Accuracy analytical determination.

To prove that the developed model is second order accurate we must demonstrate that the numerical approximations to derivatives used in the discrete form of the Shallow-water equations are all of second order.

The central operators defined in Section 4.2.3 are all of second order when applied in the difference scheme (Hirsch, 1991):

1.

Operators $D_{ox}\xi_{i\;j}$ and $D_{oy}\xi_{i\;j}$ defined in (4.10) are both centered in space, then:

$$\begin{split} & \frac{\partial \xi}{\partial x} = D_{ox}\xi_... ... \xi}{\partial y} = D_{oy}\xi_{i\;j}+O(\Delta y^2) \end{split}$$

2.

Operators $D_{-x}\xi_{i\;j}$ and $D_{oy}\xi_{i\;j}$ defined in (4.11) are always centered in space. As an example, when D_-x and D_-y are employed in (4.2), they are centered in the middle of cell, where the level is calculated, using data from the sides of the cell to make the differentiation (see figure 4.1 for an explanation of the staggered grid) and hence:

$$\begin{split} & \frac{\partial u}{\partial x} = D_{-x}u^{n+1... ...} = D_{-y}(H^n_{i\;j} \; v^n_{i\;j})+O(\Delta y^2) \end{split}$$

3.

Similarly, operators $D_{+x}\xi_{i\;j}$ and $D_{+y}\xi_{i\;j}$ defined in (4.12) are centered in space. As an example case: when they are used in equations (4.3) and (4.7) respectively, they are centered in the sides of the cell, employing data form adjacent cells to make the differentiation, therefore:

$$\begin{split} & \frac{\partial \eta}{\partial x} = D_{+x}\et... ...riptscriptstyle \frac{1}{2}}}_{i\;j}+O(\Delta y^2) \end{split}$$

4.

Approximations to the second derivative $D_{+x}D_{-x}\xi_{i\;j}$ and $D_{+y}D_{-y}\xi_{i\;j}$ introduced in (4.13), employed in the Laplacian (4.14), and hence in (4.3) and (4.7) are second order accurate. Therefore, the Laplacian is second order accurate.

$$\Delta v = \mathcal{D} v^n_{i\;j} + O(\Delta x^2)$$

Since every finite difference operator being employed and boundary conditions are second order accurate, the numerical scheme has the same accuracy.