# A 1D-2D Coupled Lattice Boltzmann Model for Shallow Water Flows in Large Scale River-Lake Systems
## 2 1D and 2D Shallow Water Submodels and Their Coupling Strategy
### 2D Shallow Water Submodel  
The 2D shallow-water equations can be expressed as  
\[\frac{\partial h}{\partial t}+\frac{\partial(hu_{j})}{\partial x_{j}}=0 \tag{10}\]  
\[\frac{\partial(hu_{i})}{\partial t}+\frac{\partial(hu_{i}u_{j})}{\partial x_{j} }=-\frac{\partial}{\partial x_{i}}\Big{(}\frac{g}{2}h^{2}\Big{)}+v\frac{ \partial^{2}(hu_{j})}{\partial x_{j}\partial x_{j}}+S_{i} \tag{11}\]  
where \(u_{i}\) and \(u_{j}\) are the depth-averaged velocities; \(S_{i}=S_{bi}-S_{fi}\), with \(S_{bi}=g_{bi}^{\frac{\partial z_{b}}{\partial x_{i}}}\) as the hydraulic gradient caused by the bed gradient, \(z_{b}\) is the bed elevation; \(S_{fi}=gu_{i}n^{2}\sqrt{u_{i}u_{j}}/\sqrt{h}\) is the hydraulic gradient caused by bed shear stresses, in which \(n\) is Manning roughness coefficient.  
The LB method has been used to numerically solve the 2D flows governed by the Equations (10) and (11). The lattice Boltzmann equation [31] can be expressed as  
\[f_{\alpha}(\mathbf{x}+\mathbf{e}_{\alpha}\Delta t,t+\Delta t)-f_{\alpha}(\mathbf{x},t)= \Omega_{\alpha}+F_{\alpha},(\alpha=0,1,2,\dots,8) \tag{12}\]  
where \(\Omega_{\alpha}\) is the collision operator; the definition of other variables are the same as Equation (3).  
The D2Q9 model has been applied, and the discrete particle velocities are  
\[\mathbf{e}_{\alpha}=c\cdot\left\{\begin{array}{cccccccc}0&1&-1&0&0&1&-1&1&-1\\ 0&0&0&1&-1&1&-1&-1&1\end{array}\right\} \tag{13}\]  
The equilibrium distribution functions are expressed as  
\[\left\{\begin{array}{l}f_{\alpha}^{eq}=h-\frac{5 gh^{2}}{6c^{2}}-\frac{2h}{3 c^{2}}\mathbf{\mu}^{2},\alpha=0\\ f_{\alpha}^{eq}=w_{\alpha}\cdot\frac{h}{c^{2}}\cdot\Big{[}\frac{3}{2}gh+3\mathbf{e }_{\alpha}\cdot\mathbf{u}+\frac{9}{2c^{2}}(\mathbf{e}_{\alpha}\cdot\mathbf{u})^{2}-\frac{ 3}{2}u^{2}\Big{]},(\alpha=1,2,\dots,8)\end{array}\right. \tag{14}\]  
where the weight factor \(w_{\alpha}=\{4/9,1/9,1/9,1/9,1/9,1/36,1/36,1/36,1/36\}\).  
Water depth and velocity of the fluid are given by Equation (5).  
The multi-relaxation time (MRT) collision operator has been adopted [33]. The particle distribution functions are transformed into moment spaces before the relaxation step: \(\Omega=-\mathbf{M}^{-1}\cdot\mathbf{S}\cdot(\mathbf{M}\cdot f-m^{eq})\), where \(\mathbf{M}\) denotes the transformation matrix from the distribution functions to the moments, and the diagonal collision matrix \(S=\mathbf{MSM}^{-1}=diag\big{[}s_{p},s_{e},s_{e},s_{j},s_{q},s_{j},s_{q},s_{v},s_{ v}\big{]}\) contains the relaxation parameters. The relaxation parameters except \(s_{v}\) are within the range [0, 2], and the optimal values for these parameters depend on the specific geometry, the initial conditions, and boundary conditions of the system. In this study, \(S=(1.0,1.8,1.8,1.0,1.2,1.0,1.2,s_{v},s_{v}),s_{v}\) is related to the kinematic viscosity \(\
u\), which is defined as  
\[s_{v}=\frac{1}{\tau}=\frac{2}{6\frac{v}{c^{2}\Delta t}+1} \tag{15}\]  
With the above setups, the 2D LB submodel can recover 2D shallow-water Equations (10) and (11) by the Chapman-Enskog expansion in the incompressible flow [34].  
Second-order treatment of the external forcing term proposed by [PERSON] [35] has been adopted. Assuming the \(A\) is the source term in the continuity Equation (10) and \(B_{i}\) is the external forcing term in the momentum Equation (11), the \(F_{\alpha}\) can be written as  
\[F_{\alpha}=\frac{1}{2}\Delta t[q_{\alpha}(\mathbf{x}+\mathbf{e}_{\alpha}\Delta t,t+ \Delta t)+q_{\alpha}(\mathbf{x},t)] \tag{16}\]  
\[q_{\alpha}=w_{\alpha}[A+3\mathbf{B}_{i}\cdot[(\mathbf{e}_{\alpha}-\mathbf{u})+3(\mathbf{e}_{ \alpha}\cdot\mathbf{u})\mathbf{e}_{\alpha}]] \tag{17}\]  
in which \(A=0\), \(\mathbf{B}_{i}=\mathbf{S}_{i}\), with \(\mathbf{g}\) regarded as a vector in \(S_{i}\).