Moments Collision#

We adopt an alternative approach to perform the collision equation, instead of colliding the populations at all velocity directions, only the moments up to second order are collided. This works due to the regularization precedure implemented, which allows all populations to be reconstructed from their moments up to second-order since [1]:

(1)#\[f_{i}^{\mathrm{eq}}=w_{i}\left\{\rho + \frac{c_{i,\alpha} \left(\rho u_{\alpha}\right)}{c_{s}^{2}} + \frac{1}{2}\frac{\left(c_{i,\alpha}c_{i,\beta}-c_{s}^{2}\delta_{\alpha\beta}\right)\left(\rho u_{\alpha}u_{\beta}\right)}{c_{s}^{4}} + \frac{1}{6}\frac{\left[c_{i,\alpha}c_{i,\beta}c_{i,\gamma}-c_{s}^{2}\left(\delta_{\alpha\beta}c_{i,\gamma} + \delta_{\alpha\gamma}c_{i,\beta} + \delta_{\beta\gamma}c_{i,\alpha}\right)\right]\left[\rho u_{\alpha}u_{\beta}u_{\gamma}\right]}{c_{s}^{6}}\right\}\]

(2)#\[f_{i}^{\mathrm{neq}}=w_{i}\left\{\frac{c_{i,\alpha}}{c_{s}}\left[-\frac{\Delta t}{2 c_{s}}F_{\alpha}\right] + \frac{1}{2}\left[\frac{c_{i,\alpha}c_{i,\beta}-c_{s}^{2}\delta_{\alpha\beta}}{c_{s}^{2}}\right]\left[\frac{1}{c_{s}^{2}}\Pi_{\alpha\beta}^{\mathrm{neq}}\right] + \frac{1}{6}\left[\frac{c_{i,\alpha}c_{i,\beta}c_{i,\gamma} - c_{s}^{2}\left(\delta_{\alpha\beta}c_{i,\gamma} + \delta_{\alpha\gamma}c_{i,\beta} + \delta_{\beta\gamma}c_{i,\alpha}\right)}{c_{s}^{3}}\right]\left[\frac{u_{\gamma}\Pi_{\alpha\beta}^{\mathrm{neq}} + \left(u_{\alpha}\Pi_{\beta\gamma}^{\mathrm{neq}} + u_{\beta}\Pi_{\alpha\gamma}^{\mathrm{neq}}\right)}{c_s^{3}}\right] \right\}\]

and \(f_{i}=f_{i}^{\mathrm{eq}} + f_{i}^{\mathrm{neq}}\). Which shows that any population in a estabilished velocity set could be reconstructed from \(\rho\), \(u_{\alpha}\) and \(\Pi_{\alpha\beta}^{\mathrm{neq}}\). Being the equilibrium population moments [2]:

(3)#\[\sum_{i}f_{i}^{\mathrm{eq}}=\rho\]

(4)#\[\sum_{i}f_{i}^{\mathrm{eq}}c_{i,\alpha}=\rho u_{\alpha}\]

(5)#\[\sum_{i}f_{i}^{\mathrm{eq}}c_{i,\alpha}c_{i,\beta}=\Pi_{\alpha\beta}^{\mathrm{eq}}=\rho \left(u_{\alpha}u_{\beta} + c_{s}^{2}\delta_{\alpha\beta}\right)\]

and non-equilibrium population moments:

(6)#\[\sum_{i}f_{i}^{\mathrm{neq}}=0\]

(7)#\[\sum_{i}f_{i}^{\mathrm{neq}}c_{i,\alpha}=-\Delta t F_{\alpha}\]

(8)#\[\sum_{i}f_{i}^{\mathrm{neq}}c_{i,\alpha}c_{i,\beta}=\Pi_{\alpha\beta}^{\mathrm{neq}}\]

The collision equation \(f_{i}^{*} = \left(1 - \omega\right)f_{i}^{\mathrm{neq}} + f_i^\mathrm{eq} + \Delta t \left(1 - \frac{\omega}{2}\right) F_i\) is summated over all populations up to its second order moment, which can be arranged in the following equations:

(9)#\[\rho^{*} = \rho\]

(10)#\[u_{\alpha}^{*} = \frac{\rho u_{\alpha} + \Delta t F_{\alpha}}{\rho^{*}}\]

(11)#\[\Pi_{\alpha\beta}^{\mathrm{neq}*} = \left(1- \omega\right)\Pi_{\alpha\beta}^{\mathrm{neq}} + \left(\Pi_{\alpha\beta}^{\mathrm{eq}}- \Pi_{\alpha\beta}^{\mathrm{eq}*}\right) + \left(1 - \frac{\omega}{2}\right)\left(F_{\alpha}u_{\beta} + F_{\beta}u_{\alpha}\right)\]

where the post-collision equilibrium moment \(\Pi_{\alpha\beta}^{\mathrm{eq}*}\) can be calculated normally using \(\rho^{*}\) and \(u_{\alpha}^{*}\). From the above moments it is possible to reconstruct the post-collision equilibrium and non equilibrium populations through the regularization, and consequently the post-collision populations are given with \(f_{i}^{*}=f_{i}^{\mathrm{eq}*} + f_{i}^{\mathrm{neq}*}\). The streaming is then performed with those populations.

The main advantage of this method is that it allows to represent the state of the fluid only through moments, requiring populations to be built only for streaming. It’s also less expensive computationally to collide moments than it’s to collide populations.