Consistent boundary conditions of the multiple-relaxation-time lattice Boltzmann method for convection-diffusion equations

In this work, the Dirichlet, Neumann and linear Robin conditions for the convection-diffusion equation (CDE) lattice Boltzmann (LB) method is investigated and a second-order boundary scheme is proposed for the D2Q9 multiple-relaxation-time (MRT) LB model. With the proposed scheme, consistent implementations are developed for the three kinds of macroscopic boundary constraints considered at both straight and curved boundaries. The second-order accuracy of the present boundary scheme is firstly demonstrated by the theoretical derivations and then confirmed by the numerical validations. Notably, the advantages of the present boundary scheme lie in its locality and consistency, i.e., no information from the neighboring fluid nodes is required in the practical treatments, and all three kinds of boundary conditions are directly implemented without degrading the Robin condition to the Dirichlet or Neumann condition.