- The initial evolution of the schemes (Fig. 28) is rather similar.
- However, with the non-conservative scheme,
the shock moves with the wrong speed
and the non-conservation of the area under is obvious.
- Remember:
the derivation of the Rankine-Hugoniot jump condition in Sect. 3.1.5
used the conservation of across the shock
to derive the shock speed.

Lax-Wendroff theorem:
If the numerical solution of a conservative scheme converges,
it converges towards a weak solution.
- Note:
an alternative to
shock capturing
where the scheme itself is able to handle discontinuities
is
shock tracking
where the positions of shocks are explicitly followed
to allow a different numerical treatment in smooth regions
and near discontinuities.