Burgers' equation

(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.

• A more careful implementation of the fully inviscid case allows us to see shock formation. Explore this in this interactive inviscid shock simulation. You can also click to introduce different initial conditions, or check out this simulation of shock wave interaction, where shocks of different profiles overtake each other due to different maximal speeds.

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