Second Difference

One-Sentence Idea / 一句话

A second difference measures how consecutive first differences change.

二阶差分是相邻一阶差分的差，也等价于测量中间值相对于两侧邻居平均值的偏离。

Core Form

$${\Delta }^2{T}_i=T_{i+1}-2T_i+T_{i-1}.$$

若网格间距为 $h$：

$$\frac{{\Delta }^2{T}_i}{h^2}=\frac{T(x+h)-2T(x)+T(x-h)}{h^2}⟶T^{\prime \prime }(x).$$

它是 discrete model 与 Second Derivative 之间的桥。

Sign Intuition

• ${\Delta }^2{T}_i>0$：中间值低于邻居平均值。
• ${\Delta }^2{T}_i=0$：三个点局部落在一条直线趋势上。
• ${\Delta }^2{T}_i<0$：中间值高于邻居平均值。

Connections

• Heat Equation
• Second Derivative
• Concavity
• Euler Method
• 02 - But What Is a Partial Differential Equation > Knowledge Point 8: Neighbor Averaging Becomes a Second Difference

Search and Source Mentions

• Main source: 02 - But What Is a Partial Differential Equation