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