  The discrete effect on the boundary condition has been a fundamental topic
for the lattice Boltzmann method in simulating heat and mass transfer problems.
In previous works based on the halfway anti-bounce-back (ABB) boundary
condition for convection-diffusion equations (CDEs), it is reported that the
discrete effect cannot be commonly removed in the Bhatnagar-Gross-Krook (BGK)
model except for a special value of relaxation time. Targeting this point in
the present paper, we still proceed within the framework of BGK model for
two-dimensional CDEs, and analyze the discrete effect on a non-halfway ABB
boundary condition which incorporates the effect of the distance ratio. By
analyzing an unidirectional diffusion problem with a parabolic distribution,
the theoretical derivations with three different discrete velocity models show
that the numerical slip is a combined function of the relaxation time and the
distance ratio. Different from previous works, we definitely find that the
relaxation time can be freely adjusted by the distance ratio in a proper range
to eliminate the numerical slip. Some numerical simulations are carried out to
validate the theoretical derivations, and the numerical results for the cases
of straight and curved boundaries confirm our theoretical analysis. Finally, it
should be noted that the present analysis can be extended from the BGK model to
other lattice Boltzmann (LB) collision models for CDEs, which can broaden the
parameter range of the relaxation time to approach 0.5.
