# 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
### The Coupling Strategy of 1D and 2D Submodels  
The coupling strategy of the 1D-2D interfaces is implemented at the mesoscopic level, in which the distribution functions are used to couple 1D and 2D submodels. The coupling interface is in the form of a node in the 1D computational domain, and in the form of a line in the 2D computational domain. In this study, the coupling interface was treated as an interior lattice node of the 1D domain, with three unknown distribution functions to be calculated. These unknown distribution functions at the coupling interface were calculated by the distribution functions and macroscopic variables from the adjacent 1D and 2D lattice nodes. The reformulation of the unknown distribution function is depicted in Figure 1, in which \(g_{\alpha}\) denote distribution functions of the 1D domain, \(f_{\alpha}\) denote distribution functions of the 2D domain, \(k\) refers to the position of the coupling node at the 1D domain, \(j\) refers to time, and \(m\) denotes the position of the coupling node at the 2D domain. The processing steps for dealing coupling interface can be expressed as follows:  
1. Calculate the three unknown distribution functions of the coupling interface;
2. Calculate the macroscopic parameters (water depth and velocity) of the coupling interface by the distribution functions solved in the first step;
3. Calculate the unknown distribution functions at the coupling boundaries of the 2D domain by the above calculated macroscopic parameters. To be specific, the unknown distribution functions \(g_{0}(k,j),g_{1}(k,j),g_{2}(k,j)\) of the coupling interface are calculated at first. \(g_{1}(k,j)\) can be expressed as  
\[g_{1}(k,j)=g_{1}(k-1,j-1)-\frac{1}{\tau}[g_{1}(k-1,j-1)-g_{1}^{\alpha}(k-1,j- 1)]+F_{1} \tag{18}\]  
\(g_{2}(k,j)\) can be calculated by the distribution functions at the adjacent 2D nodes and expressed as  
\[g_{2}(k,j)=g_{2}(m+1,j-1)-\frac{1}{\tau}[g_{2}(m+1,j-1)-g_{2}^{\alpha}(m+1,j- 1)]+F_{2} \tag{19}\]  
where \(g_{2}(m+1,j-1)=f_{2}(m+1,j-1)+f_{6}(m+1,j-1)+f_{8}(m+1,j-1)\), with \(f_{2}(m+1,j-1)\), \(f_{6}(m+1,j-1)\), and \(f_{8}(m+1,j-1)\) as the averaged values of distribution functions at position \(m+1\). \(g_{0}(k,j)\) can be calculated by  
\[g_{0}(k,j)=g_{0}(k,j-1)-\frac{1}{\tau}[g_{0}(k,j-1)-g_{0}^{\alpha}(k,j-1)]+F_{0} \tag{20}\]  
Then, the water depth and velocity at the coupling interface can be obtained by  
\[h(k,j)=g_{1}(k,j)+g_{2}(k,j)+g_{0}(k,j) \tag{21}\]  
\[u(k,j)=(g_{1}(k,j)-g_{2}(k,j))/\mathrm{h}(k,j) \tag{22}\]  
Figure 1: Sketch of the 1D–2D coupling boundary. Finally, the unknown distribution functions of lattice nodes at the inlet boundary of the 2D domain can be disposed of as following [36]: given \(h_{in}=h(k,j)\), \(u_{x}=u(k,j)\), \(u_{y}=0\), assuming that \(f_{1}-f_{1}{}^{\alpha q}=f_{2}-f_{2}{}^{\alpha q}\). The unknown distribution functions \(f_{1}\), \(f_{5}\), and \(f_{7}\) at the inlet boundary of the 2D domain can be calculated by Equation (23). \[\left\{ \begin{array}{l}f_{1}=f_{2}+\frac{2}{3}h_{in}u_{x}/c\\ f_{5}=f_{6}-\frac{1}{2}(f_{3}-f_{4})+\frac{1}{2}h_{in}u_{y}/c+\frac{1}{6}h_{in }u_{x}/c\\ f_{7}=f_{8}+\frac{1}{2}(f_{3}-f_{4})-\frac{1}{2}h_{in}u_{y}/c+\frac{1}{6}h_{in }u_{x}/c\end{array} \right. \tag{23}\]  
Similarly, the unknown distribution functions \(f_{2}\), \(f_{6}\), and \(f_{8}\) at the outlet boundary of the 2D domain can be solved by Equation (24). Other boundaries of the 2D domain are regarded as no-slip wall boundary conditions and treated by a non-equilibrium extrapolation scheme [37]. \[\left\{ \begin{array}{l}f_{2}=f_{1}-\frac{2}{3}h_{out}u_{x}/c\\ f_{6}=f_{5}+\frac{1}{2}(f_{3}-f_{4})-\frac{1}{2}h_{out}u_{y}/c-\frac{1}{6}h_{ out}u_{x}/c\\ f_{8}=f_{7}-\frac{1}{2}(f_{3}-f_{4})+\frac{1}{2}h_{out}u_{y}/c-\frac{1}{6}h_{ out}u_{x}/c\end{array} \right. \tag{24}\]  
The above coupling strategy is aimed at the condition that the 1D domain is at the upstream, and the 2D domain is at the downstream, with the coupling interface at the inlet boundary of the 2D domain.