A Grid Representation and Finite Differencing

A Regular Grid Representation

✅ 把场定义在标准格子上的好处：(1)把物理量定义在格子的中心（2）计算导数或利用导数进行微分计算变得容易了。
✅ 上节课grid用1D来表示2D，2D表示3D，不是真正的grid方法。
🔎 Central Differencing：L10.

The grid is very friendly with central differencing.

$$\frac{∂f_{i+0.5,j}}{∂x}≈\frac{f_{i+1,j}−f_{i,j}}{ℎ}$$

The grid is very friendly with central differencing.

We can then obtain the discretized Laplacian operator on grid.

$$ \frac{∂^2f_{i,j}}{∂x^2}≈\frac{\frac{∂f_{i−0.5,j}}{∂x}−\frac{∂f_{i+0.5,j}}{∂x}}{ℎ}≈\frac{f_{i−1,j}+f_{i+1,j}−2f_{i,j}}{ℎ^2} $$

$$ \frac{∂^2f_{i,j}}{∂y^2}≈\frac{\frac{∂f_{i,j+0.5}}{∂y}−\frac{∂f_{i,j−0.5}}{∂y}}{ℎ} ≈\frac{f_{i,j−1}+f_{i,j+1}−2f_{i,j}}{ℎ^2} $$

$$∆f_{i,j}=\frac{∂^2f_{i,j}}{∂x^2}+\frac{∂^2f_{i,j}}{∂y^2}≈\frac{f_{i−1,j}+f_{i+1,j}+f_{i,j−1}+f_{i,j+1−4}f_{i,j}}{ℎ^2} $$

✅ 网格上的Laplace算子。

The boundary condition specifies $f_{i−1,j}$ if it’s outside.

A Dirichlet boundary: $f_{i−1,j}=C$

$$ ∆f_{i,j}≈\frac{C+f_{i+1,j}+f_{i,j−1}+f_{i,j+1}−4f_{i,j}}{ℎ^2}$$

A Neumann boundary: $f_{i−1,j}=f_{i,j}$

$$∆f_ {i,j} ≈ \frac{f_ {i+1,j}+f_ {i,j−1}+f_ {i,j+1}−3f_{i,j}}{ℎ^2}$$

✅ 至少有一个边界使用Dirithlet．否则会全部收缩为0．
✅ Neumann是约束相对关系，没有绝对数值，会有无穷多解。

P12

The process of applying Laplacian smoothing is called diffusion.

✅ Laplace的本质是与邻居做平均。

本文出自CaterpillarStudyGroup，转载请注明出处。

https://caterpillarstudygroup.github.io/GAMES103_mdbook/