# A 1D-2D Coupled Lattice Boltzmann Model for Shallow Water Flows in Large Scale River-Lake Systems
## 5 Conclusions  
In this work, a 1D-2D coupled model was developed for shallow water simulations in large scale river-lake systems, in which both 1D and 2D submodels were numerically solved by the LB method. Unlike any other existing coupling approaches, the coupling strategy of the proposed model was carried out at the mesoscopic level. The distribution functions of the LB equation replaced the role of water depth and discharged to bridge the 1D and 2D submodels. The coupling interface was treated as an interior lattice node of the 1D domain, and its unknown distribution functions were calculated by the macroscopic variables and the distribution functions from the adjacent 1D and 2D lattice nodes. The proposed approach was rather simple and of high efficiency because there was no iterative process.  
Numerical tests on two benchmark cases showed that the numerical solutions using the proposed model were in excellent agreement with the analytical solutions and commercial software results, and second-order convergence was observed, indicating the proposed model had good accuracy. The application in simulating the surge wave propagation and reflection phenomena in the reservoir of a run-of-river hydropower station had shown that the proposed model could achieve better performances on finer meshes, indicating its high computational efficiency in the computational intensive problem. Therefore, the 1D-2D coupled model had a huge potential to simulate flows in large-scale river-lake systems.  
The proposed model retained all the merits of the LB method, which were simple, high accuracy, convenient processing complex boundaries, suitable for parallel computing, and wide application range. Thus, it has great value for applications. Further studies of the 1D-2D coupled model would  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Lattice** & **2D Grid** & **1D Grid** & **Consumed Time (h)** & **Performance (MNUPS)** & **Speedup Ratio** \\ \hline \(\Delta x=17.28\) & \(400\times 300\) & 2026 & 0.71 & 17.09 & - \\ \(\Delta x=8.64\) & \(800\times 600\) & 4051 & 1.90 & 25.54 & 1.46 \\ \(\Delta x=4.32\) & \(1600\times 1200\) & 8102 & 4.47 & 43.12 & 2.52 \\ \(\Delta x=2.16\) & \(320\times 2400\) & 16204 & 15.42 & 49.92 & 2.92 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance of the 1D-2D coupled model under different mesh sizes.  
focus on its applications on simulations of water temperature, water quality, and sediment transport in large scale river-lake systems.  
Conceptualization, W.M. and Y.C.; methodology, W.M. and J.W.; software, W.M. and J.W.; validation, W.M.; writing--original draft, W.M. and C.Z.; writing--review and editing, W.M., Y.C., C.Z. and L.X. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (NSFC) (Grant No. 51839008 and 51579187), and Natural Science Foundation Project of CQ CSTC (Grant No. KJQN201800712).  
Our grateful thanks to [PERSON] for his valuable advice for revising the manuscript.  
The authors declare no conflict of interest.