# Development of a Two-Dimensional Hybrid Sediment-Transport Model
## 3 Numerical Method
### FVM Discretization  
Following the work of [PERSON] et al. [36], the discretization method is illustrated by the integral form of the suspended-load transport equation in the typical cells shown in Figure 1, where one triangle cell and one quadrilateral cell are neighbored with each other. In each cell, the centroid nodes (denoted by the superscript \"\(c\)\") and the vertex nodes (denoted by the superscript \"\(v\)\") are counterclockwise numbered. A similar discretization procedure can be carried out for the bed-load transport equation.  
For the time integral, the first-order Euler forward scheme was used:  
\[\int\limits_{\Omega}\frac{\partial(hC_{k}/\beta_{sk})}{\partial t}d\Omega \approx\frac{\left(hC_{k}/\beta_{sk}\right)^{n+1}-\left(hC_{k}/\beta_{sk} \right)^{n}}{\Delta t}\cdot\Omega \tag{19}\]  
where the superscript \"\(n\)\" denotes the time level.  
For the convective fluxes,  
\[\int\limits_{\Omega}\left(\frac{\partial(uhC_{k})}{\partial x}+\frac{\partial( hC_{k})}{\partial y}\right)d\Omega\approx\sum F_{0-m}\left(hC_{k}\right)_{0-m} \tag{20}\]  
where \(F_{0-m}=u_{0-m}\Delta y_{i-j}-v_{0-m}\Delta x_{i-j}\) is the flux; the subscripts \"\(i\)\" and \"\(j\)\" denote the two vertices of the edge \"\(0-m\)\" ordered counterclockwise. Note that for coordinate calculations, the subscript \"\(i-j\)\" denotes the subtraction of coordinate \"\(i\)\" from \"\(j\)\"; that is, \(\Delta x_{i-j}=x_{j}-x_{i}\) and \(\Delta y_{i-j}=y_{j}-y_{i}\).  
The second-order upwind scheme was used for the edge center \"\(0-m\)\",  
\[\phi_{0-m}\approx\left\{\begin{array}{ll}\phi_{0}+\left(\phi_{x}\right)_{0 }\Delta x_{0-F}+\left(\phi_{y}\right)_{0}\Delta y_{0-F}&\left(F_{0-m}>0\right) \\ \phi_{m}+\left(\phi_{x}\right)_{m}\Delta x_{m-F}+\left(\phi_{y}\right)_{m} \Delta y_{m-F}&\left(F_{0-m}<0\right)\end{array}\right. \tag{21}\]  
where \(\phi_{0-m}=\left(hC_{k}\right)_{0-m}\) and, \(\phi_{x}\) and \(\phi_{y}\) represent the first-order derivatives with respect to \(x\) and \(y\).  
In Equation (21), the first term on the RHS will be treated implicitly, while the second term with gradients is treated as the source term.  
\[S^{\phi}=\sum_{m}\{F^{+}_{0-m}\cdot[(\phi_{x})_{0}\Delta x_{0-F}+(\phi_{y})_{0} \Delta y_{0-F}]+F^{-}_{0-m}\cdot[(\phi_{x})_{m}\Delta x_{m-F}+(\phi_{y})_{m} \Delta y_{m-F})]\} \tag{22a}\]  
\[F^{\pm}_{0-m}=\frac{1}{2}(F_{0-m}\pm|F_{0-m}|) \tag{22b}\]  
For the diffusion fluxes, the Green Theorem was applied so that one can obtain:  
\[\int\limits_{\Omega_{0-m}}\big{[}\frac{\partial}{\partial x}[h\varepsilon_{s} \frac{\partial C_{k}}{\partial x})+\frac{\partial}{\partial y}(h\varepsilon_{ s}\frac{\partial C_{k}}{\partial y})]d\Omega\approx\sum_{m}\big{[}(h \varepsilon_{s}\frac{\partial C_{k}}{\partial x})_{0-m}\cdot\Delta y_{i-j}-( h\varepsilon_{s}\frac{\partial C_{k}}{\partial y})_{0-m}\cdot\Delta x_{i-j} \big{]} \tag{23}\]  
where \(\Omega_{0-m}\) is the area of the quadrilateral at the edge \"\(0-m\)\" (refer to the quadrilateral 0\"-1\"-1\"-2\" in Figure 1).  
For the source terms, one can obtain:  
\[\int\limits_{\Omega}\big{[}\alpha\omega_{sk}(C_{*k}-C_{k})-S^{\phi}\big{]}d \Omega\approx[-\alpha\omega_{sk}C_{k}+\alpha\omega_{sk}C_{*k}-S^{\phi}]\cdot\Omega \tag{24}\]  
where the first term with \(C_{k}\) at RHS will be treated implicitly.  
Figure 1: Typical triangle and quadrilateral cells: the black circle denotes the cell centers; the gray circle denotes the vertices; the big white circles denote the edge centers; the small white circle denotes the interception point of the edge and the line of two neighboring cell centers; the blank quadrilateral (0”-1”-1”-2”) with the dash-dot lines is formed by two vertices of the edge and two neighboring cell centers; and the quadrilateral filled with gray color is formed by four neighboring edge centers (E\({}_{0-1}\), E\({}_{0-3}\), E\({}_{1-4}\), and E\({}_{1-6}\)) with E\({}_{0-1}\) as centroid.