## 2 Numerical Model
### Mathematical Equations
#### 2.1.2 Integration of the Topography  
The method of solving the equations by decomposition, which considers the system as homogeneous, as seen above, could not be used in the context of topography because such an approach yields poor results when \(U\) corresponds to a stationary state. It can be expressed as:  
\[\begin{cases}u=0\\ \frac{\partial(1\frac{2}{\mathrm{g}}\mathrm{h}^{2})}{\partial x}=-\mathrm{g} \mathrm{h}\frac{\partial Z}{\partial x}\end{cases}\;. \tag{13}\]  
The following Equation (11) can be resumed as a constant, namely:  
\[H=h+Z \tag{14}\]  
where H is the water level.  
H is a constant; this state physically corresponds to the state of equilibrium of the watercourse or steady state. The hydrostatic reconstruction proposed by [38], coupled with a positive numerical flux, allows us to verify important mathematical and physical properties, such as the positivity of the water height, and, thus, to avoid instabilities when dealing with dry zones. The equations are as follows:  
\[\left\{\begin{aligned} &\mathrm{h}_{\mathrm{i}+\frac{1}{2}\mathrm{L}}= \mathrm{max}(\mathrm{h}_{\mathrm{i}}+\mathrm{Z}_{\mathrm{i}}-\mathrm{max}( \mathrm{Z}_{\mathrm{i}},\mathrm{Z}_{i+1}),0)\\ &\mathrm{h}_{\mathrm{i}+\frac{1}{2}\mathrm{R}}=\mathrm{max}( \mathrm{h}_{\mathrm{i}+1}+\mathrm{Z}_{\mathrm{i}+1}-\mathrm{max}(\mathrm{Z}_{ \mathrm{i}},\mathrm{Z}_{\mathrm{i}+1}),0)\\ &\mathrm{U}_{\mathrm{i}+\frac{1}{2}\mathrm{L}}=\left(\mathrm{h}_{ \mathrm{i}+\frac{1}{2}\mathrm{L}},\mathrm{h}_{\mathrm{i}+\frac{1}{2}\mathrm{L} }\ast\mathrm{u}_{\mathrm{i}}\right)\\ &\mathrm{U}_{\mathrm{i}+\frac{1}{2}\mathrm{R}}=\left(\mathrm{h}_{ \mathrm{i}-\frac{1}{2}\mathrm{R}},\mathrm{h}_{\mathrm{i}-\frac{1}{2}\mathrm{R} }\ast\mathrm{u}_{\mathrm{i}}\right)\end{aligned}\right. \tag{15}\]  
The water height computed on the left and right sides of the cell, respectively, are expressed as \(\mathrm{h}_{\mathrm{i}+\frac{1}{2}\mathrm{L}}\) and \(\mathrm{h}_{\mathrm{i}+\frac{1}{2}\mathrm{R}}\); the general solutions found at the left and right sides of each cell are represented respectively as \(\mathrm{U}_{\mathrm{i}+\frac{1}{2}\mathrm{L}}\) and \(\mathrm{U}_{\mathrm{i}+\frac{1}{2}\mathrm{R}}\). The numerical schema can now be written as:  
\[\mathrm{U}_{\mathrm{i}}^{\mathrm{n}+1}-\mathrm{U}_{\mathrm{i}}^{\mathrm{n}}+ \frac{\mathrm{\Delta t}}{\mathrm{\Delta x}}\left(\mathrm{F}_{\mathrm{i}+\frac{ 1}{2}\mathrm{L}}^{\mathrm{n}}-\mathrm{F}_{\mathrm{i}-\frac{1}{2}\mathrm{R}}^{ \mathrm{n}}\right)=0 \tag{16}\]  
where:  
\[\mathrm{F}_{\mathrm{i}-\frac{1}{2}\mathrm{R}}^{\mathrm{n}}=\mathrm{F}_{\mathrm{ i}-\frac{1}{2}}^{\mathrm{n}}+\mathrm{S}_{\mathrm{i}-\frac{1}{2}\mathrm{R}}\text{ and }\mathrm{F}_{\mathrm{i}+\frac{1}{2}\mathrm{L}}^{\mathrm{n}}=\mathrm{F}_{\mathrm{i}+ \frac{1}{2}}^{\mathrm{n}}+\mathrm{S}_{\mathrm{i}+\frac{1}{2}\mathrm{L}} \tag{17}\]  
\[\mathrm{S}_{\mathrm{i}-\frac{1}{2}\mathrm{R}}=\begin{pmatrix}0\\ \mathrm{P}(\mathrm{h}_{\mathrm{i}}^{\mathrm{n}})-\mathrm{P}\Big{(}\mathrm{h}_{ \mathrm{i}+\frac{1}{2}\mathrm{L}}^{\mathrm{n}}\Big{)}\end{pmatrix}\text{ and }\mathrm{S}_{\mathrm{i}+\frac{1}{2}\mathrm{L}}=\begin{pmatrix}0\\ \mathrm{P}(\mathrm{h}_{\mathrm{i}}^{\mathrm{n}})-\mathrm{P}\Big{(}\mathrm{h}_{ \mathrm{i}+\frac{1}{2}\mathrm{L}}^{\mathrm{n}}\Big{)}\end{pmatrix}, \tag{18}\]  
where the hydrostatic pressure is given by \(\mathrm{P}(\mathrm{h})=\frac{1}{2}\mathrm{gh}^{2}\).