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

Also, recall that
K1=E(η n , u n 1 ),
KK1=F(η n , u n 1 ),
u 1k1 =Ψ(η n + 0.5∆tK1, V n + 0.5∆tKK1),
K2=E(η n + 0.5∆tK1, u 1k1 ),
KK2=F(η n + 0.5∆tK1, u 1k1 ),
u 1k2 =Ψ(η n + 0.5∆tK2, V n + 0.5∆tKK2),
K3=E(η n + 0.5∆tK2, u 1k2 ),
KK3=F(η n + 0.5∆tK2, u 1k2 ),
u 1k3 =Ψ(η n +∆tK3, V n +∆tKK3),
K4=E(η n +∆tK3, u 1k3 ),
KK4=F(η n +∆tK3, u 1k3 ).
Notice that the projection from (η, V) back to u 1 must be done after each stage.
These intermediate values are denoted by u 1ki , i=1, 2, 3.
Now we will specify how to deal with the operatorΨin order to recover u 1 in
terms of V andη. Since the Fast Fourier Transform (FFT) of a Hilbert transform
is easily computed, as well as the FFT for aξ-derivative, we can go to frequency
space and solve u 1 in terms of V for the linear and constant coefficients relation
that appears in Table 4.2 for both LFM and WNFM. In Fourier space we have
̂V(k)=
(
1+ √ β ρ ( ))
2 kπ
ρ 1 l coth kπh2
û 1 (k)
lL
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