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

Consider the initial condition
η 0 (ξ)=0.5e −a(ξ−2π)2 /64 , ξ ∈ Π[0, 16π],
with a=50,η t (ξ, 0)=u 10ξ (ξ)=−η 0 ξ(ξ), whereΠ[0, 16π] is the interval [0, 16π]
with periodic boundary conditions. If we set u 1 initially to−η 0 (ξ) then the Riemann
invariants for the nondispersive LFM, A=η+u 1 and B=η−u 1 , will be
A(ξ, t)=0 and B(ξ, t)=η(ξ− t, 0). Therefore
η= A+ B
2
=η(ξ− t, 0)
will be a right-travelling wave as shown in Fig. 4.3, where the numerical solution
obtained with the AM4 scheme is shown until time t=39.2699. The solution for
the final time is also plotted in Fig. 4.4, the absolute error is 0.00098872, a little
less than with RK4 (0.0013). Nevertheless, with RK4 we are able to advance much
more in time, until t=151.1891, while with AM4 instabilities set for t=49.0874.
AM3 is unstable as early as t=9.8175.
Example 4.2. Another interesting example from the wave equation is that of the
fission of the wave. Take the initial condition forηas
η 0 (ξ)=0.5e −50(ξ−8π)2 /64 , ξ ∈ Π[0, 16π],
and u 10
= 0. If we set u 1 initially to zero then the Riemann invariants will be
A(ξ, t)=η(ξ+ t, 0) and B(ξ, t)=η(ξ− t, 0). Therefore
η= A+ B
2
=
η(ξ+ t, 0)+η(ξ− t, 0)
2
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