Shallow Water Model – Shallow Water 2D

Mathematical model

SW2D solves the 2D shallow water equations:

$\frac{\partial \mathbf{U}}{\partial t} + \frac{\partial \mathbf{F}}{\partial x} + \frac{\partial \mathbf{G}}{\partial y} = \mathbf{S}$

$\mathbf{U} = \left[\begin{array}{c}u \\ q \\ r\end{array}\right]\,\,\mathbf{F} = \left[\begin{array}{c}q \\ \displaystyle{\frac{q^2}{h}} + \frac{1}{2}gh^2 \\ \displaystyle{\frac{qr}{h}}\end{array}\right]\,\,\mathbf{G} = \left[\begin{array}{c}r \\ \displaystyle{\frac{qr}{h}}\\ \displaystyle{\frac{r^2}{h}} + \frac{1}{2}gh^2\end{array}\right]\,\,\mathbf{S} = \left[\begin{array}{c}0 \\ gh\left(S_{0,x}-S_{f,x}\right) \\ gh\left(S_{0,y}-S_{f,y}\right)\end{array}\right]$

where $t$ (resp. $x$ and $y$ ) is the time (resp. space), $g$ the gravitational acceleration, $h$ represents the water depth, $q$ (resp. $r$ ) the unit-discharge in the $x$ (resp. $y$ ) direction, $S_{0,x}$ (resp. $S_{0,y}$ ) the bottom slope in the $x$ (resp. $y$ ) direction and $S_{f,x}$ (resp. $S_{f,y}$ ) the friction slope in the $x$ (resp. $y$ ) direction estimated using the classical Strickler formulation:

$S_{0,x} = - \frac{\partial z_b}{\partial x} \quad S_{f,x} = -g h \frac{(u^2 + v^2)^{1/2}}{K^2 h^{4/3}}u$

$S_{0,y} = - \frac{\partial z_b}{\partial y} \quad S_{f,y} = -g h \frac{(u^2 + v^2)^{1/2}}{K^2 h^{4/3}}v$

Numerical scheme

The shallow water equations are discretized using a finite volume approach:

$\mathbf{U}_{i}^{n+1} = \mathbf{U}_{i}^{n} - \frac{\Delta t}{A_{i}}\left[\sum_j^{N\left(i\right)}L_j \mathbf{F}_j n_j + \mathbf{S}\left(\mathbf{U}_{i}^{n}\right)\right]$

The fluxes and the source terms are estimated using an explicit method. The fluxes through the interfaces $\mathbf{F}_j$ are estimated using the HLL/HLLC Riemann solver.