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

• T is the convolution matrix forT [0,2l] . The composition of convolution and
differentiation can be computed with the help of the Fourier transform matrix
F,
F jl = w ( j−1)(l−1) , w=e 2πi/N ,
leading to the operator with symbol matrix
⎧
⎪⎨
Λ i j =
⎪⎩
− jπ coth( )
jπh 2
l lL , i= j=1,..., N/2,
− ( j−N)π coth ( ( j−N)πh 2
l lL
0 elsewhere,
)
, i= j=N/2+1,..., N− 1,
where
T D=DT= 1 N FΛF.
Although the original expression (4.6) is nonlinear, the relation (4.7) is a linear
algebraic system to be solved for u 1 sinceηand V are already known at the current
time step n+1. So at this stage, by using a spectral matrix instead of an FFT, we
are only paying a price in complexity but not in accuracy. Table 4.4 sumarizes the
discretizations used for each model, including the discretizations for the LFM and
WNFM cases. These cases were implemented with the help of the FFT and also
using a matrix formulation as a way to validate the method for the other models.
We also implemented in Matlab two predictor-corrector schemes, one that uses
a third order, three step, explicit Adams-Bashforth solver (AB3) as predictor and
a fourth order, three step implicit Adams-Moulton (AM3) as corrector. The other
scheme uses a fourth order, four step Adams-Bashforth solver (AB4) as predictor
and a fifth order, four step implicit Adams-Moulton (AM4) as corrector.
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