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

inary axis (where the eigenvalues of the linearized spatial discretization operators
lie), RK4 is less restrictive, allowing larger time steps and numerical evolution
over longer time intervals, as we will see in the experiments below. For a better
visualization, in Fig. 4.2 we compute the amplification factor|R(z)| where z=λ∆t
andλrepresents the largest (in the sense of absolute value) eigenvalue of the linearized
spatial discretization operator, see for example [2]. The AM3 scheme
has the smallest stability interval along the imaginary axis since the amplification
factor becomes greater than one very quickly. This implies a more restrictive
stability condition when using the AM3 scheme to solve the bidirectional wave
equation, for example. However, for the regularized dispersive systems considered
here, we obtained less instability since the phase velocity actually decreases
as the wavenumber grows accommodating high wave-numbers better than in the
hyperbolic case (see the dispersion relation in Section 2.3). Dispersion comes in
as a physical regularization in comparison with its underlying hyperbolic counterpart.
Still, the classical fourth order Runge-Kutta seems to be the best choice.
The spectral approximation of theξ-derivative is more accurate than the five
point formula exhibited above. However, we cannot use numerical velocity one
(∆t = ∆ξ) with it, in correspondence with the theoretical velocity, because of
stability restrictions of the method of lines already discussed. The reason for
choosing the Courant number as one is to avoid a numerical delay of the travelling
wave speed that could interfere with the expected delay that may result from the
interaction of the wave with a rapidly-varying bottom profile. This particular issue
will be investigated in the near future. For the time being, we choose the five point
formula (4.5) in all the numerical experiments presented in this chapter.
The time domain is discretized as t n = n∆t, n=0,...,T final with∆t=∆ξ.
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