概要

Poisson方程式を差分法で離散化したときの離散化形式と，Jacobi法で反復計算を行うときの式を導出する．
実装の必要が生じたときには忘れていることが多いので，取り急ぎ記しておく．

そのうち導出とかを加筆する．

Poisson方程式

数値計算において，Poisson方程式は様々な場面で現れてくる．

1次元

\frac{\partial^2 f}{\partial x^2} = g

2次元

\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2} = \frac{\rho}{\Delta t}\theta^*

3次元

\frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}+\frac{\partial^2 \psi}{\partial z^2} = -\omega

離散化

1次元

\frac{f_{i+1}-2f_{i}+f_{i-1}}{\Delta x^2} = g_i

2次元

\frac{p_{i+1,j}-2p_{i,j}+p_{i-1,j}}{\Delta x^2}+\frac{p_{i,j+1}-2p_{i,j}+p_{i,j-1}}{\Delta y^2}  = \frac{\rho}{\Delta t}\theta^*_{i,j}

3次元

\frac{\psi_{i+1,j,z}-2\psi_{i,j,z}+\psi_{i-1,j,z}}{\Delta x^2}+\frac{\psi_{i,j+1,z}-2\psi_{i,j,z}+\psi_{i,j-1,z}}{\Delta y^2}+\frac{\psi_{i,j,z+1}-2\psi_{i,j,z}+\psi_{i,j,z-1}}{\Delta z^2} = -\omega_{i,j,k}

漸化式

1次元

f_{i}^{(m+1)} = \frac{f_{i+1}^{(m)}+f_{i-1}^{(m)}}{2} - \frac{\Delta x^2}{2}g_i

2次元

p_{i,j}^{(m+1)} = \frac{\Delta y^2(p_{i+1,j}^{(m)}+p_{i-1,j}^{(m)}) +\Delta x^2(p_{i,j+1}^{(m)}+p_{i,j-1}^{(m)})-\frac{\rho}{\Delta t}\theta^*_{i,j}\Delta x^2\Delta y^2}{2(\Delta x^2+\Delta y^2)}

3次元

\psi_{i,j,k}^{(m+1)} = \left.\left[\Delta y^2\Delta z^2(\psi_{i+1,j,k}^{(m)}+\psi_{i-1,j,k}^{(m)}) +\Delta x^2\Delta z^2(\psi_{i,j+1,k}^{(m)}+\psi_{i,j-1,k}^{(m)})+\Delta y^2\Delta z^2(\psi_{i,j,k+1}^{(m)}+\psi_{i,j,k-1}^{(m)})-(-\omega_{i,j,k})\Delta x^2\Delta y^2\Delta z^2\right] \middle/ 2(\Delta y^2\Delta z^2+\Delta x^2\Delta z^2+\Delta x^2\Delta y^2)\right.

準調和方程式

\nabla\cdot\left(\frac{1}{\rho}\nabla p\right) = \frac{1}{\Delta t}\theta^*

離散化

\frac{1}{\Delta x}\left(\frac{2}{\rho_{i+1,j}+\rho_{i,j}}\frac{p_{i+1,j}-p_{i,j}}{\Delta x}-\frac{2}{\rho_{i,j}+\rho_{i-1,j}}\frac{p_{i,j}-p_{i-1,j}}{\Delta x}\right)+\frac{1}{\Delta y}\left(\frac{2}{\rho_{i,j+1}+\rho_{i,j}}\frac{p_{i,j+1}-p_{i,j}}{\Delta y}-\frac{2}{\rho_{i,j}+\rho_{i,j-1}}\frac{p_{i,j}-p_{i,j-1}}{\Delta y}\right) = \frac{1}{\Delta t}\theta^*_{i,j}

漸化式

p_{i,j}^{(m+1)} = \frac{ \frac{1}{\Delta x^2} \left(\frac{p_{i+1,j}^{(m)}}{\rho_{i+\frac{1}{2},j}}+\frac{p_{i-1,j}^{(m)}}{\rho_{i-\frac{1}{2},j}}\right) + \frac{1}{\Delta y^2} \left( \frac{p_{i,j+1}^{(m)}}{\rho_{i,j+\frac{1}{2}}}+\frac{p_{i,j-1}^{(m)}}{\rho_{i,j-\frac{1}{2}}} \right) -\frac{\theta^*_{i,j}}{\Delta t}}{\frac{1}{\Delta x^2} \left(\frac{1}{\rho_{i+\frac{1}{2},j}}+\frac{1}{\rho_{i-\frac{1}{2},j}}\right) + \frac{1}{\Delta y^2} \left( \frac{1}{\rho_{i,j+\frac{1}{2}}}+\frac{1}{\rho_{i,j-\frac{1}{2}}} \right)}

\begin{eqnarray}
\rho_{i+\frac{1}{2},j} &=& \frac{\rho_{i+1,j}+\rho_{i,j}}{2}\\
\rho_{i-\frac{1}{2},j} &=& \frac{\rho_{i,j}+\rho_{i-1,j}}{2}\\
\rho_{i,j+\frac{1}{2}} &=& \frac{\rho_{i,j+1}+\rho_{i,j}}{2}\\
\rho_{i,j-\frac{1}{2}} &=& \frac{\rho_{i,j}+\rho_{i,j-1}}{2}
\end{eqnarray}