driven cavity of aspect ratio $L/H=2$ and $L/W=4$ at a Reynolds number
$\mbox{Re}=100$ computed using 3DCCM-LBM with the cubic lattice $(r=s=1)$ and
a cuboid lattice with the following three different choices o fthe grid aspect
ratios: $(r,s)=(0.406,1.3)$, $(r,s)=(0.375,0.75)$, and $(r,s)=(0.375,1.0)$.
(a) $u$ component along the $y$ direction at $x=0.5L$ and $z=0.5W$, and (b)
$v$ component along the $x$ direction at $y=0.5H$ and $z=0.5W$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: Comparison of streamline patterns along the mid-planes of a 3D
shallow lid driven cavity of aspect ratio $L/H=2$ and $L/W=4$ at a Reynolds
number $\mbox{Re}=100$ computed using 3DCCM-LBM with the cubic lattice
($(r,s)=(1.0,1.0)$) using a grid resolution of $256\times 128\times 64$ shown
in sub-figures (a), (c) and (e) and compared with the 3DCCM-LBM with the
cuboid lattice ($(r,s)=(0.406,1.3)$) using a grid resolution of $104\times
64\times 40$ shown in sub-figures (b), (d) and (f).

##  7 Numerical stability test results of raw moment and central moment based
LB algorithms on the D3Q27 cuboid lattice

The algorithmic details given in Sec. 3.2 is general and readily extends to
other collision models. For example, if instead of central moments, the raw
moments are used in performing the relaxations under collision with the same
corrections in the equilibria as before and omitting the steps involving the
mappings between the central moments and raw moments (i.e., involving
$\bm{\mathsf{\mathcal{F}}}$ and $\bm{\mathsf{\mathcal{F}}}^{-1}$), the
algorithm reduces to a 3D cuboid raw moment LBM, which is referred to as the
3DCRM-LBM in what follows. Such a 3DCRM-LBM would be significantly better than
other prior 3D cuboid LB raw moment based formulations, as it avoids the
orthogonalization of the moment basis, uses better forms of the equilibria
obtained from matching with those of the continuous Maxwell distribution
function and the elimination of the non-GI cubic velocity errors, and with
simpler expressions for the corrections and transport coefficients. However,
it would be interesting to see how the resulting 3DCRM-LBM compares with the
3DCCM-LBM based on central moments that has been used as the method of choice
in this work and validated in detail earlier. While the 3DCRM-LBM incurs a
slightly lower computational cost than the 3DCCM-LBM because of the absence of
the additional mappings indicated above, the 3DCCM-LBM, which executes the
collision step in the local moving frame of reference, is expected to be more
robust with better numerical stability characteristics in simulating shear
flows at larger characteristic velocities or lower viscosities than the 3DCRM-
LBM, as observed in the recent 2D studies involving square [31] and
rectangular [26] lattice sets. Let’s verify if this is indeed observed in 3D
simulations of the lid-driven cubic cavity flow using the cuboid lattice. In
this regard, we first perform a stability test in which, for a chosen grid
resolution (or the grid aspect ratio), and by varying the relaxation parameter
$\omega_{\nu}$ controlling the shear viscosity, we determine the maximum
possible lid velocity $U$, i.e., $U_{max}$ that maintains stable simulation
for 100,000 steps in each case. We chose two different grid resolutions of
$30\times 40\times 30$ and $30\times 60\times 30$ corresponding to the grid
ratios of $r=0.75$ and $s=1.0$ and $r=0.5$ and $s=1.0$, respectively. In both
cases, we use $c_{s}^{2}=0.08$, which is the minimum of the optimal values for
the two grid aspect ratios, in order to have the same reference speed of sound
in calculating the Mach number. All the relaxation parameters, other than that
for the shear viscosity, are set to $1.0$ for simplicity. The results are
shown in Fig. 12. Clearly, the 3DCCM-LBM is able to support larger magnitudes
of shear consistently than the 3DCRM-LBM for all the choices of the grid
resolution used.

Figure 12: Numerical stability test results showing the maximum threshold
velocity of the lid $U$ in a 3D lid driven cubic cavity flow at different
values of the relaxation parameter corresponding to the shear viscosity for
the 3DCCM-LBM (i.e., based on central moments) and 3DCRM-LBM (i.e., based on
central moments) at grid aspect ratios of $r=0.75$, $s=1.0$ (upper two
curves), and $r=0.5$, $s=1.0$ (lower two curves).

Moreover, it was pointed out earlier that in our 3D cuboid LB algorithms, the
trace of the diagonal components of the second order moments is evolved
separately from the other moments with its own relaxation parameter
$\omega_{\xi}$ after applying $\bm{\mathsf{B}}$, and the individual components
are subsequently recovered via applying $\bm{\mathsf{B}}^{-1}$, so that the
bulk viscosity can be specified independently from shear viscosity. It is
known that for the common LB schemes based on the cubic lattice sets that a
higher value of the bulk viscosity (via decreasing $\omega_{\xi}$) enhances
stability by suppressing the spurious pressure waves. We will now verify if
this also holds true for the cuboid LB formulation based on central moments.
Repeating the stability test mentioned above by choosing
$\omega_{\xi}=0.5,1.0$, and $1.4$ at $30\times 60\times 30$ and
$c_{s}^{2}=0.08$, Fig. 13 shows the resulting maximum threshold velocity of
the lid $U$ for numerical stability. This figure clarifies that the robustness
of the 3DCCM-LBM is significantly improved by choosing lower $\omega_{\xi}$ or
higher bulk viscosity.

Figure 13: Numerical stability results of the 3DCCM-LBM showing the maximum
threshold velocity of the lid $U$ in a 3D lid driven cubic cavity flow at
different values of the relaxation parameter corresponding to the shear
viscosity obtained for three different values of the relaxation parameter
controlling the bulk viscosity, i.e., $\omega_{\xi}=0.5,1.0$, and $1.4$.

In addition, we now perform a different type of stability study, where the
grid resolution and lid velocity are fixed, and the lowest possible shear
viscosity or the maximum Reynolds number that maintains stable simulations for
3DCCM-LBM and 3DCRM-LBM are determined. In this regard, we set $U=0.2$,
$c_{s}^{2}=0.08$, $r=0.5$ and $s=1.0$, and three different choices of the grid
resolutions $40\times 80\times 40$, $50\times 100\times 50$ and $90\times
180\times 90$ are used to determine the maximum Reynolds number. The central
moment formulation is found to be more stable and allows larger Reynolds
number than that due to the raw moment formulation, which again support the
superior numerical characteristics of the 3DCCM-LBM in simulating shear flows
on the cuboid lattice.

Figure 14: Comparison of the maximum Reynolds number for numerical stability
of 3DCCM-LBM and 3DCRM-LBM at different grid resolutions with cuboid grid
aspect ratios of $r=0.5$ and $s=1.0$ for a fixed lid velocity of $U=0.2$ and
$c_{s}^{2}=0.08$.

##  8 Summary and Conclusions

In this paper, we have presented a new 3D LB algorithm based on central
moments for the D3Q27 lattice using a cuboid grid, which is parameterized by
two grid aspect ratios that are related to the ratios of the particle speeds
with respect to that along a reference coordinate direction. The additional
degrees of freedom introduced by the flexible specification of the cuboid
lattice grid enables the method to naturally and more efficiently compute
flows having different characteristic length scales in different directions.
It is constructed to simulate the Navier-Stokes equations consistently via
introducing counteracting corrections to the second order moment equilibria
that eliminate the associated grid anisotropy and the non-Galilean invariant
third order velocity errors obtained via a Chapman-Enskog analysis. Such
corrections are related to the diagonal components of the velocity gradient
tensor, which are computed locally using non-equilibrium moments in our
approach and depend on the grid aspect ratios. The implementation is shown to
be compact and modular, with an interpretation based on special matrices,
admitting ready extension of the standard algorithm for the cubic lattice to
the cuboid lattice via appropriate scaling of moments based on grid aspect
ratios before and after collision step and equilibria corrections. The
resulting formulation is general in that the same grid corrections developed
for the D3Q27 lattice for recovering the correct viscous stress tensor is
applicable for other lattice subsets, and a variety of collision models,
including those based on the relaxation of raw moments, central moments and
cumulants, as well as their special case involving the distribution functions.
The cuboid central moment LBM was validated against a variety of benchmark
flows, and when used in lieu of the corresponding raw moment formulation for
simulating shear flows, we show that it results in significant improvements in
numerical stability. Finally, we demonstrated that our cuboid LB approach is
efficient in simulating anisotropic shear flow problems with significant
savings in computational cost and memory storage when compared to that based
on the cubic lattice. Future improvements are expected by extending the
present formulation as a multiblock cuboid LB algorithm, which will be
considered in a future study.

## Acknowledgements

Parts of this work were presented virtually at the SIAM Conference on
Computational Science and Engineering (CSE-21) in March 2021 and at the
International Conference on Mesoscopic Methods in Engineering Science (ICMMES
2021) in July 2021. The first author (EY) thanks the Department of Mechanical
Engineering at the University of Colorado Denver for financial support. The
second (WS) and third (KNP) authors would like to acknowledge the support of
the US National Science Foundation (NSF) under Grant CBET-1705630. The third
author (KNP) would like to also thank the NSF for support of the development
of a computer cluster infrastructure through the project “CC* Compute:
Accelerating Science and Education by Campus and Grid Computing” under Award
2019089.

##  Appendix A Mapping pre-collision distribution functions to raw moments

The intermediate raw moments with respect to the moment basis for the D3Q27
cubic lattice computed from $\mathbf{m}=\bm{\mathsf{P}}\mathbf{f}$ are
expressed in the following. While the full expressions are shown in this
appendix and all the ones that follow for better clarity, in actual
implementations they should be optimized by grouping together the common sub-
expressions when possible.

$\displaystyle k^{{}^{\prime}}_{\scriptscriptstyle 000}=f_{\scriptscriptstyle
0}+f_{\scriptscriptstyle 1}+f_{\scriptscriptstyle 2}+f_{\scriptscriptstyle
3}+f_{\scriptscriptstyle 4}+f_{\scriptscriptstyle 5}+f_{\scriptscriptstyle
6}+f_{\scriptscriptstyle 7}+f_{\scriptscriptstyle 8}+f_{\scriptscriptstyle
9}+f_{\scriptscriptstyle 10}+f_{\scriptscriptstyle 11}+f_{\scriptscriptstyle
12}+f_{\scriptscriptstyle 13}+f_{\scriptscriptstyle 14}+f_{\scriptscriptstyle
15}+f_{\scriptscriptstyle 16}+f_{\scriptscriptstyle 17}+f_{\scriptscriptstyle
18}$ $\displaystyle\quad\quad\quad\;\;\;+f_{\scriptscriptstyle
19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle 21}+f_{\scriptscriptstyle
22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle 24}+f_{\scriptscriptstyle
25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 100}=f_{\scriptscriptstyle
1}-f_{\scriptscriptstyle 2}+f_{\scriptscriptstyle 7}-f_{\scriptscriptstyle
8}+f_{\scriptscriptstyle 9}-f_{\scriptscriptstyle 10}+f_{\scriptscriptstyle
11}-f_{\scriptscriptstyle 12}+f_{\scriptscriptstyle 13}-f_{\scriptscriptstyle
14}+f_{\scriptscriptstyle 19}-f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}-f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 010}=f_{\scriptscriptstyle
3}-f_{\scriptscriptstyle 4}+f_{\scriptscriptstyle 7}+f_{\scriptscriptstyle
8}-f_{\scriptscriptstyle 9}-f_{\scriptscriptstyle 10}+f_{\scriptscriptstyle
15}-f_{\scriptscriptstyle 16}+f_{\scriptscriptstyle 17}-f_{\scriptscriptstyle
18}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle
21}-f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}-f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 001}=f_{\scriptscriptstyle
5}-f_{\scriptscriptstyle 6}+f_{\scriptscriptstyle 11}+f_{\scriptscriptstyle
12}-f_{\scriptscriptstyle 13}-f_{\scriptscriptstyle 14}+f_{\scriptscriptstyle
15}+f_{\scriptscriptstyle 16}-f_{\scriptscriptstyle 17}-f_{\scriptscriptstyle
18}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}-f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle
24}-f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 110}=f_{\scriptscriptstyle
7}-f_{\scriptscriptstyle 8}-f_{\scriptscriptstyle 9}+f_{\scriptscriptstyle
10}+f_{\scriptscriptstyle 19}-f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle
24}-f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 101}=f_{\scriptscriptstyle
11}-f_{\scriptscriptstyle 12}-f_{\scriptscriptstyle 13}+f_{\scriptscriptstyle
14}+f_{\scriptscriptstyle 19}-f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}-f_{\scriptscriptstyle 22}-f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}-f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 011}=f_{\scriptscriptstyle
15}-f_{\scriptscriptstyle 16}-f_{\scriptscriptstyle 17}+f_{\scriptscriptstyle
18}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle
21}-f_{\scriptscriptstyle 22}-f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 200}=f_{\scriptscriptstyle
1}+f_{\scriptscriptstyle 2}+f_{\scriptscriptstyle 7}+f_{\scriptscriptstyle
8}+f_{\scriptscriptstyle 9}+f_{\scriptscriptstyle 10}+f_{\scriptscriptstyle
11}+f_{\scriptscriptstyle 12}+f_{\scriptscriptstyle 13}+f_{\scriptscriptstyle
14}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 020}=f_{\scriptscriptstyle
3}+f_{\scriptscriptstyle 4}+f_{\scriptscriptstyle 7}+f_{\scriptscriptstyle
8}+f_{\scriptscriptstyle 9}+f_{\scriptscriptstyle 10}+f_{\scriptscriptstyle
15}+f_{\scriptscriptstyle 16}+f_{\scriptscriptstyle 17}+f_{\scriptscriptstyle
18}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 002}=f_{\scriptscriptstyle
5}+f_{\scriptscriptstyle 6}+f_{\scriptscriptstyle 11}+f_{\scriptscriptstyle
12}+f_{\scriptscriptstyle 13}+f_{\scriptscriptstyle 14}+f_{\scriptscriptstyle
15}+f_{\scriptscriptstyle 16}+f_{\scriptscriptstyle 17}+f_{\scriptscriptstyle
18}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 120}=f_{\scriptscriptstyle
7}-f_{\scriptscriptstyle 8}+f_{\scriptscriptstyle 9}-f_{\scriptscriptstyle
10}+f_{\scriptscriptstyle 19}-f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}-f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 102}=f_{\scriptscriptstyle
11}-f_{\scriptscriptstyle 12}+f_{\scriptscriptstyle 13}-f_{\scriptscriptstyle
14}+f_{\scriptscriptstyle 19}-f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}-f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 210}=f_{\scriptscriptstyle
7}+f_{\scriptscriptstyle 8}-f_{\scriptscriptstyle 9}-f_{\scriptscriptstyle
10}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle
21}-f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}-f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 012}=f_{\scriptscriptstyle
15}-f_{\scriptscriptstyle 16}+f_{\scriptscriptstyle 17}-f_{\scriptscriptstyle
18}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle
21}-f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}-f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 201}=f_{\scriptscriptstyle
11}+f_{\scriptscriptstyle 12}-f_{\scriptscriptstyle 13}-f_{\scriptscriptstyle
14}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}-f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle
24}-f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 021}=f_{\scriptscriptstyle
15}+f_{\scriptscriptstyle 16}-f_{\scriptscriptstyle 17}-f_{\scriptscriptstyle
18}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}-f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle
24}-f_{\scriptscriptstyle 25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 111}=f_{\scriptscriptstyle
19}-f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle 21}+f_{\scriptscriptstyle
22}-f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle 24}+f_{\scriptscriptstyle
25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 220}=f_{\scriptscriptstyle
7}+f_{\scriptscriptstyle 8}+f_{\scriptscriptstyle 9}+f_{\scriptscriptstyle
10}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 202}=f_{\scriptscriptstyle
11}+f_{\scriptscriptstyle 12}+f_{\scriptscriptstyle 13}+f_{\scriptscriptstyle
14}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 022}=f_{\scriptscriptstyle
15}+f_{\scriptscriptstyle 16}+f_{\scriptscriptstyle 17}+f_{\scriptscriptstyle
18}+f_{\scriptscriptstyle 19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle
21}+f_{\scriptscriptstyle 22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle
24}+f_{\scriptscriptstyle 25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 211}=f_{\scriptscriptstyle
19}+f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle 21}-f_{\scriptscriptstyle
22}-f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle 24}+f_{\scriptscriptstyle
25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 121}=f_{\scriptscriptstyle
19}-f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle 21}-f_{\scriptscriptstyle
22}-f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle 24}-f_{\scriptscriptstyle
25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 112}=f_{\scriptscriptstyle
19}-f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle 21}+f_{\scriptscriptstyle
22}+f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle 24}-f_{\scriptscriptstyle
25}+f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 122}=f_{\scriptscriptstyle
19}-f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle 21}-f_{\scriptscriptstyle
22}+f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle 24}+f_{\scriptscriptstyle
25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 212}=f_{\scriptscriptstyle
19}+f_{\scriptscriptstyle 20}-f_{\scriptscriptstyle 21}-f_{\scriptscriptstyle
22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle 24}-f_{\scriptscriptstyle
25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 221}=f_{\scriptscriptstyle
19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle 21}+f_{\scriptscriptstyle
22}-f_{\scriptscriptstyle 23}-f_{\scriptscriptstyle 24}-f_{\scriptscriptstyle
25}-f_{\scriptscriptstyle 26},$ $\displaystyle
k^{{}^{\prime}}_{\scriptscriptstyle 222}=f_{\scriptscriptstyle
19}+f_{\scriptscriptstyle 20}+f_{\scriptscriptstyle 21}+f_{\scriptscriptstyle
22}+f_{\scriptscriptstyle 23}+f_{\scriptscriptstyle 24}+f_{\scriptscriptstyle
25}+f_{\scriptscriptstyle 26}.$

##  Appendix B Scaling pre-collision raw moments by grid aspect ratios

In order to obtain the pre-collision raw moments for the cuboid lattice from
those for the cubic lattice, we scale the elements of the latter, i.e.,
$k_{mnp}^{\prime}$ by the grid aspect ratios using the factor $r^{n}s^{p}$ for
the moment of order $(m+n+p)$. Thus, the operations involving
$\mathbf{m}\leftarrow\bm{\mathsf{S}}\mathbf{m}$ can be written as follows:

$\begin{matrix}k^{{}^{\prime}}_{\scriptscriptstyle
000}=k^{{}^{\prime}}_{\scriptscriptstyle
000}&k^{{}^{\prime}}_{\scriptscriptstyle
100}=k^{{}^{\prime}}_{\scriptscriptstyle
100}&k^{{}^{\prime}}_{\scriptscriptstyle
010}=rk^{{}^{\prime}}_{\scriptscriptstyle 010},&&&\\\
k^{{}^{\prime}}_{\scriptscriptstyle 001}=sk^{{}^{\prime}}_{\scriptscriptstyle
001}&k^{{}^{\prime}}_{\scriptscriptstyle
200}=k^{{}^{\prime}}_{\scriptscriptstyle
200}&k^{{}^{\prime}}_{\scriptscriptstyle
102}=s^{2}k^{{}^{\prime}}_{\scriptscriptstyle
102}&k^{{}^{\prime}}_{\scriptscriptstyle
021}=r^{2}sk^{{}^{\prime}}_{\scriptscriptstyle
021}&k^{{}^{\prime}}_{\scriptscriptstyle
022}=r^{2}s^{2}k^{{}^{\prime}}_{\scriptscriptstyle
022}&k^{{}^{\prime}}_{\scriptscriptstyle
122}=r^{2}s^{2}k^{{}^{\prime}}_{\scriptscriptstyle 122},\\\
k^{{}^{\prime}}_{\scriptscriptstyle 110}=rk^{{}^{\prime}}_{\scriptscriptstyle
110}&k^{{}^{\prime}}_{\scriptscriptstyle
020}=r^{2}k^{{}^{\prime}}_{\scriptscriptstyle
020}&k^{{}^{\prime}}_{\scriptscriptstyle
210}=rk^{{}^{\prime}}_{\scriptscriptstyle
210}&k^{{}^{\prime}}_{\scriptscriptstyle
111}=rsk^{{}^{\prime}}_{\scriptscriptstyle
111}&k^{{}^{\prime}}_{\scriptscriptstyle
211}=rsk^{{}^{\prime}}_{\scriptscriptstyle
211}&k^{{}^{\prime}}_{\scriptscriptstyle
212}=rs^{2}k^{{}^{\prime}}_{\scriptscriptstyle 212},\\\
k^{{}^{\prime}}_{\scriptscriptstyle 101}=sk^{{}^{\prime}}_{\scriptscriptstyle
101}&k^{{}^{\prime}}_{\scriptscriptstyle
002}=s^{2}k^{{}^{\prime}}_{\scriptscriptstyle
002}&k^{{}^{\prime}}_{\scriptscriptstyle
012}=rs^{2}k^{{}^{\prime}}_{\scriptscriptstyle
012}&k^{{}^{\prime}}_{\scriptscriptstyle
220}=r^{2}k^{{}^{\prime}}_{\scriptscriptstyle
220}&k^{{}^{\prime}}_{\scriptscriptstyle
121}=r^{2}sk^{{}^{\prime}}_{\scriptscriptstyle
121}&k^{{}^{\prime}}_{\scriptscriptstyle
221}=r^{2}sk^{{}^{\prime}}_{\scriptscriptstyle 221},\\\
k^{{}^{\prime}}_{\scriptscriptstyle 011}=rsk^{{}^{\prime}}_{\scriptscriptstyle
011}&k^{{}^{\prime}}_{\scriptscriptstyle
120}=r^{2}k^{{}^{\prime}}_{\scriptscriptstyle
120}&k^{{}^{\prime}}_{\scriptscriptstyle
201}=sk^{{}^{\prime}}_{\scriptscriptstyle
201}&k^{{}^{\prime}}_{\scriptscriptstyle
202}=s^{2}k^{{}^{\prime}}_{\scriptscriptstyle
202}&k^{{}^{\prime}}_{\scriptscriptstyle
112}=rs^{2}k^{{}^{\prime}}_{\scriptscriptstyle
112}&k^{{}^{\prime}}_{\scriptscriptstyle
222}=r^{2}s^{2}k^{{}^{\prime}}_{\scriptscriptstyle 222}.\end{matrix}$

##  Appendix C Mapping pre-collision raw moments to central moments

By writing the central moments for all the components for the D3Q27 lattice in
terms of the raw moments via the binomial expansions, represented by
$\mathbf{m}^{c}=\bm{\mathsf{\mathcal{F}}}\mathbf{m}$, the following
expressions are obtained:

$k_{\scriptscriptstyle 000}=k^{{}^{\prime}}_{\scriptscriptstyle 000},$
$k_{\scriptscriptstyle 100}=k^{{}^{\prime}}_{\scriptscriptstyle
100}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $k_{\scriptscriptstyle
010}=k^{{}^{\prime}}_{\scriptscriptstyle
010}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $k_{\scriptscriptstyle
001}=k^{{}^{\prime}}_{\scriptscriptstyle
001}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $k_{\scriptscriptstyle
110}=k^{{}^{\prime}}_{\scriptscriptstyle
110}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
010}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
100}+u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle 000},$
$k_{\scriptscriptstyle 101}=k^{{}^{\prime}}_{\scriptscriptstyle
101}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
001}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
100}+u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 000},$
$k_{\scriptscriptstyle 011}=k^{{}^{\prime}}_{\scriptscriptstyle
011}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
001}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
010}+u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 000},$
$k_{\scriptscriptstyle 200}=k^{{}^{\prime}}_{\scriptscriptstyle
200}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle 100}+u_{x}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $k_{\scriptscriptstyle
020}=k^{{}^{\prime}}_{\scriptscriptstyle
020}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle 010}+u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $k_{\scriptscriptstyle
002}=k^{{}^{\prime}}_{\scriptscriptstyle
002}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 001}+u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $k_{\scriptscriptstyle
120}=k^{{}^{\prime}}_{\scriptscriptstyle
120}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
020}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
110}+2u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
010}+u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
100}-u_{x}u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $k_{\scriptscriptstyle 102}=k^{{}^{\prime}}_{\scriptscriptstyle
102}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
002}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
101}+2u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
001}+u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
100}-u_{x}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $k_{\scriptscriptstyle 210}=k^{{}^{\prime}}_{\scriptscriptstyle
210}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
200}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle 110}+u_{x}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
010}+2u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
100}-u_{x}^{\scriptscriptstyle 2}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $k_{\scriptscriptstyle 012}=k^{{}^{\prime}}_{\scriptscriptstyle
012}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
002}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 011}+u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
010}+2u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
001}-u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $k_{\scriptscriptstyle 201}=k^{{}^{\prime}}_{\scriptscriptstyle
201}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
200}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle 101}+u_{x}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
001}+2u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
100}-u_{x}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $k_{\scriptscriptstyle 021}=k^{{}^{\prime}}_{\scriptscriptstyle
021}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
020}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle 011}+u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
001}+2u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
010}-u_{y}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $k_{\scriptscriptstyle 111}=k^{{}^{\prime}}_{\scriptscriptstyle
111}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
011}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
101}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
110}+u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
001}+u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
010}+u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
100}-u_{x}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 000},$
$k_{\scriptscriptstyle 220}=k^{{}^{\prime}}_{\scriptscriptstyle
220}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
210}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle 120}+u_{x}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 020}+u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
200}+4u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
110}-2u_{x}^{\scriptscriptstyle 2}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
010}-2u_{x}u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $k_{\scriptscriptstyle
202}=k^{{}^{\prime}}_{\scriptscriptstyle
202}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
201}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle 102}+u_{x}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 002}+u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
200}+4u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
101}-2u_{x}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
001}-2u_{x}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
100}u_{x}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $k_{\scriptscriptstyle
022}=k^{{}^{\prime}}_{\scriptscriptstyle
022}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
021}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle 012}+u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 020}+u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
002}+4u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
011}-2u_{y}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
001}-2u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
010}+u_{y}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $\displaystyle
k_{\scriptscriptstyle 211}$ $\displaystyle=k^{{}^{\prime}}_{\scriptscriptstyle
211}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
111}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
201}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 210}+u_{x}^{\scriptscriptstyle
2}k_{\scriptscriptstyle 011}+2u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
101}+u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
200}+2u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;-u_{x}^{\scriptscriptstyle
2}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle 001}-u_{x}^{\scriptscriptstyle
2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
010}-2u_{x}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $\displaystyle k_{\scriptscriptstyle 121}$
$\displaystyle=k^{{}^{\prime}}_{\scriptscriptstyle
121}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
111}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
021}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
120}+u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
020}+2u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
011}+u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
101}+2u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;-u_{x}u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
001}-2u_{x}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
010}-u_{y}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
100}+u_{x}u_{y}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $\displaystyle k_{\scriptscriptstyle 112}$
$\displaystyle=k^{{}^{\prime}}_{\scriptscriptstyle
112}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
111}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
012}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
102}+u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
002}+2u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
011}+2u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
101}+u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;-2u_{x}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
001}-u_{x}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
010}-u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
100}+u_{x}u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
000},$ $\displaystyle k_{\scriptscriptstyle 122}$
$\displaystyle=k^{{}^{\prime}}_{\scriptscriptstyle
122}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
112}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
121}-u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
022}+4u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
111}+2u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
021}+2u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
012}+u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle 102}$
$\displaystyle\;\;\;\;+u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 120}-u_{x}u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 002}-u_{x}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle
020}-4u_{x}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
011}-2u_{y}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
101}-2u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
110}$ $\displaystyle\;\;\;\;+2u_{x}u_{y}^{\scriptscriptstyle
2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
001}+2u_{x}u_{y}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 010}+u_{y}^{\scriptscriptstyle
2}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
100}-u_{x}u_{y}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 000},$ $\displaystyle
k_{\scriptscriptstyle 212}$ $\displaystyle=k^{{}^{\prime}}_{\scriptscriptstyle
212}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
112}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
211}-u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
202}+4u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
111}+2u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
201}+u_{x}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
012}+u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle 210}$
$\displaystyle\;\;\;\;+2u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
102}-u_{x}^{\scriptscriptstyle 2}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
002}-u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
200}-2u_{x}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
011}-4u_{x}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
101}-2u_{x}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
110}$ $\displaystyle\;\;\;\;+2u_{x}^{\scriptscriptstyle
2}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 001}+u_{x}^{\scriptscriptstyle
2}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
010}+2u_{x}u_{y}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 100}-u_{x}^{\scriptscriptstyle
2}u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle 000},$
$\displaystyle k_{\scriptscriptstyle 221}$
$\displaystyle=k^{{}^{\prime}}_{\scriptscriptstyle
221}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
121}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
211}-u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
220}+4u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
111}+u_{x}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
021}+u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
201}+2u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 210}$
$\displaystyle\;\;\;\;+2u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
120}-u_{x}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
020}-u_{y}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
200}-2u_{x}^{\scriptscriptstyle 2}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
011}-2u_{x}u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
101}-4u_{x}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;+u_{x}^{\scriptscriptstyle 2}u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 001}+2u_{x}^{\scriptscriptstyle
2}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
010}+2u_{x}u_{y}^{\scriptscriptstyle
2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 100}-u_{x}^{\scriptscriptstyle
2}u_{y}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 000},$
$\displaystyle k_{\scriptscriptstyle 222}$
$\displaystyle=k^{{}^{\prime}}_{\scriptscriptstyle
222}-2u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
221}-2u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
212}-2u_{x}k^{{}^{\prime}}_{\scriptscriptstyle
122}+4u_{x}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
112}+4u_{x}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 121}$
$\displaystyle\;\;\;\;+4u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
211}+u_{x}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
022}+u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
202}+u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
220}-8u_{x}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
111}-2u_{x}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
021}$ $\displaystyle\;\;\;\;-2u_{y}^{\scriptscriptstyle
2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 201}-2u_{x}^{\scriptscriptstyle
2}u_{y}k^{{}^{\prime}}_{\scriptscriptstyle
012}-2u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
210}-2u_{x}u_{y}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
102}-2u_{x}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
120}+u_{x}^{\scriptscriptstyle 2}u_{y}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 002}$
$\displaystyle\;\;\;\;+u_{x}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 020}+u_{y}^{\scriptscriptstyle
2}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
200}+4u_{x}^{\scriptscriptstyle
2}u_{y}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
011}+4u_{x}u_{y}^{\scriptscriptstyle
2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle
101}+4u_{x}u_{y}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 110}-2u_{x}^{\scriptscriptstyle
2}u_{y}^{\scriptscriptstyle 2}u_{z}k^{{}^{\prime}}_{\scriptscriptstyle 001}$
$\displaystyle\;\;\;\;-2u_{x}^{\scriptscriptstyle
2}u_{y}u_{z}^{\scriptscriptstyle 2}k^{{}^{\prime}}_{\scriptscriptstyle
010}-2u_{x}u_{y}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 100}+u_{x}^{\scriptscriptstyle
2}u_{y}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}k^{{}^{\prime}}_{\scriptscriptstyle 000}.$

##  Appendix D Compute post-collision central moments via relaxations under
collision using extended equilibria for cuboid lattice

First, reflecting the original moment basis in Eq. (4), we apply
$\bm{\mathsf{B}}$ and combine the selected pre-collision central moments of
2nd, 3rd and 4th orders as follows:

$\displaystyle\;\;k_{2s}=k_{200}+k_{020}+k_{002},\quad
k_{2d1}=k_{200}-k_{020},\quad\quad k_{2d2}=k_{200}-k_{002}$ $\displaystyle
k_{3s1}=k_{120}+k_{102},\quad k_{3m1}=k_{120}-k_{102},$ $\displaystyle
k_{3s2}=k_{210}+k_{012},\quad k_{3m2}=k_{210}-k_{012},$ $\displaystyle
k_{3s3}=k_{201}+k_{021},\quad k_{3m3}=k_{201}-k_{021},$ $\displaystyle
k_{4s}=k_{220}+k_{202}+k_{022},\quad
k_{4d1}=k_{220}+k_{202}-k_{022},\quad\quad k_{4d2}=k_{220}-k_{202}.$ (78)

Next, from Sec. 2, we can write the effective equilibrium central moments as
follows:

$\displaystyle
k_{000}^{eq}=\rho,\quad\;\;k_{100}^{eq}=k_{010}^{eq}=k_{001}^{eq}=0,\quad\;\;k_{110}^{eq}=k_{101}^{eq}=k_{011}^{eq}=0,$
$\displaystyle k_{2s}^{eq}$ $\displaystyle=$ $\displaystyle
k_{200}^{eq}+k_{020}^{eq}+k_{002}^{eq}=3c_{s}^{2}\rho+\Big{(}\theta_{bx}\partial_{x}u_{x}+\theta_{by}\partial_{y}u_{y}+\theta_{bz}\partial_{z}u_{z}+\lambda_{bx}\partial_{x}\rho+\lambda_{by}\partial_{y}\rho+\lambda_{bz}\partial_{z}\rho\Big{)}\Delta
t,$ $\displaystyle k_{2d1}^{eq}$ $\displaystyle=$ $\displaystyle
k_{200}^{eq}-k_{020}^{eq}=\Big{(}\theta_{sx}\partial_{x}u_{x}-\theta_{sy}\partial_{y}u_{y}+\lambda_{sx}\partial_{x}\rho+\lambda_{sy}\partial_{y}\rho\Big{)}\Delta
t,$ $\displaystyle k_{2d2}^{eq}$ $\displaystyle=$ $\displaystyle
k_{200}^{eq}-k_{002}^{eq}=\Big{(}\theta_{sx}\partial_{x}u_{x}-\theta_{sz}\partial_{z}u_{z}+\lambda_{sx}\partial_{x}\rho+\lambda_{sz}\partial_{z}\rho\Big{)}\Delta
t,$
$\displaystyle\begin{matrix}k_{3s1}^{eq}=k_{120}^{eq}+k_{102}^{eq}=0,&k_{3m1}^{eq}=k_{120}^{eq}-k_{102}^{eq}=0,&\\\
k_{3s2}^{eq}=k_{210}^{eq}+k_{012}^{eq}=0,&k_{3m2}^{eq}=k_{210}^{eq}-k_{012}^{eq}=0,&\\\
k_{3s3}^{eq}=k_{201}^{eq}+k_{021}^{eq}=0,&k_{3m3}^{eq}=k_{201}^{eq}-k_{021}^{eq}=0,&k_{111}^{eq}=0,\\\
k_{211}^{eq}=k_{121}^{eq}=k_{112}^{eq}=0,&k_{122}^{eq}=k_{212}^{eq}=k_{221}^{eq}=0,&k_{222}^{eq}=c_{s}^{6}\rho,\\\
k_{4s}^{eq}=k_{220}^{eq}+k_{202}^{eq}+k_{022}^{eq}=3c_{s}^{4}\rho,&k_{4d1}^{eq}=k_{220}^{eq}+k_{202}^{eq}-k_{022}^{eq}=c_{s}^{4}\rho,&k_{4d2}^{eq}=k_{220}^{eq}-k_{202}^{eq}=0.\\\
\end{matrix}$ (79)

Note that in Eq. (79), we have extended the combinations of the diagonal parts
of the second order moments by including the corrections due to grid
anisotropy for the cuboid grid and non-GI velocity errors due to aliasing on
the D3Q27 lattice using the results from Sec. 2.5. Moreover, in a slight
economy for the notations, in the following, we will introduce the subscript
‘b’ for the coefficients related to corrections associated with bulk viscosity
and the subscript ‘s’ for those other components influencing the shear
viscosity (rather than the numerical subscripts used in Sec. 2.5):

$\displaystyle\begin{matrix}\theta_{sx}=-\rho\left(3u_{x}^{2}+3c_{s}^{2}-1\right)\left(\dfrac{1}{\omega_{\nu}}-\dfrac{1}{2}\right),&\quad\quad\lambda_{sx}=-\left(3c_{s}^{2}-1\right)\left(\dfrac{1}{\omega_{\nu}}-\dfrac{1}{2}\right)u_{x},\\\
\theta_{bx}=-\rho\left(3u_{x}^{2}+3c_{s}^{2}-1\right)\left(\dfrac{1}{\omega_{\xi}}-\dfrac{1}{2}\right),&\quad\quad\lambda_{bx}=-\left(3c_{s}^{2}-1\right)\left(\dfrac{1}{\omega_{\xi}}-\dfrac{1}{2}\right)u_{x},\\\
\theta_{sy}=-\rho\left(3u_{y}^{2}+3c_{s}^{2}-r^{2}\right)]\left(\dfrac{1}{\omega_{\nu}}-\dfrac{1}{2}\right),&\quad\quad\lambda_{sy}=+(3c_{s}^{2}-r^{2})\left(\dfrac{1}{\omega_{\nu}}-\dfrac{1}{2}\right)u_{y},\\\
\theta_{by}=-\rho\left(3u_{y}^{2}+3c_{s}^{2}-r^{2}\right)\left(\dfrac{1}{\omega_{\xi}}-\dfrac{1}{2}\right),&\quad\quad\lambda_{by}=-(3c_{s}^{2}-r^{2})\left(\dfrac{1}{\omega_{\xi}}-\dfrac{1}{2}\right)u_{y},\\\
\theta_{sz}=-\rho\left(3u_{z}^{2}+3c_{s}^{2}-s^{2}\right)\left(\dfrac{1}{\omega_{\nu}}-\dfrac{1}{2}\right),&\quad\quad\lambda_{sz}=+(3c_{s}^{2}-s^{2})\left(\dfrac{1}{\omega_{\nu}}-\dfrac{1}{2}\right)u_{z},\\\
\theta_{bz}=-\rho\left(3u_{z}^{2}+3c_{s}^{2}-s^{2}\right)\left(\dfrac{1}{\omega_{\xi}}-\dfrac{1}{2}\right),&\quad\quad\lambda_{bz}=-(3c_{s}^{2}-s^{2})\left(\dfrac{1}{\omega_{\xi}}-\dfrac{1}{2}\right)u_{z}.\\\
\\\ \end{matrix}$

Here, $\omega_{\xi}$ is the relaxation parameter associated with the trace of
the second order moments $k_{2s}$ and $\omega_{\nu}$ for the other two
diagonal components $k_{2d1}$ and $k_{2d2}$ defined in Eq. (D) (as well as the
off-diagonal components $k_{110}$, $k_{101}$ and $k_{011}$). As we will show
later that this would enable independent specification of the bulk viscosity
from the shear viscosity. For the associated equilibria in Eq. (79), the
density gradients $\partial_{x}\rho$, $\partial_{y}\rho$ and
$\partial_{z}\rho$ will be obtained from a second order (isotropic) finite
difference scheme, while the diagonal components of the velocity gradient
tensor $\partial_{x}u_{x}$, $\partial_{y}u_{y}$ and $\partial_{z}u_{z}$ will
be computed locally from non-equilibrium moments (using the results in Eqs.
(41)-(46) derived in Sec. 2.5) as follows:

Defining

$\displaystyle A=\frac{1}{2}(3c_{s}^{2}-1)u_{x},\quad\quad
B=\frac{1}{2}(3c_{s}^{2}-r^{2})u_{y},\quad\quad
C=\frac{1}{2}(3c_{s}^{2}-s^{2})u_{z},$ $\displaystyle
e_{sr1}=-A\;\partial_{x}\rho+B\;\partial_{y}\rho,\quad\quad
e_{sr2}=-A\;\partial_{x}\rho+C\;\partial_{z}\rho,\quad\quad
e_{br}=-A\;\partial_{x}\rho-B\;\partial_{y}\rho-C\;\partial_{z}\rho$
$\displaystyle R_{b}$ $\displaystyle=$
$\displaystyle\left({k_{200}}+{k_{020}}+{k_{002}}\right)-3c_{s}^{2}\rho+e_{br}=k_{2s}-3c_{s}^{2}\rho+e_{br},$
$\displaystyle R_{s1}$ $\displaystyle=$
$\displaystyle\left({k_{200}}-{k_{020}}\right)+e_{sr1}=k_{2d1}+e_{sr1},$
$\displaystyle R_{s2}$ $\displaystyle=$
$\displaystyle\left({k_{200}}-{k_{002}}\right)+e_{sr2}=k_{2d2}+e_{sr2}.$
$\displaystyle
C_{sx}=\left[-\dfrac{2c_{s}^{2}}{\omega_{\nu}}+\dfrac{\left(3c_{s}^{2}-1\right)}{2}+\dfrac{3u_{x}^{2}}{2}\right]\rho,$
$\displaystyle
C_{bx}=\left[-\dfrac{2c_{s}^{2}}{\omega_{\xi}}+\dfrac{\left(3c_{s}^{2}-1\right)}{2}+\dfrac{3u_{x}^{2}}{2}\right]\rho,$
$\displaystyle
C_{sy}=\left[\dfrac{2c_{s}^{2}}{\omega_{\nu}}-\dfrac{\left(3c_{s}^{2}-r^{2}\right)}{2}-\dfrac{3u_{y}^{2}}{2}\right]\rho,$
$\displaystyle
C_{by}=\left[-\dfrac{2c_{s}^{2}}{\omega_{\xi}}+\dfrac{\left(3c_{s}^{2}-r^{2}\right)}{2}+\dfrac{3u_{y}^{2}}{2}\right]\rho,$
$\displaystyle
C_{sz}=\left[\dfrac{2c_{s}^{2}}{\omega_{\nu}}-\dfrac{\left(3c_{s}^{2}-s^{2}\right)}{2}-\dfrac{3u_{z}^{2}}{2}\right]\rho,$
$\displaystyle
C_{bz}=\left[-\dfrac{2c_{s}^{2}}{\omega_{\xi}}+\dfrac{\left(3c_{s}^{2}-s^{2}\right)}{2}+\dfrac{3u_{z}^{2}}{2}\right]\rho,$

we can then write the local expressions for the velocity gradients as

$\displaystyle\partial_{x}u_{x}={\cfrac{\left[-C_{sz}C_{by}R_{s1}-C_{sy}\left(C_{bz}R_{s2}-C_{sz}R_{b}\right)\right]}{\left[-C_{sx}C_{sz}C_{by}-C_{sy}\left(C_{sx}C_{bz}-C_{sz}C_{bx}\right)\right]}},$
$\displaystyle\partial_{y}u_{y}=\cfrac{1}{C_{sy}}\left[R_{s1}-C_{sx}\;\partial_{x}u_{x}\right],\;\;\;\;\;\;\partial_{z}u_{z}=\cfrac{1}{C_{sz}}\left[R_{s2}-C_{sx}\;\partial_{x}u_{x}\right].$

Based on these quantities, we can then prescribe the post-collision central
moments via relaxations under collision as follows:

$\displaystyle\tilde{k}_{000}$ $\displaystyle=$ $\displaystyle k_{000},$
$\displaystyle\tilde{k}_{100}$ $\displaystyle=$ $\displaystyle
k_{100}+\omega_{1}(k_{100}^{eq}-k_{100})+(1-\omega_{1}/2)\sigma_{100}\Delta
t,$ $\displaystyle\tilde{k}_{010}$ $\displaystyle=$ $\displaystyle
k_{010}+\omega_{1}(k_{010}^{eq}-k_{010})+(1-\omega_{1}/2)\sigma_{010}\Delta
t,$ $\displaystyle\tilde{k}_{001}$ $\displaystyle=$ $\displaystyle
k_{001}+\omega_{1}(k_{001}^{eq}-k_{001})+(1-\omega_{1}/2)\sigma_{001}\Delta
t,$ $\displaystyle\tilde{k}_{110}$ $\displaystyle=$ $\displaystyle
k_{110}+\omega_{\nu}(k_{110}^{eq}-k_{110}),$ $\displaystyle\tilde{k}_{101}$
$\displaystyle=$ $\displaystyle k_{101}+\omega_{\nu}(k_{101}^{eq}-k_{101}),$
$\displaystyle\tilde{k}_{011}$ $\displaystyle=$ $\displaystyle
k_{011}+\omega_{\nu}(k_{011}^{eq}-k_{011}),$ $\displaystyle\tilde{k}_{2d1}$
$\displaystyle=$ $\displaystyle k_{2d1}+\omega_{\nu}(k_{2d1}^{eq}-k_{2d1}),$
$\displaystyle\tilde{k}_{2d2}$ $\displaystyle=$ $\displaystyle
k_{2d2}+\omega_{\nu}(k_{2d2}^{eq}-k_{2d2}),$ $\displaystyle\tilde{k}_{2s}$
$\displaystyle=$ $\displaystyle k_{2s}+\omega_{\xi}(k_{2s}^{eq}-k_{2s}),$
$\displaystyle\tilde{k}_{3s1}$ $\displaystyle=$ $\displaystyle
k_{3s1}+\omega_{3s1}(k_{3s1}^{eq}-k_{3s1}),$ $\displaystyle\tilde{k}_{3s2}$
$\displaystyle=$ $\displaystyle k_{3s2}+\omega_{3s2}(k_{3s2}^{eq}-k_{3s2}),$
$\displaystyle\tilde{k}_{3s3}$ $\displaystyle=$ $\displaystyle
k_{3s3}+\omega_{3s3}(k_{3s3}^{eq}-k_{3s3}),$ $\displaystyle\tilde{k}_{3m1}$
$\displaystyle=$ $\displaystyle k_{3m1}+\omega_{3m1}(k_{3m1}^{eq}-k_{3m1}),$
$\displaystyle\tilde{k}_{3m2}$ $\displaystyle=$ $\displaystyle
k_{3m2}+\omega_{3m2}(k_{3m2}^{eq}-k_{3m2}),$ $\displaystyle\tilde{k}_{3m3}$
$\displaystyle=$ $\displaystyle k_{3m3}+\omega_{3m3}(k_{3m3}^{eq}-k_{3m3}),$
$\displaystyle\tilde{k}_{111}$ $\displaystyle=$ $\displaystyle
k_{111}+\omega_{111}(k_{111}^{eq}-k_{111}),$ $\displaystyle\tilde{k}_{211}$
$\displaystyle=$ $\displaystyle k_{211}+\omega_{211}(k_{211}^{eq}-k_{211}),$
$\displaystyle\tilde{k}_{121}$ $\displaystyle=$ $\displaystyle
k_{121}+\omega_{121}(k_{121}^{eq}-k_{121}),$ $\displaystyle\tilde{k}_{112}$
$\displaystyle=$ $\displaystyle k_{112}+\omega_{112}(k_{112}^{eq}-k_{112}),$
$\displaystyle\tilde{k}_{4s}$ $\displaystyle=$ $\displaystyle
k_{4s}+\omega_{4s}(k_{4s}^{eq}-k_{4s}),$ $\displaystyle\tilde{k}_{4d1}$
$\displaystyle=$ $\displaystyle k_{4d1}+\omega_{4d1}(k_{4d1}^{eq}-k_{4d1}),$
$\displaystyle\tilde{k}_{4d2}$ $\displaystyle=$ $\displaystyle
k_{4d2}+\omega_{4d2}(k_{4d2}^{eq}-k_{4d2}),$ $\displaystyle\tilde{k}_{122}$
$\displaystyle=$ $\displaystyle k_{122}+\omega_{122}(k_{122}^{eq}-k_{122}),$
$\displaystyle\tilde{k}_{212}$ $\displaystyle=$ $\displaystyle
k_{212}+\omega_{212}(k_{212}^{eq}-k_{212}),$ $\displaystyle\tilde{k}_{221}$
$\displaystyle=$ $\displaystyle k_{221}+\omega_{221}(k_{221}^{eq}-k_{221}),$
$\displaystyle\tilde{k}_{222}$ $\displaystyle=$ $\displaystyle
k_{222}+\omega_{222}(k_{222}^{eq}-k_{222}).$

Here, the relaxation parameters for the second order moments $\omega_{\xi}$
and $\omega_{\nu}$ independently specify the bulk viscosity $\xi$ and shear
viscosity $\nu$, respectively, and are related to one another via

$\displaystyle\nu=c_{s}^{2}\left(\dfrac{1}{\omega_{\nu}}-\dfrac{1}{2}\right)\Delta
t,\quad\xi=\dfrac{2c_{s}^{2}}{3}\left(\dfrac{1}{\omega_{\xi}}-\dfrac{1}{2}\right)\Delta
t,$

so that, in conjunction the moment equilibria corrections, the 3DCCM-LBM
simulates the 3D NS equations. The rest of the relaxation parameters for the
evolution of other central moments under collision are set to unity in this
work. That is,
$\omega_{1}=\omega_{3s1}=\omega_{3s2}=\omega_{3s3}=\omega_{3m1}=\omega_{3m2}=\omega_{3m3}=\omega_{111}=\omega_{211}=\omega_{121}=\omega_{112}=\omega_{4s}=\omega_{4d1}=\omega_{4d2}=\omega_{122}=\omega_{212}=\omega_{221}=\omega_{222}=1.0$.
Finally, we segregate the post-collision values of those selectively combined
central moments in the above, which is represented by $\bm{\mathsf{B}}^{-1}$,
as

$\displaystyle\tilde{k}_{200}=\dfrac{1}{3}\left(\tilde{k}_{2s}+\tilde{k}_{2d1}+\tilde{k}_{2d2}\right),\quad\;\;\;\;\tilde{k}_{120}=\dfrac{1}{2}\left(\tilde{k}_{3s1}+\tilde{k}_{3m1}\right),\quad\;\;\;\;\tilde{k}_{102}=\dfrac{1}{2}\left(\tilde{k}_{3s1}-\tilde{k}_{3m1}\right)$
$\displaystyle\tilde{k}_{020}=\dfrac{1}{3}\left(\tilde{k}_{2s}-2\tilde{k}_{2d1}+\tilde{k}_{2d2}\right),\quad\;\;\;\;\tilde{k}_{210}=\dfrac{1}{2}\left(\tilde{k}_{3s2}+\tilde{k}_{3m2}\right)\quad\;\;\;\;\tilde{k}_{012}=\dfrac{1}{2}\left(\tilde{k}_{3s2}-\tilde{k}_{3m2}\right),$
$\displaystyle\tilde{k}_{002}=\dfrac{1}{3}\left(\tilde{k}_{2s}+\tilde{k}_{2d1}-2\tilde{k}_{2d2}\right),\quad\;\;\;\;\tilde{k}_{201}=\dfrac{1}{2}\left(\tilde{k}_{3s3}+\tilde{k}_{3m3}\right),\quad\;\;\;\;\tilde{k}_{021}=\dfrac{1}{2}\left(\tilde{k}_{3s3}-\tilde{k}_{3m3}\right),$
$\displaystyle\tilde{k}_{220}=\dfrac{1}{4}\left(\tilde{k}_{4s}+\tilde{k}_{4d1}+2\tilde{k}_{4d2}\right),$
$\displaystyle\tilde{k}_{202}=\dfrac{1}{4}\left(\tilde{k}_{4s}+\tilde{k}_{4d1}-2\tilde{k}_{4d2}\right),$
$\displaystyle\tilde{k}_{022}=\dfrac{1}{2}\left(\tilde{k}_{4s}-\tilde{k}_{4d1}\right).$

As a result, at the end of this computation, all the linearly independent
post-collision central moments $\tilde{k}_{mnp}$ of the D3Q27 lattice have
been computed.

##  Appendix E Mapping post-collision central moments to raw moments

The post-collision raw moments can be obtained from the central moments via
$\tilde{\mathbf{m}}=\bm{\mathsf{\mathcal{F}}}^{-1}\tilde{\mathbf{m}}^{c}$,
where the inverse of the frame transformation matrix, i.e.,
$\bm{\mathsf{\mathcal{F}}}^{-1}$ is given by
$\bm{\mathsf{\mathcal{F}}}^{-1}=\bm{\mathsf{\mathcal{F}}}(-u_{x},-u_{y},-u_{z})$.
As noted in our previous work [26], this follows from the property of the
binomial transforms in their generating function representation. In other
words, by using the expressions presented in C and performing the exchanges
$k_{mnp}\leftrightarrow k_{mnp}^{\prime}$ and, $u_{x}\leftrightarrow-u_{x}$,
$u_{y}\leftrightarrow-u_{y}$ and $u_{z}\leftrightarrow-u_{z}$, and then
applying $\tilde{(\cdot)}$ over the moments as they represent the post-
collision states, we obtain the required mapping relations which are listed as
follows:

$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
000}=\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
100}=\tilde{k}_{\scriptscriptstyle 100}+u_{x}\tilde{k}_{\scriptscriptstyle
000},$ $\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
010}=\tilde{k}_{\scriptscriptstyle 010}+u_{y}\tilde{k}_{\scriptscriptstyle
000},$ $\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
001}=\tilde{k}_{\scriptscriptstyle 001}+u_{z}\tilde{k}_{\scriptscriptstyle
000},$ $\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
110}=\tilde{k}_{\scriptscriptstyle 110}+u_{x}\tilde{k}_{\scriptscriptstyle
010}+u_{y}\tilde{k}_{\scriptscriptstyle
100}+u_{x}u_{y}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
101}=\tilde{k}_{\scriptscriptstyle 101}+u_{x}\tilde{k}_{\scriptscriptstyle
001}+u_{z}\tilde{k}_{\scriptscriptstyle
100}+u_{x}u_{z}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
011}=\tilde{k}_{\scriptscriptstyle 011}+u_{y}\tilde{k}_{\scriptscriptstyle
001}+u_{z}\tilde{k}_{\scriptscriptstyle
010}+u_{y}u_{z}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
200}=\tilde{k}_{\scriptscriptstyle 200}+2u_{x}\tilde{k}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
020}=\tilde{k}_{\scriptscriptstyle 020}+2u_{y}\tilde{k}_{\scriptscriptstyle
010}+u_{y}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
002}=\tilde{k}_{\scriptscriptstyle 002}+2u_{z}\tilde{k}_{\scriptscriptstyle
001}+u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120}=\tilde{k}_{\scriptscriptstyle 120}+u_{x}\tilde{k}_{\scriptscriptstyle
020}+2u_{y}\tilde{k}_{\scriptscriptstyle
110}+2u_{x}u_{y}\tilde{k}_{\scriptscriptstyle 010}+u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 100}+u_{x}u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102}=\tilde{k}_{\scriptscriptstyle 102}+u_{x}\tilde{k}_{\scriptscriptstyle
002}+2u_{z}\tilde{k}_{\scriptscriptstyle
101}+2u_{x}u_{z}\tilde{k}_{\scriptscriptstyle 001}+u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 100}+u_{x}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210}=\tilde{k}_{\scriptscriptstyle 210}+u_{y}\tilde{k}_{\scriptscriptstyle
200}+2u_{x}\tilde{k}_{\scriptscriptstyle 110}+u_{x}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 010}+2u_{x}u_{y}\tilde{k}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}u_{y}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
012}=\tilde{k}_{\scriptscriptstyle 012}+u_{y}\tilde{k}_{\scriptscriptstyle
002}+2u_{z}\tilde{k}_{\scriptscriptstyle 011}+u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 010}+2u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
001}+u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201}=\tilde{k}_{\scriptscriptstyle 201}+u_{z}\tilde{k}_{\scriptscriptstyle
200}+2u_{x}\tilde{k}_{\scriptscriptstyle 101}+u_{x}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 001}+2u_{x}u_{z}\tilde{k}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
021}=\tilde{k}_{\scriptscriptstyle 021}+u_{z}\tilde{k}_{\scriptscriptstyle
020}+2u_{y}\tilde{k}_{\scriptscriptstyle 011}+u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 001}+2u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
010}+u_{y}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}=\tilde{k}_{\scriptscriptstyle 111}+u_{x}\tilde{k}_{\scriptscriptstyle
011}+u_{y}\tilde{k}_{\scriptscriptstyle
101}+u_{z}\tilde{k}_{\scriptscriptstyle
110}+u_{x}u_{y}\tilde{k}_{\scriptscriptstyle
001}+u_{x}u_{z}\tilde{k}_{\scriptscriptstyle
010}+u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
100}+u_{x}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}=\tilde{k}_{\scriptscriptstyle 220}+2u_{y}\tilde{k}_{\scriptscriptstyle
210}+2u_{x}\tilde{k}_{\scriptscriptstyle 120}+u_{x}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 020}+u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 200}+4u_{x}u_{y}\tilde{k}_{\scriptscriptstyle
110}+2u_{x}^{\scriptscriptstyle 2}u_{y}\tilde{k}_{\scriptscriptstyle
010}+2u_{x}u_{y}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}=\tilde{k}_{\scriptscriptstyle 202}+2u_{z}\tilde{k}_{\scriptscriptstyle
201}+2u_{x}\tilde{k}_{\scriptscriptstyle 102}+u_{x}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 002}+u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 200}+4u_{x}u_{z}\tilde{k}_{\scriptscriptstyle
101}+2u_{x}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
001}+2u_{x}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 000},$
$\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}=\tilde{k}_{\scriptscriptstyle 022}+2u_{z}\tilde{k}_{\scriptscriptstyle
021}+2u_{y}\tilde{k}_{\scriptscriptstyle 012}+u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 020}+u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 002}+4u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
011}+2u_{y}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
001}+2u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
010}+u_{y}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 000},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 211}$
$\displaystyle=\tilde{k}_{\scriptscriptstyle
211}+2u_{x}\tilde{k}_{\scriptscriptstyle
111}+u_{y}\tilde{k}_{\scriptscriptstyle
201}+u_{z}\tilde{k}_{\scriptscriptstyle 210}+u_{x}^{\scriptscriptstyle
2}k_{\scriptscriptstyle 011}+2u_{x}u_{y}\tilde{k}_{\scriptscriptstyle
101}+u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
200}+2u_{x}u_{z}\tilde{k}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;+u_{x}^{\scriptscriptstyle
2}u_{y}\tilde{k}_{\scriptscriptstyle 001}+u_{x}^{\scriptscriptstyle
2}u_{z}\tilde{k}_{\scriptscriptstyle
010}+2u_{x}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
000},$ $\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 121}$
$\displaystyle=\tilde{k}_{\scriptscriptstyle
121}+2u_{y}\tilde{k}_{\scriptscriptstyle
111}+u_{x}\tilde{k}_{\scriptscriptstyle
021}+u_{z}\tilde{k}_{\scriptscriptstyle
120}+u_{x}u_{z}\tilde{k}_{\scriptscriptstyle
020}+2u_{x}u_{y}\tilde{k}_{\scriptscriptstyle 011}+u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 101}+2u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
110}$ $\displaystyle\;\;\;\;-u_{x}u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle
001}+2u_{x}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
010}+u_{y}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
100}+u_{x}u_{y}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
000},$ $\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 112}$
$\displaystyle=\tilde{k}_{\scriptscriptstyle
112}+2u_{z}\tilde{k}_{\scriptscriptstyle
111}+u_{x}\tilde{k}_{\scriptscriptstyle
012}+u_{y}\tilde{k}_{\scriptscriptstyle
102}+u_{x}u_{y}\tilde{k}_{\scriptscriptstyle
002}+2u_{x}u_{z}\tilde{k}_{\scriptscriptstyle
011}+2u_{y}u_{z}\tilde{k}_{\scriptscriptstyle 101}+u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;+2u_{x}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
001}+u_{x}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
010}+u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
100}+u_{x}u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
000},$ $\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 122}$
$\displaystyle=\tilde{k}_{\scriptscriptstyle
122}+2u_{y}\tilde{k}_{\scriptscriptstyle
112}+2u_{z}\tilde{k}_{\scriptscriptstyle
121}+u_{x}\tilde{k}_{\scriptscriptstyle
022}+4u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
111}+2u_{x}u_{z}\tilde{k}_{\scriptscriptstyle
021}+2u_{x}u_{y}\tilde{k}_{\scriptscriptstyle 012}+u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 102}$
$\displaystyle\;\;\;\;+u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 120}+u_{x}u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 002}+u_{x}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle
020}+4u_{x}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
011}+2u_{y}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
101}+2u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;+2u_{x}u_{y}^{\scriptscriptstyle
2}u_{z}\tilde{k}_{\scriptscriptstyle 001}+2u_{x}u_{y}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 010}+u_{y}^{\scriptscriptstyle
2}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
100}+u_{x}u_{y}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 000},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 212}$
$\displaystyle=\tilde{k}_{\scriptscriptstyle
212}+2u_{x}\tilde{k}_{\scriptscriptstyle
112}+2u_{z}\tilde{k}_{\scriptscriptstyle
211}+u_{y}\tilde{k}_{\scriptscriptstyle
202}+4u_{x}u_{z}\tilde{k}_{\scriptscriptstyle
111}+2u_{y}u_{z}\tilde{k}_{\scriptscriptstyle 201}+u_{x}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 012}+u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 210}$
$\displaystyle\;\;\;\;+2u_{x}u_{y}\tilde{k}_{\scriptscriptstyle
102}+u_{x}^{\scriptscriptstyle 2}u_{y}\tilde{k}_{\scriptscriptstyle
002}+u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
200}+2u_{x}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
011}+4u_{x}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
101}+2u_{x}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;+2u_{x}^{\scriptscriptstyle
2}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle 001}+u_{x}^{\scriptscriptstyle
2}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
010}+2u_{x}u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
100}+u_{x}^{\scriptscriptstyle 2}u_{y}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 000},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 221}$
$\displaystyle=\tilde{k}_{\scriptscriptstyle
221}+2u_{x}\tilde{k}_{\scriptscriptstyle
121}+2u_{y}\tilde{k}_{\scriptscriptstyle
211}+u_{z}\tilde{k}_{\scriptscriptstyle
220}+4u_{x}u_{y}\tilde{k}_{\scriptscriptstyle 111}+u_{x}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 021}+u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 201}+2u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
210}$ $\displaystyle\;\;\;\;+2u_{x}u_{z}\tilde{k}_{\scriptscriptstyle
120}+u_{x}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
020}+u_{y}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
200}+2u_{x}^{\scriptscriptstyle 2}u_{y}\tilde{k}_{\scriptscriptstyle
011}+2u_{x}u_{y}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
101}+4u_{x}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle 110}$
$\displaystyle\;\;\;\;+u_{x}^{\scriptscriptstyle 2}u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 001}+2u_{x}^{\scriptscriptstyle
2}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle 010}+2u_{x}u_{y}^{\scriptscriptstyle
2}u_{z}\tilde{k}_{\scriptscriptstyle 100}+u_{x}^{\scriptscriptstyle
2}u_{y}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle 000},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}$
$\displaystyle=\tilde{k}_{\scriptscriptstyle
222}+2u_{z}\tilde{k}_{\scriptscriptstyle
221}+2u_{y}\tilde{k}_{\scriptscriptstyle
212}+2u_{x}\tilde{k}_{\scriptscriptstyle
122}+4u_{x}u_{y}\tilde{k}_{\scriptscriptstyle
112}+4u_{x}u_{z}\tilde{k}_{\scriptscriptstyle 121}$
$\displaystyle\;\;\;\;+4u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
211}+u_{x}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
022}+u_{y}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
202}+u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
220}+8u_{x}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
111}+2u_{x}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle 021}$
$\displaystyle\;\;\;\;+2u_{y}^{\scriptscriptstyle
2}u_{z}\tilde{k}_{\scriptscriptstyle 201}+2u_{x}^{\scriptscriptstyle
2}u_{y}\tilde{k}_{\scriptscriptstyle 012}+2u_{y}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 210}+2u_{x}u_{y}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 102}+2u_{x}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 120}+u_{x}^{\scriptscriptstyle
2}u_{y}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle 002}$
$\displaystyle\;\;\;\;+u_{x}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 020}+u_{y}^{\scriptscriptstyle
2}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
200}+4u_{x}^{\scriptscriptstyle 2}u_{y}u_{z}\tilde{k}_{\scriptscriptstyle
011}+4u_{x}u_{y}^{\scriptscriptstyle 2}u_{z}\tilde{k}_{\scriptscriptstyle
101}+4u_{x}u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
110}+2u_{x}^{\scriptscriptstyle 2}u_{y}^{\scriptscriptstyle
2}u_{z}\tilde{k}_{\scriptscriptstyle 001}$
$\displaystyle\;\;\;\;+2u_{x}^{\scriptscriptstyle
2}u_{y}u_{z}^{\scriptscriptstyle 2}\tilde{k}_{\scriptscriptstyle
010}+2u_{x}u_{y}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 100}+u_{x}^{\scriptscriptstyle
2}u_{y}^{\scriptscriptstyle 2}u_{z}^{\scriptscriptstyle
2}\tilde{k}_{\scriptscriptstyle 000}.$

##  Appendix F Inverse scaling post-collision raw moments by grid aspect
ratios

The inverse scaling of the post-collision raw moments, i.e.,
$\tilde{\mathbf{m}}\leftarrow\bm{\mathsf{S}}^{-1}\tilde{\mathbf{m}}$ involves
dividing the post-collision raw moments $\tilde{k}_{mnp}^{\prime}$ of the
order $(m+n+p)$ by $r^{n}s^{p}$. That is,

$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
000}=\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
000},\quad\quad\;\;\;\;\;\;\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
100}=\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
100},\quad\;\;\;\;\;\;\;\;\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
010}=r^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 010},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
001}=s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
001},\quad\;\;\;\;\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
200}=\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
200},\quad\;\;\;\;\;\;\;\;\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102}=s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102},\quad\;\;\;\;\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
021}=r^{-2}s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 021},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
110}=r^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
110},\quad\;\;\;\;\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
020}=r^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
020},\quad\;\;\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210}=r^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210},\quad\;\;\;\;\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}=r^{-1}s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 111},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
101}=s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
101},\quad\;\;\;\;\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
002}=s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
002},\quad\;\;\;\quad\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}=r^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220},\quad\;\quad\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
012}=r^{-1}s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 012},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
011}=r^{-1}s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
011},\;\;\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120}=r^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120},\quad\;\;\;\quad\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201}=s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201},\quad\;\;\;\;\;\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}=s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 202},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}=r^{-2}s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022},\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}=r^{-1}s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211},\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}=r^{-2}s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121},\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}=r^{-1}s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 112},$
$\displaystyle\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}=r^{-2}s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122},\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}=r^{-1}s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212},\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}=r^{-2}s^{-1}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221},\quad\;\;\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
222}=r^{-2}s^{-2}\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}.$

##  Appendix G Mapping post-collision raw moments to distribution functions

The post-collision distribution functions can be obtained from the raw moments
via $\tilde{\mathbf{f}}=\bm{\mathsf{P}}^{-1}\tilde{\mathbf{m}}$, where
$\bm{\mathsf{P}}^{-1}$ is the inverse of the simpler moment basis for the
cubic lattice presented in Eq. (3.1). Thus, we get

$\displaystyle\tilde{f}_{\scriptscriptstyle
0}=\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
000}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
200}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
020}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
002}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222},$
$\displaystyle\tilde{f}_{\scriptscriptstyle
1}=\frac{1}{2}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
100}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
200}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
2}=\frac{1}{2}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
100}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
200}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
3}=\frac{1}{2}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
010}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
020}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
012}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
4}=-\frac{1}{2}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
010}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
020}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
012}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
5}=\frac{1}{2}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
001}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
002}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
021}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
6}=\frac{1}{2}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
001}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
002}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
021}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
7}=\frac{1}{4}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
110}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
8}=\frac{1}{4}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
110}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
9}=\frac{1}{4}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
110}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
10}=\frac{1}{4}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
110}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
120}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
210}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
220}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
11}=\frac{1}{4}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
101}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
12}=\frac{1}{4}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
101}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
13}=\frac{1}{4}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
101}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
14}=\frac{1}{4}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
101}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
102}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
201}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
202}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
15}=\frac{1}{4}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
011}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
012}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
021}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
16}=\frac{1}{4}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
011}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
012}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
021}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
17}=\frac{1}{4}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
011}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
012}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
021}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
18}=\frac{1}{4}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
011}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
012}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
021}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
022}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
19}=\frac{1}{8}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
20}=\frac{1}{8}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
21}=\frac{1}{8}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
22}=\frac{1}{8}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
23}=\frac{1}{8}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
24}=\frac{1}{8}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
25}=\frac{1}{8}\left(\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right),$
$\displaystyle\tilde{f}_{\scriptscriptstyle
26}=\frac{1}{8}\left(-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
111}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
211}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
121}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
112}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
122}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
212}-\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle
221}+\tilde{k}^{{}^{\prime}}_{\scriptscriptstyle 222}\right).$

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