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

TL;DR

This paper introduces a semi-Lagrangian implicit BGK collision model for the finite-volume discrete Boltzmann method, improving accuracy and efficiency by tracing particle distributions along characteristic paths and simplifying the evaluation process.

Contribution

The paper proposes a novel implicit BGK collision model using semi-Lagrangian tracing, enhancing accuracy and computational efficiency over existing models.

Findings

Improves accuracy by nearly an order of magnitude.

Reduces computational cost slightly.

Extends the ${ riangle}t/{ au}$ limit significantly.

Abstract

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.