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

In order to increase the accuracy of temporal solutions, reduce the computational cost of time marching, and improve the stability associated with collisions for the finite-volume discrete Boltzmann method, an advanced implicit Bhatnagar-Gross-Krook (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 proposed 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_{α}^{eq,n+1} by f_{α}^{eq,n} no matter where it appears. After comparing the proposed model with the existing models under different numerical conditions (e.g., different flux schemes and time-marching schemes) and using the proposed model to successfully modify the variable transformation technique, three conclusions can be drawn. First, the proposed model can improve the accuracy by almost an order of magnitude. Second, it can slightly reduce the computational cost. Therefore, the proposed scheme improves accuracy without extra cost. Finally, the proposed model can significantly improve the Δt/τ limit compared to the temporal interpolation model while having the same Δt/τ limit as the variable transformation approach. The proposed 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 developed 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.