  We follow the mathematical framework proposed by Bouchut and present in this
contribution a dual entropy approach for determining equilibrium states of a
lattice Boltzmann scheme. This method is expressed in terms of the dual of the
mathematical entropy relative to the underlying conservation law. It appears as
a good mathematical framework for establishing a "H-theorem" for the system of
equations with discrete velocities. The dual entropy approach is used with D1Q3
lattice Boltzmann schemes for the Burgers equation. It conducts to the
explicitation of three different equilibrium distributions of particles and
induces naturally a nonlinear stability condition. Satisfactory numerical
results for strong nonlinear shocks and rarefactions are presented. We prove
also that the dual entropy approach can be applied with a D1Q3 lattice
Boltzmann scheme for systems of linear and nonlinear acoustics and we present a
numerical result with strong nonlinear waves for nonlinear acoustics. We
establish also a negative result: with the present framework, the dual entropy
approach cannot be used for the shallow water equations.
