物理学

Last updated Nov 30, 2022 Edit Source

ベクトル解析ノート

波動方程式の離散化

$$ \begin{aligned} \frac{\partial^2u}{\partial t^2} &= v^2\left(\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2}\right) \\ \frac{u_{i,j}^{n+1}-2u_{i,j}^n+u_{i,j}^{n-1}}{\Delta t^2} &= v^2\left(\frac{u_{i,j}^{n+1}-2u_{i,j}^n+u_{i,j}^{n-1}}{\Delta x^2} + \frac{u_{i,j}^{n+1}-2u_{i,j}^n+u_{i,j}^{n-1}}{\Delta y^2}\right) \\ u_{i,j}^{n+1} &= 2u_{i,j}^n - u_{i,j}^{n-1} + \Delta t^2v^2\left(\frac{u_{i,j}^{n+1}-2u_{i,j}^n+u_{i,j}^{n-1}}{\Delta x^2} + \frac{u_{i,j}^{n+1}-2u_{i,j}^n+u_{i,j}^{n-1}}{\Delta y^2}\right) \end{aligned} $$