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

which has the same eigenvalues of C with double multiplicity.
Another choice to approximate theξ-derivatives is to use the spectralξ-derivative,
whose corresponding matrix is
D=
⎡
⎢⎣
− 1 2
− 1 2
0 − 1 2
cot(
1∆ξ
2 ) ...
cot(
1∆ξ
2 )
...
1 2∆ξ
cot( )
2 2
1 2∆ξ
cot( ) ... − 1 3∆ξ
cot( )
2 2 2 2
− 1 2
3∆ξ
cot( ) ...
2
.
...
... − 1 2
cot(
(N−1)∆ξ
2
) 0
.
cot(
(N−1)∆ξ
2
)
⎤
⎥⎦
.
Here we also have a skew-symmetric, Toeplitz, circulant matrix with imaginary
eigenvalues ik, k=−N/2+1,..., N/2−1, with zero having multiplicity 2. With
it, the discretization of the non-dispersive LFM leads to an ODEs system forηand
u 1 involving the block matrix
⎡
⎢⎣
D
D
⎤
⎥⎦ .
This block matrix has the same imaginary eigenvalues of D with double multiplicity.
The rule of thumb for stability (valid for normal matrices) is [28]: the method
of lines is stable if the eigenvalues of the linearized spatial discretization operator,
scaled by∆t, lie in the stability region of the time-discretization operator.
In Fig. 4.1 we depict the stability regions for the fourth order Runge-Kutta integration
scheme (RK4), the fifth order, four step Adams-Moulton scheme (AM4)
and the fourth order, three step Adams-Moulton scheme (AM3). Along the imag-
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