P13
Problem with Central Differencing
Central differencing gives the derivative in the middle.
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The cell doesn’t exist at (i+0.5, j).
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To get \( \frac{∂f_ {i,j}}{∂x} \), we need \(f_{i−1,j}\) and \(f_{i+1,j}\). But this is weird, because \(f_{i,j}\) is unused.
✅ 前面假设所有物理量定义在格子的中间。但此处算出来的一阶微分量不在格子中间。
P14
Solution: Staggered Grid
✅ 不规定所有物理量都定义在格子中间，也可以定义在墙上。
We define some physical quantities on faces, specifically velocities.
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The x-part of the velocity is defined on vertical faces.
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The y-part of the velocity is defined on horizonal faces.
✅ 把速度定义在墙上的好处量，速度是矢量、可以用不同方向的墙表达不同方向上的速度、直观。
- Intuitively, they represent the flow speed between two cells. For example, we write the volume changing speed at cell (i,j) as:
| $$u_{i+1,j}+v_{i,j+1}−u_{i,j}−v_{i,j}$$ |
|---|
✅ 通过四面墙上的速度计算当前格子的净流出（注意正负号）
P15
Divergence-Free Condition
No volume change is equal to say the fluid is incompressible. This can be formally written as a divergence-free velocity field.
❓ 这一页没听懂、净流入流出为0，水面还怎么动呢？
✅ 由于格子不可压，每个格子的净流出（入）应该为0．
✅ \(\nabla\)为散度符号，见前面课程。
✅ 公式1为直观理解，公式2为数学推导，本质上是一致的。
P16
Bilinear Interpolation
🔎 双线性插值：见GAMES 101
P17
We use bilinear interpolation to interpolate staggered velocities as well.
本文出自CaterpillarStudyGroup，转载请注明出处。
https://caterpillarstudygroup.github.io/GAMES103_mdbook/