Lattice Boltzmann simulation of some nonlinear convection–diffusion equations

In this work we proposed a lattice Boltzmann model for the nonlinear convection–diffusion equation (NCDE) with anisotropic diffusion. The constraints on the model for correctly recovering macroscopic equation are also carefully analyzed, which are ignored in some existing work. Detailed simulations of some 1D/2D NCDEs, including the nonlinear Schrödinger equation (NLSE), Buckley–Leverett equation with discontinuous initial data, NCDE with anisotropic diffusion, and generalized Zakharov system, are performed. The numerical results obtained by the proposed model agree well with the analytical solutions and/or the numerical solutions reported in previous studies. It is also found that, for complex-valued NLSE, the model using a complex distribution function is superior to that using two real distribution functions for the real and imaginary parts of the NLSE separately.