  We consider the problem of ''energy conserving'' lattice Boltzmann models. A
major difficulty observed in previous studies is the coupling between the
viscous and thermal waves even at moderate wave numbers. We propose a
theoretical framework based founded on the knowledge of the partial equivalent
equations of the lattice Boltzmann scheme at several orders of precision. With
the help of linearized models (inviscid and dissipative advective acoustics and
classical acoustics), we suggest natural sets of relations for the parameters
of lattice Boltzmann schemes. The application is proposed for three
two-dimensional schemes. Numerical test cases for simple linear and nonlinear
waves establish that the main difficulty in the previous contributions can now
be overcome.
