Curved Grid
\Delta \xi = 1, \ \Delta \eta = 1
With:
\xi = \xi(x) = \frac{x - x_u}{x_o - x_u}
\eta = \eta(x,y) = \frac{y - h_u(x)}{h_o(x) - h_u(x)}
And respectively:
x = x(\xi) = x_u + \xi \cdot (x_o - x_u)
y = y(\xi, \eta) = h_u(x(\xi)) + \eta \cdot \left( h_o(x(\xi)) - h_u(x(\xi)) \right)
Derivatives:
\left(\frac{\delta \xi}{\delta x}\right)_{i,j} = \frac{\xi(x_{i, j} + \Delta x, y_{i, j}) - \xi(x_{i, j} - \Delta x, y_{i, j})}{2 \Delta x}
\left(\frac{\delta \xi}{\delta y}\right)_{i,j} = \frac{\xi(x_{i, j}, y_{i, j} + \Delta y) - \xi(x_{i, j}, y_{i, j} - \Delta y)}{2 \Delta y}
\left(\frac{\delta \eta}{\delta x}\right)_{i,j} = \frac{\eta(x_{i, j} + \Delta x, y_{i, j}) - \eta(x_{i, j} - \Delta x, y_{i, j})}{2 \Delta x}
\left(\frac{\delta \eta}{\delta y}\right)_{i,j} = \frac{\eta(x_{i, j}, y_{i, j} + \Delta y) - \eta(x_{i, j}, y_{i, j} - \Delta y)}{2 \Delta y}
Conture Conditions
\frac{dh}{dx} = \frac{h(x_{i+1,j}) - h(x_{i-1,j})}{x_{i+1,j} - x_{i-1,j}} = \frac{\frac{\delta \Phi}{\delta y}}{\frac{\delta \Phi}{\delta x}} = \frac{\left( \frac{\delta \Phi}{\delta \xi} \frac{\delta \xi}{\delta y} + \frac{\delta \Phi}{\delta \eta} \frac{\delta \eta}{\delta y} \right)}{\left( \frac{\delta \Phi}{\delta \xi} \frac{\delta \xi}{\delta x} + \frac{\delta \Phi}{\delta \eta} \frac{\delta \eta}{\delta x} \right)}
South, North
\left( \frac{\delta \Phi}{\delta \xi} \right)_{i,j} = \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \xi}
South Conture:
\left( \frac{\delta \Phi}{\delta \eta} \right)_{i,j} = \frac{- \Phi_{i,j+2} + 4 \Phi_{i,j+1} - 3 \Phi_{i,j}}{2 \Delta \eta}
North Conture:
\left( \frac{\delta \Phi}{\delta \eta} \right)_{i,j} = \frac{\Phi_{i, j-2} - 4 \Phi_{i, j-1} + 3 \Phi_{i, j} }{2 \Delta \eta}
East, West
\left( \frac{\delta \Phi}{\delta \eta} \right)_{i,j} = \frac{\Phi_{i,j+1} - \Phi_{i,j-1}}{2 \Delta \eta}
West Conture:
\left( \frac{\delta \Phi}{\delta \xi} \right)_{i,j} = \frac{- \Phi_{i+2,j} + 4 \Phi_{i+1,j} - 3 \Phi_{i,j}}{2 \Delta \xi}
East Conture:
\left( \frac{\delta \Phi}{\delta \xi} \right)_{i,j} = \frac{\Phi_{i-2, j} - 4 \Phi_{i-1, j} + 3 \Phi_{i, j} }{2 \Delta \xi}