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Slide : [ dispersion || VIDEO login]
The harmonic ansatz in space and time shows that the phase velocity
is larger for short wavelengths (large propagate faster) than long wavelengths (small propagate slower). In the Korteweg-DeVries 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 under-estimates 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.
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.
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