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1.3.3 Dispersion
Slide : [
dispersion 
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Dispersion occurs when different wavelengths propagate with different
phase velocities. Take for example a third order dispersion equation

(1.3.3#eq.1) 
The harmonic ansatz in space and time
shows that
the phase velocity

(1.3.3#eq.2) 
is larger for short wavelengths (large
propagate faster) than long
wavelengths (small
propagate slower).
In the KortewegDeVries equation, this will explain why
large amplitude solitons with short wavelengths propagate more
rapidly than low amplitudes solitons having long wavelengths.
Unfortunately, dispersion does not always have a physical origin: remember
(1.2.4#eq.3), showing how a centered finite difference approximation
underestimates first order derivatives of short wavelengths.
This is exactly what happens in the 3 levels scheme for
the advection equation, where different Fourier components included in the
initial square box function propagate with different velocities.
JBONE applet: press Start/Stop
to observe how the numerical dispersion induced from a finite
difference approximation affects the advection of a square pulse.

Numerical dispersion affects almost every approximation that will be
discussed in this course. It is therefore important to ensure that
numerical solutions reproduce the physical dispersion and not the
spurious numerical dispersion introduced by the discretisation.
SYLLABUS Previous: 1.3.2 Diffusion
Up: 1.3 Prototype problems
Next: 1.3.4 Wavebreaking
