Euler Method

Core Idea

Euler’s method approximates a trajectory by repeatedly following its current derivative for a short finite time step.

若 $\dot{\mathbf{s}}=F(\mathbf{s})$，则

$${\mathbf{s}}_{n+1}={\mathbf{s}}_n+\Delta tF({\mathbf{s}}_n).$$

每一步用当前 tangent direction 代替短时间内的 curved trajectory。

Trade-off

较小的 $\Delta t$ 通常减少 discretization error，却需要更多 steps。Euler method 展示基本思想，但实际 numerical solvers 会用更高阶方法改善 accuracy 与 efficiency 的平衡。

Connections

• Differential Equations provide $F$.
• Initial Conditions provide ${\mathbf{s}}_0$.
• Vector Field gives the direction used at every step.
• Tiny Nudges explains the local linear approximation behind the update.

Search and Source Mentions

• 01 - Differential Equations, a Tourist’s Guide > 7. Formula Map / 公式地图
• 微积分BC第10次正课 · Euler’s Method