Viscous Burgers' equation
Parabolic  Waves
(Viscous) Burgers’ equation:
\[\pd{u}{t} =u\pd{u}{x}+\varepsilon \pdd{u}{x}.\]
Load the interactive simulation. Locally in space, the wave is translating to the right with a speed $u$, and hence larger initial amplitudes have greater speed.

In the limit of $\varepsilon \to 0$, the solution forms discontinuous shock solutions. These can be approximated with small $\varepsilon$ (as these solutions will be smooth), though advection will cause numerical difficulties (e.g. oscillations near the front of the wave). Nonzero $\varepsilon$ leads to some loss of amplitude/height of the wave, but otherwise roughly captures the limiting shock behaviour as long as it is sufficiently small.