A simple and efficient curved boundary scheme of the lattice Boltzmann method for Robin boundary conditions of convection–diffusion equations

Abstract

This paper is concerned with boundary schemes of the lattice Boltzmann method for Robin boundary conditions of convection–diffusion equations on curved boundaries. For such boundary conditions, all the existing boundary schemes suffer from the possibility that the denominator in the scheme may become zero, which will lead to numerical instability. To avoid this possibility, we propose a boundary scheme by approximating the gradient along the outgoing discrete velocity at the boundary with the given Robin boundary condition and the gradient at the interior point next to the boundary. With this approximated gradient at the boundary, the classical bounce back scheme for Neumann-type boundary conditions is employed to obtain the unknown distribution function at the interior point. The scheme obtained has the first-order accuracy for curved boundaries and its advantages are: (1) the scheme is simple in form so that it can be easily implemented; (2) it avoids the denominator in the scheme to be zero, and (3) the scheme is single-node, i.e., it only involves the information at the present point. Numerical examples demonstrate the designed accuracy and good stability of our scheme for complex boundaries.