  A new implicit BGK collision model using a semi-Lagrangian approach is
proposed in this paper. Unlike existing models, in which the implicit BGK
collision is resolved either by a temporal extrapolation or by a variable
transformation, the new model removes the implicitness by tracing the particle
distribution functions (PDFs) back in time along their characteristic paths
during the collision process. An interpolation scheme is needed to evaluate the
PDFs at the traced-back locations. By using the first-order interpolation, the
resulting model allows for the straightforward replacement of
${f_{\alpha}}^{eq,n+1}$ by ${f_{\alpha}}^{eq,n}$ no matter where it appears.
After comparing the new model with the existing models under different
numerical conditions (e.g. different flux schemes and time marching schemes)
and using the new model to successfully modify the variable transformation
technique, three conclusions can be drawn. First, the new model can improve the
accuracy by almost an order of magnitude. Second, it can slightly reduce the
computational cost. Therefore, the new scheme improves accuracy without extra
cost. Finally, the new model can significantly improve the ${\Delta}t/{\tau}$
limit compared to the temporal interpolation model while having the same
${\Delta}t/{\tau}$ limit as the variable transformation approach. The new
scheme with a second-order interpolation is also developed and tested; however,
that technique displays no advantage over the simple first-order interpolation
approach. Both numerical and theoretical analyses are also provided to explain
why the new implicit scheme with simple first-order interpolation can
outperform the same scheme with second-order interpolation, as well as the
existing temporal extrapolation and variable transformation schemes.
