b) Formulieren Sie eine algebraische Gleichung für die numerische Lösung der Potentialgleichung oder Stromfunktionsgleichung! (Krummlinig)
f```math \frac{\delta^2 \Phi}{\delta x^2} = \frac{\delta}{\delta x} \left( \frac{\delta \xi}{\delta x} \frac{\delta}{\delta \xi} + \frac{\delta \eta}{\delta x} \frac{\delta}{\delta \eta} \right) \Phi
```math
= \left( \left( \frac{\delta^2 \xi}{\delta x^2}\frac{\delta}{\delta \xi} + \frac{\delta^2 \eta}{\delta x^2}\frac{\delta}{\delta \eta}\right) + \left( \left( \frac{\delta \xi}{\delta x}\right)^2 \frac{\delta^2}{\delta \xi^2} + 2 \frac{\delta \xi}{\delta x}\frac{\delta \eta}{\delta x} \frac{\delta^2}{\delta \xi \delta \eta} + \left( \frac{\delta \eta}{\delta x}\right)^2 \frac{\delta^2}{\delta \eta^2}\right) \right) \Phi
\frac{\delta^2 \Phi}{\delta y^2} = \frac{\delta}{\delta y} \left( \frac{\delta \xi}{\delta y} \frac{\delta}{\delta \xi} + \frac{\delta \eta}{\delta y} \frac{\delta}{\delta \eta} \right) \Phi
= \left( \left( \frac{\delta^2 \xi}{\delta y^2}\frac{\delta}{\delta \xi} + \frac{\delta^2 \eta}{\delta y^2}\frac{\delta}{\delta \eta}\right) + \left( \left( \frac{\delta \xi}{\delta y}\right)^2 \frac{\delta^2}{\delta \xi^2} + 2 \frac{\delta \xi}{\delta y}\frac{\delta \eta}{\delta y} \frac{\delta^2}{\delta \xi \delta \eta} + \left( \frac{\delta \eta}{\delta y}\right)^2 \frac{\delta^2}{\delta \eta^2}\right) \right) \Phi
\frac{\delta^2 \Phi}{\delta x^2} + \frac{\delta^2 \Phi}{\delta y^2} = 0
\rightarrow \left( \alpha_1 \frac{\delta^2}{\delta \xi^2} + \alpha_2 \frac{\delta^2}{\delta \eta^2} + \alpha_3 \frac{\delta^2}{\delta \xi \delta \eta} + \alpha_4 \frac{\delta}{\delta \xi} + \alpha_5 \frac{\delta}{\delta \eta} + \alpha_6 \right) \Phi = 0
Daraus folgt:
\alpha_1 = \left(\frac{\delta \xi}{\delta x}\right)^2 + \left(\frac{\delta \xi}{\delta y}\right)^2
\alpha_2 = \left(\frac{\delta \eta}{\delta x}\right)^2 + \left(\frac{\delta \eta}{\delta y}\right)^2
\alpha_3 = 2 \left(\frac{\delta \xi}{\delta x}\frac{\delta \eta}{\delta x} + \frac{\delta \xi}{\delta y}\frac{\delta \eta}{\delta y} \right)
\alpha_4 = \frac{\delta^2 \xi}{\delta x^2} + \frac{\delta^2 \xi}{\delta y^2}
\alpha_5 = \frac{\delta^2 \eta}{\delta x^2} + \frac{\delta^2 \eta}{\delta y^2}
Mit:
\frac{\delta \xi}{\delta x} = \frac{\xi_{i+1, j} - \xi_{i-1,j}}{x_{i+1,j} - x_{i-1,j}}
\frac{\delta \xi}{\delta y} = \frac{\xi_{i, j+1} - \xi_{i,j-1}}{y_{i,j+1} - y_{i,j-1}}
\frac{\delta^2 \xi}{\delta x^2} = \frac{\xi_{i+1, j} - 2 \xi_{i,j} + \xi_{i-1,j}}{\left(\frac{x_{i+1,j} - x_{i-1,j}}{2}\right)^2}
\frac{\delta^2 \xi}{\delta y^2} = \frac{\xi_{i, j+1} - 2 \xi_{i,j} + \xi_{i,j-1}}{\left(\frac{y_{i,j+1} - y_{i,j-1}}{2}\right)^2}
\frac{\delta \eta}{\delta x} = \frac{\eta_{i+1, j} - \eta_{i-1,j}}{x_{i+1,j} - x_{i-1,j}}
\frac{\delta \eta}{\delta y} = \frac{\eta_{i, j+1} - \eta_{i,j-1}}{y_{i,j+1} - y_{i,j-1}}
\frac{\delta^2 \eta}{\delta x^2} = \frac{\eta_{i+1, j} - 2 \eta_{i,j} + \eta_{i-1,j}}{\left(\frac{x_{i+1,j} - x_{i-1,j}}{2}\right)^2}
\frac{\delta^2 \eta}{\delta y^2} = \frac{\eta_{i, j+1} - 2 \eta_{i,j} + \eta_{i,j-1}}{\left(\frac{y_{i,j+1} - y_{i,j-1}}{2}\right)^2}
Und schließlich folgt:
\alpha_1 \frac{\Phi_{i+1,j} - 2 \Phi_{i,j} + \Phi_{i-1,j}}{\Delta \xi^2} +
\alpha_2 \frac{\Phi_{i,j+1} - 2 \Phi_{i,j} + \Phi_{i,j-1}}{\Delta \eta^2} +
\alpha_3 \frac{\Phi_{i+1,j+1} - \Phi_{i-1,j+1} - \Phi_{i+1,j-1} + \Phi_{i-1,j-1}}{4 \Delta \xi \Delta \eta} +
\alpha_4 \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \xi} + \alpha_5 \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \eta} + \alpha_6 = 0
Zwischenschritte
2 \Phi_{i,j} \left( \frac{\alpha_1}{\Delta \xi^2} + \frac{\alpha_2}{\Delta \eta^2} \right) = \alpha_1 \frac{\Phi_{i+1,j} + \Phi_{i-1,j}}{\Delta \xi^2} +
\alpha_2 \frac{\Phi_{i,j+1} + \Phi_{i,j-1}}{\Delta \eta^2} +
\alpha_3 \frac{\Phi_{i+1,j+1} - \Phi_{i-1,j+1} - \Phi_{i+1,j-1} + \Phi_{i-1,j-1}}{4 \Delta \xi \Delta \eta} +
\alpha_4 \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \xi} + \alpha_5 \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \eta} + \alpha_6
\Phi_{i,j} = \frac{\alpha_1 \frac{\Phi_{i+1,j} + \Phi_{i-1,j}}{\Delta \xi^2} +
\alpha_2 \frac{\Phi_{i,j+1} + \Phi_{i,j-1}}{\Delta \eta^2} +
\alpha_3 \frac{\Phi_{i+1,j+1} - \Phi_{i-1,j+1} - \Phi_{i+1,j-1} + \Phi_{i-1,j-1}}{4 \Delta \xi \Delta \eta} +
\alpha_4 \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \xi} + \alpha_5 \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \eta} + \alpha_6}{2 \left( \frac{\alpha_1}{\Delta \xi^2} + \frac{\alpha_2}{\Delta \eta^2} \right)}
Lösung:
\Phi_{i,j} = \frac{\alpha_1 \frac{\Phi_{i+1,j} + \Phi_{i-1,j}}{\Delta \xi^2} +
\alpha_2 \frac{\Phi_{i,j+1} + \Phi_{i,j-1}}{\Delta \eta^2} +
\alpha_3 \frac{\Phi_{i+1,j+1} - \Phi_{i-1,j+1} - \Phi_{i+1,j-1} + \Phi_{i-1,j-1}}{4 \Delta \xi \Delta \eta} +
\alpha_4 \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \xi} + \alpha_5 \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \eta} + \alpha_6}{2 \left( \frac{\alpha_1}{\Delta \xi^2} + \frac{\alpha_2}{\Delta \eta^2} \right)}