# A 1D-2D Coupled Lattice Boltzmann Model for Shallow Water Flows in Large Scale River-Lake Systems
## 3 Numerical Experiments
### The Dam Break Flow  
The dam break flow in a frictionless and flat rectangular channel was simulated, and its analytic solutions are given by [PERSON] [38]. As shown in Figure 2, the dam was placed across the channel at the middle plane of the domain, and the initial water depths at the upstream and downstream sides were set to 8 m and 2 m, respectively. In simulations, the channel was divided into three domains, with the middle domain simulated by the 2D submodel, and the two outer domains simulated by the 1D submodel. The upstream and downstream boundaries were set to constant water depths. The lateral boundaries of the 2D domain were treated as no-slip wall boundary conditions. \(x_{1}\) and \(x_{2}\) were the coupling interfaces, which were located at the inlet and outlet boundaries of the 2D domain, respectively. The initial velocity was zero. When the dam collapsed completely in a sudden, two bores propagating in the upstream and downstream directions, respectively, were generated. In simulations, unless mentioned otherwise, the values of the relaxation parameters for the 1D and 2D submodels were taken to be \(\tau_{1}=1.0\) and \(\tau_{2}=2.0\), respectively. We verified the accuracy and stability of the 1D-2D coupled model by analyzing its grid convergence, errors, and convergence rate. Firstly, the grid convergence of the 1D-2D coupled model was analyzed, and the lattice spacing ranged from 0.05 m to 0.4 m, with each one doubled from the former. As shown in Figure 3, the numerical results approached the analytic solution as the mesh became finer. But as the mesh became very fine, such as lattice spacing 0.05 m and 0.1 m, the result differences between them were very small. To get an accurate solution and to improve the computational efficiency, the lattice spacing of 0.1 m was reasonable, which was adopted in the following simulation. Then, the accuracy of the proposed model was analyzed by comparing its results with those of the 1D model, 2D model, and analytic solution. Note that the results of the 2D part were the average values in the \(y\)-direction. In Figure 4, water depth and velocity profiles of the three numerical models almost coincided with each other and agreed well with the analytic solutions. Figure 4: Water depth and velocity profiles of different models for the dam break flow (\(\tau_{1}=1.0\), \(\tau_{2}=2.0\), \(\Delta x=0.1\), \(Fr\leq 0.691\)): (**a**) water depth and velocity profiles at \(t=4\) s, and (**b**) water depth and velocity histories at coupling interface \(x_{1}\). Figure 3: Water depth and velocity profiles of the 1D–2D coupled model at \(t=4\) s under different lattice spacing. Figure 2: Computational domain of the dam break case. Table 1 shows the errors of three numerical models at \(x_{1}\) and \(x_{2}\), respectively. The errors of the proposed model were closed to those of the 2D model and slightly larger than those of the 1D model, but still quite small. The absolute errors for water depth were within 0.04 m, and the relative errors were within 0.5%. The errors for velocity was a little larger, with a maximum absolute error of 0.045 m/s, and the corresponding relative error of 4.97%, which is acceptable in practical engineering. Finally, the accuracy of the 1D-2D coupled model was studied by measuring the convergence rate of the relative global error as the lattice spacing changes, where the relative global error is defined as  
\[E=\frac{\sqrt{\sum{(u-u_{a})}^{2}}}{\sqrt{\sum{u_{a}}^{2}}} \tag{26}\]  
where \(u\) is the numerical solution, \(u_{a}\) is the analytic solution. The relaxation parameters of 1D and 2D submodels calculated by Equations (6) and (15), respectively, were divided into three cases: (1) \(\tau_{1}=0.9\) and \(\tau_{2}=1.7\), (2) \(\tau_{1}=1.0\) and \(\tau_{2}=2.0\), and (3) \(\tau_{1}=1.5\) and \(\tau_{2}=3.5\). For each case, four lattice spacing 0.05 m, 0.1 m, 0.2 m, and 0.4 m were tested. In Figure 5, the relative global errors were plotted natural logarithmically as functions of the lattice spacing.