Lattice Boltzmann models for the convection-diffusion equation: D2Q5 vs D2Q9

A comparison study between the two most popular lattice Boltzmann (LB) models - D2Q5 and D2Q9 in two dimensions with five and nine discrete lattice velocities, respectively - for the convection-diffusion equation (CDE) for scalar transport is presented in this work. The local equilibria in both the discrete velocity space and the moment space are provided for the LB models examined, including the generalized D2Q5 model without high-order velocity terms and four various D2Q9 models frequently used in the literature. The boundary condition treatment is discussed with particular Dirichlet and Neumann boundary schemes presented for each LB model. In particular, the non-uniqueness in the splitting of the normal flux into three lattice directions in D2Q9 model for Neumann conditions is highlighted. Five representative numerical tests are presented to assess the performance of the D2Q5 and D2Q9 LB models. Performance metrics include the numerical accuracy and convergence orders of the interior distribution of the scalar variable ϕ and its derivatives as well as the boundary flux, boundary scalar values, and total boundary heat/mass transfer rate. The results demonstrate that (i) without the boundary effect, both the D2Q5 and D2Q9 models have second-order convergence for ϕ and its derivatives, and the D2Q5 model has smaller numerical errors at low and moderate Péclet numbers while the D2Q9 models have better accuracy at high Péclet numbers; (ii) for the CDE with specific boundary conditions on straight walls, the D2Q5 model with the interpolation based boundary schemes is able to preserve the second-order accuracy for all the quantities of interest for both Dirichlet and Neumann problems; when the D2Q9 models are applied, these boundary schemes result in second- and first-order accurate ϕ field for the Dirichlet and Neumann problems, respectively; in addition, the convergence orders for the interior derivatives and the boundary flux or boundary values are also degraded for both types of problems; (iii) when curved geometry is encountered, the D2Q5 model also shows better accuracy and/or higher convergence orders than the D2Q9 models. Clearly, D2Q5 is more appealing to D2Q9 for the CDE when the convection is not very strong and the boundary effect is significant. The same behavior in the respective 3-D numerical solutions is observed when the performance of D3Q7 and D3Q19 LB models are compared.