c) Programmieren Sie alle notwendigen Routinen zur Lösung eines Laplace-Problems! (Krummlinig)
Randbedingungen:
\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)}
Mit:
\frac{\delta \Phi}{\delta \eta} = \frac{- \Phi_{i,j+2} + 4 \Phi_{i,j+1} - 3 \Phi_{i,j}}{2 \Delta \eta}
\frac{\delta \Phi}{\delta \xi} = \frac{\Phi_{i+1,j} - \Phi_{i-1,j}}{2 \Delta \xi}
\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}
Herleitung
\frac{dh}{dx} = \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)}
\frac{dh}{dx} \left( \frac{\delta \Phi}{\delta \xi} \frac{\delta \xi}{\delta x} + \frac{\delta \Phi}{\delta \eta} \frac{\delta \eta}{\delta x} \right) = \left( \frac{\delta \Phi}{\delta \xi} \frac{\delta \xi}{\delta y} + \frac{\delta \Phi}{\delta \eta} \frac{\delta \eta}{\delta y} \right)
\frac{dh}{dx}\left(\frac{\delta \Phi}{\delta \eta} \frac{\delta \eta}{\delta x} \right) - \left( \frac{\delta \Phi}{\delta \eta} \frac{\delta \eta}{\delta y} \right) = \left( \frac{\delta \Phi}{\delta \xi} \frac{\delta \xi}{\delta y}\right) - \frac{dh}{dx} \left(\frac{\delta \Phi}{\delta \xi} \frac{\delta \xi}{\delta x}\right)
\frac{\delta \Phi}{\delta \eta} \left(\frac{\delta \eta}{\delta x}\frac{dh}{dx} - \frac{\delta \eta}{\delta y} \right) = \frac{\delta \Phi}{\delta \xi} \left( \frac{\delta \xi}{\delta y} - \frac{\delta \xi}{\delta x}\frac{dh}{dx} \right)
\frac{\delta \Phi}{\delta \eta} = - \frac{\delta \Phi}{\delta \xi} \frac{\left( \frac{\delta \xi}{\delta y} - \frac{\delta \xi}{\delta x}\frac{dh}{dx} \right)}{\left( \frac{\delta \eta}{\delta y} - \frac{\delta \eta}{\delta x}\frac{dh}{dx} \right)}
\frac{- \Phi_{i,j+2} + 4 \Phi_{i,j+1} - 3 \Phi_{i,j}}{2 \Delta \eta} = - \frac{\delta \Phi}{\delta \xi} \frac{\left( \frac{\delta \xi}{\delta y} - \frac{\delta \xi}{\delta x}\frac{dh}{dx} \right)}{\left( \frac{\delta \eta}{\delta y} - \frac{\delta \eta}{\delta x}\frac{dh}{dx} \right)}
- 3 \Phi_{i,j} = \Phi_{i,j+2} - 4 \Phi_{i,j+1} - 2 \Delta \eta \frac{\delta \Phi}{\delta \xi} \frac{\left( \frac{\delta \xi}{\delta y} - \frac{\delta \xi}{\delta x}\frac{dh}{dx} \right)}{\left( \frac{\delta \eta}{\delta y} - \frac{\delta \eta}{\delta x}\frac{dh}{dx} \right)}
\Phi_{i,j} = \frac{1}{3} \left(4 \Phi_{i,j+1} - \Phi_{i,j+2} + 2 \Delta \eta \frac{\delta \Phi}{\delta \xi} \frac{\left( \frac{\delta \xi}{\delta y} - \frac{\delta \xi}{\delta x}\frac{dh}{dx} \right)}{\left( \frac{\delta \eta}{\delta y} - \frac{\delta \eta}{\delta x}\frac{dh}{dx} \right)} \right)
folgt für die untere Schranke:
\Phi_{i,j} = \frac{1}{3} \left( 4 \Phi_{i,j+1} - \Phi_{i,j+2} + 2 \Delta \eta \frac{\delta \Phi}{\delta \xi} \left(\frac{ \frac{\delta \xi}{\delta y} - \frac{\delta \xi}{\delta x} \frac{h(x_{i+1,j}) - h(x_{i-1,j})}{x_{i+1,j} - x_{i-1,j}} }{\frac{\delta \eta}{\delta y} - \frac{\delta \eta}{\delta x} \frac{h(x_{i+1,j}) - h(x_{i-1,j})}{x_{i+1,j} - x_{i-1,j}}}\right)\right)
Für die obere Schranke lässt sich gleichermaßen folgende Beziehung herleiten (siehe Kommentar):
\Phi_{i, j} = \frac{1}{3} \left(4 \Phi_{i, j-1} - \Phi_{i, j-2} - 2 \Delta \eta \frac{\delta \Phi}{\delta \xi} \frac{\left( \frac{\delta \xi}{\delta y} - \frac{\delta \xi}{\delta x}\frac{dh}{dx} \right)}{\left( \frac{\delta \eta}{\delta y} - \frac{\delta \eta}{\delta x}\frac{dh}{dx} \right)} \right)
Edited by Moritz Leibauer