  The stability of Lattice Boltzmann Equations modelling Shallow Water
Equations in the special case of reduced gravity is investigated theoretically.
A stability notion is defined as applied in incompressible Navier-Stokes
equations in Banda, M. K., Yong, W.- A. and Klar, A: A stability notion for
lattice Boltzmann equations. SIAM J. Sci. Comput. {\bf 27(6)}, 2098-2111
(2006). It is found that to maintain stability a careful choice of the value of
the reduced gravity must be made. The stability notion is employed to
investigate different shallow water lattice Boltzmann Equations. Results are
tested using the Lattice Boltzmann Method for various values of the governing
parameters of the flow. It is observed that even for the discrete model the
reduced gravity has a significant effect on the stability.
