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1.3.4 Wave-breaking

Slide : [ wave-breaking - shock-waves - solitons || VIDEO 99) echo " modem - ISDN - LAN "; else echo "login";?> ]

Wave-breaking is a non-linearity that is particularly nicely understood when surfing on a see shore, where shallow waters steepen the waves until they break. The process can be modeled theoretically from the advection equation by choosing the velocity proportional to the amplitude $ u=f$ :

$\displaystyle \frac{\partial f}{\partial t} + f\frac{\partial f}{\partial x} = 0$ (1.3.4#eq.1)

Since a local maximum (large $ f$ ) propagates faster than a local minimum (small $ f$ ), the top of the wave tries to over-take the bottom. The function soon becomes multi-valued causing the wave (and our numerical schemes) to break. Sometimes, the wave-breaking is balanced by a competing mechanism. This is the case of example in the Burger equation for shock-waves

$\displaystyle \frac{\partial f}{\partial t} + f\frac{\partial f}{\partial x} -D \frac{\partial^2 f}{\partial x^2} = 0$ (1.3.4#eq.2)

where the creation of a shock front (with short wavelengths) is physically limited by diffusion, which damps the short wavelengths (1.3.2#eq.4). Here is an example of a shock formation computed using a 2-levels explicit finite difference scheme from chapter 2.

JBONE applet:  press Start/Stop to simulate the propagation of a shock front using the Burger equation, where the wave-breaking non-linearity is balanced by a finite diffusion.
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Numerical experiments: shock waves
  1. Vary the Diffusion parameter in the interval $ D\in[0.4; 1]$ where the results are more or less satisfactory at the beginning of the simulation when the time reaches $ T=10$ . Click in the plot window to measure the propagation speed and count the density of mesh points to evaluate the steepness of the shock front as a function of the diffusion.
  2. Observe what happens if the Diffusion parameter is chosen outside the interval $ D\in[0.4; 1]$ .
  3. Without spending much effort on a topic that will be discussed further soon, try to modify the simulation parameters to perform a physically relevant simulation using a small Diffusion=0.1.

Another type of non-linear equations where the wave-breaking is balanced by dispersion leads to the Korteweg-DeVries equation for solitons

$\displaystyle \frac{\partial f}{\partial t} + f\frac{\partial f}{\partial x} +b \frac{\partial^3 f}{\partial x^3} = 0$ (1.3.4#eq.3)

The evolution below shows how large amplitudes solitons (short wavelengths) propagate faster than lower amplitudes (long wavelength), in agreement with the dispersion analysis previously performed in sect.1.3.3.

JBONE applet:  press Start/Stop to simulate the propagation of solitons using the KdV equation, where the wave-breaking non-linearity is balanced by a finite dispersion.
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Numerical experiments: solitons
  1. Study the collision that occurs when the time reaches around $ T=104$ . Monitor precisely the peak of each pulse as a function of time, to verify that solitons in fact never get superposed - even if the fast pulse ends up in front and eventually propagates independently of the slower pulse.
  2. Initialize a Gaussian pulse and observe how it decays into a so-called train of solitons that undergo multiple collisions in a periodic domain.

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