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A new implicit BGK collision model using a semi-Lagrangian approach isproposed in this paper. Unlike existing models, in which the implicit BGKcollision is resolved either by a temporal extrapolation or by a variabletransformation, the new model removes the implicitness by tracing the particledistribution functions (PDFs) back in time along their characteristic pathsduring the collision process. An interpolation scheme is needed to evaluate thePDFs at the traced-back locations. By using the first-order interpolation, theresulting 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 differentnumerical conditions (e.g. different flux schemes and time marching schemes)and using the new model to successfully modify the variable transformationtechnique, three conclusions can be drawn. First, the new model can improve theaccuracy by almost an order of magnitude. Second, it can slightly reduce thecomputational cost. Therefore, the new scheme improves accuracy without extracost. 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 newscheme with a second-order interpolation is also developed and tested; however,that technique displays no advantage over the simple first-order interpolationapproach. Both numerical and theoretical analyses are also provided to explainwhy the new implicit scheme with simple first-order interpolation canoutperform the same scheme with second-order interpolation, as well as theexisting temporal extrapolation and variable transformation schemes.

Semi-Lagrangian implicit Bhatnagar-Gross-Krook collision model for the finite-volume discrete Boltzmann method