# Preprint 2007-014

# Energy-preserving and stable approximations for the two-dimensional shallow water equations

## Eitan Tadmor and Weigang Zhong

**Abstract:**
We present a systematic development
of energy-stable approximations
of the two-dimensional shallow water (SW) equations,
which are based on the general framework of
*entropy conservative* schemes introduced in
[E. Tadmor, Acta Numerica 12, (2003) pp. 451-512] and
[E. Tadmor and W. Zhong, JHDEs 3 (2006) pp. 529-559].
No artificial numerical viscosity is involved:
stability is dictated solely by eddy viscosity.
In particular, in the absence of any dissipative mechanism,
the resulting numerical schemes *precisely*
preserve the total energy,
which serves as an entropy function for the SW equations.
We demonstrate the dispersive nature
of such entropy conservative schemes
with a series of scalar examples,
interesting for their own sake.
We then turn to the SW equations.
Numerical experiments of the partial-dam-break problem
with energy-preserving and energy stable schemes,
successfully simulate the propagation of circular shock
and the vortices formed on the both sides of the breach.