## 2 Methodology
### ALE-LBFS for Fluid Solver
#### 2.2.1 Macroscopic Conservation Laws
Forced on \(f_{a}(\mathbf{0},t)\) at the cell interface, its formulation is as  
\[f_{a}(\mathbf{0},t)=g_{\alpha}(\mathbf{0},t)+f_{\alpha}^{neq}(\mathbf{0},t) \tag{24}\]  
Following the expansion of the Chapman-Enskog analysis, the nonequilibrium part \(f_{\alpha}^{neq}(\mathbf{0},t)\) truncated by the first term is discretized by the finite difference scheme as  
\[f_{\alpha}^{neq}(\mathbf{0},t)=-\frac{\tau}{\delta t}(g_{\alpha}(\mathbf{0},t )-g_{\alpha}(-\xi_{\alpha}\delta t,t-\delta t)) \tag{25}\]  
Substituting Equation (25) into Equation (22), the flux \(\bar{\mathbf{F}}_{c}\) at the interface can be calculated by  
\[\bar{\mathbf{F}}_{c}=\sum_{\alpha=1}^{N}\xi_{\alpha}\boldsymbol{\varphi}_{ \alpha}g_{\alpha}(0,t)-\frac{\tau}{\delta t}\sum_{\alpha=1}^{N}\xi_{\alpha} \boldsymbol{\varphi}_{\alpha}(g_{\alpha}(0,t)-g_{\alpha}(-\xi_{\alpha}\delta t,t-\delta t)) \tag{26}\]  
where \(\tau\) is the collision time and \(\delta t\) is the streaming time step. N is the number of velocities in the lattice model and is equal to 4 for the D1Q4 lattice model. From Equation (26), it is apparent that \(\bar{\mathbf{F}}_{c}\) is determined by the density distribution function at the cell interface \(g_{\alpha}(\mathbf{0},t)\) as well as that streaming from the neighboring location \(g_{\alpha}(-\xi_{\alpha}\delta t,t-\delta t)\). In fact, this coincides with the conventional convective scheme at the interface of the cell control volume, as depicted in Figure 4. Let \(\mathbf{F}_{c}^{I}\) and \(\mathbf{F}_{c}^{II}\) be  
\[\mathbf{F}_{c}^{I}=\sum_{i=1}^{N}\xi_{\alpha}\boldsymbol{\varphi}_{\alpha}g_{ \alpha}(0,t) \tag{27}\]  
\[\mathbf{F}_{c}^{II}=\sum_{i=1}^{N}\xi_{\alpha}\boldsymbol{\varphi}_{\alpha}g_{ \alpha}(-\xi_{\alpha}\delta t,t-\delta t) \tag{28}\]  
Figure 4: Streaming process in the D1Q4 model. In order to evaluate \(\mathbf{F}_{c}^{l}\), \(g_{\alpha}(0,t)\) can be obtained using Equations (12)-(15) by the conservative variable \(\tilde{\mathbf{W}}\) at the interface. From the lattice structure in Figure 4, an upwind scheme is applied to evaluate \(g_{\alpha}(-\xi_{\alpha}\delta t,t-\delta t)\) by reconstructed variables on the left and right side of the interface, that is,  
\[g_{\alpha}(-\xi_{\alpha}\delta t,t-\delta t)= \begin{cases}g_{\alpha}^{L}\,\,if \,\,\alpha=1,3\\ g_{\alpha}^{R}\,\,if\,\,\alpha=2,4\end{cases}  \tag{29}\]  
Thus, \(\tilde{\mathbf{W}}\) can be computed by  
\[\tilde{\mathbf{W}}=\sum_{\alpha=1}^{N}\varphi_{\alpha}g_{\alpha}(0,t)=\sum_{ \alpha=1,3}\varphi_{\alpha}g_{\alpha}^{L}+\sum_{\alpha=2,4}\varphi_{\alpha}g_ {\alpha}^{R} \tag{30}\]  
where \(g_{\alpha}^{L}\) and \(g_{\alpha}^{R}\) are computed from Equations (12)-(15). Once the evaluation of \(\tilde{\mathbf{W}}\) is completed, \(g_{\alpha}(0,t)\) can be calculated by Equations (12)-(15) and then used to calculate \(\mathbf{F}_{c}^{l}\) in Equation (27). Within the calculation of \(\mathbf{F}_{c}^{lI}\) in Equation (28), \(g_{\alpha}(-\xi_{\alpha}\delta t,t-\delta t)\) can be determined according to the streaming discretion. Then,  
\[\mathbf{F}_{c}^{II}=\sum_{\alpha=1}^{N}\xi_{\alpha}\varphi_{\alpha}g_{\alpha}( -\xi_{\alpha}\delta t,t-\delta t)=\sum_{\alpha=1,3}\xi_{\alpha}\varphi_{ \alpha}g_{\alpha}^{L}+\sum_{\alpha=2,4}\xi_{\alpha}\varphi_{\alpha}g_{\alpha} ^{R} \tag{31}\]  
To complete the evaluation of \(\mathbf{F}_{c}\) in Equation (20), the terms related to \(\rho u_{n}\mathbf{u}_{\tau}\) and \(\rho u_{n}\|\mathbf{u}_{\tau}\|^{2}\) are reconstructed locally as  
\[\rho u_{n}\mathbf{u}_{\tau}=\sum_{\alpha=1,3}\xi_{\alpha}g_{\alpha}^{L} \mathbf{u}_{\tau}^{L}+\sum_{\alpha=2,4}\xi_{\alpha}g_{\alpha}^{R}\mathbf{u}_{ \tau}^{R} \tag{32}\]  
\[\rho u_{n}\|\mathbf{u}_{\tau}\|^{2}=\sum_{\alpha=1,3}\xi_{\alpha}g_{\alpha}^{ L}\Big{\|}\mathbf{u}_{\tau}^{L}\Big{\|}^{2}+\sum_{\alpha=2,4}\xi_{\alpha}g_{ \alpha}^{R}\Big{\|}\mathbf{u}_{\tau}^{R}\Big{\|}^{2} \tag{33}\]  
Substituting Equations (27), (28), (32) and (33) into Equation (20), the final convective flux \(\mathbf{F}_{c}\) is obtained.