Boundary Conditions for Kinetic Theory Based Models I: Lattice Boltzmann Models

Lattice Boltzmann method for general convection-diffusion equations: MRT model and boundary schemes

“…Moreover, we analyze the accuracy of the anti-bounceback scheme accompanying the present MRT LBM with the Maxwell iteration [43,44]. Based on an elegant relation of the collision matrix of the MRT model found in [45], we justify that the scheme accompanying the MRT model is second-order accurate when the boundary is located at the middle of two lattice nodes (in this case the scheme is usually called half-way anti-bounce-back scheme). To the best of our knowledge, this is the first time that the second-order accuracy of the half-way anti-bounce-back scheme is rigorously analyzed for the MRT model.…”

“…In this section, we base on the half-way anti-bounce-back scheme (3.29) to construct a family of single-node second-order boundary schemes with the approach in [45]. The construction in [45] focuses on the no-slip boundary conditions of the incompressible Navier-Stokes equations, here we extend it to the Dirichlet boundary condition of the general nonlinear CDE (2.1).…”

“…Remark 4.1. The above construction of the single-node scheme (4.39) is a direct application of that in [45]. The only difference is that here we use the anti-bounce-back scheme (3.29) as a central step while that in [45] is based on the bounce-back scheme for the no-slip boundary condition of the incompressible Navier-Stokes equations.…”

“…To the best of our knowledge, this is the first time that the second-order accuracy of the half-way anti-bounce-back scheme is rigorously analyzed for the MRT model. Furthermore, by using the half-way anti-bounce-back scheme as a central step, we construct some parameterized singlenode second-order schemes for curved boundaries with the approach in [45]. Finally, numerical experiments are conducted to validate the proposed model and boundary schemes, and demonstrate the advantages of the MRT model over the BGK model in terms of stability and accuracy.…”

“…Finally, numerical experiments are conducted to validate the proposed model and boundary schemes, and demonstrate the advantages of the MRT model over the BGK model in terms of stability and accuracy. We would like to point out that the work in [45] focuses on the no-slip boundary conditions of the incompressible Navier-Stokes equations, here we adapt it to the general nonlinear CDEs and find that the relation of the collision matrix and the construction of single-node schemes can be applied directly. However, the analysis in [45] relies on the assumption that the derivative of the fluid density is small as O(h 2 ), where h is the mesh size, while such an assumption is not required in this paper.…”