Unexpected convergence of lattice Boltzmann schemes

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 discrepancy 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.