  In this work, we study numerically the convergence of the scalar D2Q9 lattice
Boltzmann scheme with multiple relaxation times when the time step is
proportional to the space step and tends to zero. We do this by a combination
of theory and numerical experiment. The classical formal analysis when all the
relaxation parameters are fixed and the time step tends to zero shows that the
numerical solution converges to solutions of the heat equation, with a
constraint connecting the diffusivity, the space step and the coefficient of
relaxation of the momentum. If the diffusivity is fixed and the space step
tends to zero, the relaxation parameter for the momentum is very small, causing
a discrepency between the previous analysis and the numerical results. We
propose a new analysis of the method for this specific situation of evanescent
relaxation, based on the dispersion equation of the lattice Boltzmann scheme. A
new asymptotic partial differential equation, the damped acoustic system, is
emergent as a result of this formal analysis. Complementary numerical
experiments establish the convergence of the scalar D2Q9 lattice Boltzmann
scheme with multiple relaxation times and acoustic scaling in this specific
case of evanescent relaxation towards the numerical solution of the damped
acoustic system.
