigenvalues. The same is
true for the block matrix
⎡⎢⎣
C
C
⎤
⎥⎦ ,
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