Finite differences operators

The operators D_ox and D_oy are defined by (4.10) to (4.14) (Gustafsson et al., 1995; Hirsch, 1991)^7.2.

$$D_{ox}\xi_{i\;j}=\frac{\xi_{i\;j+1}-\xi_{i\;j-1}} {2 \Delta x... ...D_{oy}\xi_{i\;j}=\frac{\xi_{i+1\;j}-\xi_{i-1\;j}} {2 \Delta y}$$ (7.7)

$$D_{-x}\xi_{i\;j}=\frac{\xi_{i\;j}-\xi_{i\;j-1}} {\Delta x} \qquad D_{-y}\xi_{i\;j}=\frac{\xi_{i\;j}-\xi_{i-1\;j}} {\Delta y}$$ (7.8)

$$D_{+x}\xi_{i\;j}=\frac{\xi_{i\;j+1}-\xi_{i\;j}} {\Delta x} \qquad D_{+y}\xi_{i\;j}=\frac{\xi_{i+1\;j}-\xi_{i\;j}} {\Delta y}$$ (7.9)

$$\begin{split} &D_{+x}D_{-x}\xi_{i\;j}=\frac{\xi_{i\;j+1} -... ...\;j} - 2 \xi_{i\;j} + \xi_{i-1\;j}} {\Delta y^2} \end{split}$$ (7.10)

$$\mathcal{D} \xi^{i\;j} = {D_{+x}D_{-x}\xi_{i\;j} + D_{+y}D_{-y}\xi_{i\;j} }$$ (7.11)