# 欧拉方法

## 欧拉方法的推导

${\displaystyle y'(t)=f(t,y(t)),\qquad y(t_{0})=y_{0},}$

${\displaystyle y_{n+1}=y_{n}+hf(t_{n},y_{n}).\qquad \qquad }$

${\displaystyle y^{(N)}(t)=f(t,y(t),y'(t),\ldots ,y^{(N-1)}(t))}$

${\displaystyle \mathbf {z} '(t)={\begin{pmatrix}z_{1}'(t)\\\vdots \\z_{N-1}'(t)\\z_{N}'(t)\end{pmatrix}}={\begin{pmatrix}y'(t)\\\vdots \\y^{(N-1)}(t)\\y^{(N)}(t)\end{pmatrix}}={\begin{pmatrix}z_{2}(t)\\\vdots \\z_{N}(t)\\f(t,z_{1}(t),\ldots ,z_{N}(t))\end{pmatrix}}}$

## 应用例题

${\displaystyle y_{n+1}=y_{n}+hf(t_{n},y_{n}).}$

${\displaystyle f(t_{0},y_{0})=f(0,1)=1.}$

${\displaystyle h\cdot f(y_{0})=1\cdot 1=1.}$
${\displaystyle y_{0}+hf(y_{0})=y_{1}=1+1\cdot 1=2.}$

${\displaystyle y_{2}=y_{1}+hf(y_{1})=2+1\cdot 2=4}$
${\displaystyle y_{3}=y_{2}+hf(y_{2})=4+1\cdot 4=8}$

${\displaystyle y_{n}}$ ${\displaystyle t_{n}}$ ${\displaystyle y'(t)}$ ${\displaystyle h}$ ${\displaystyle dy}$ ${\displaystyle y_{n+1}}$
1 0 1 1 1 2
2 1 2 1 2 4
4 2 4 1 4 8

## 局部截尾误差

${\displaystyle y_{1}=y_{0}+hf(t_{0},y_{0}).\quad }$

${\displaystyle y(t_{0}+h)=y(t_{0})+hy'(t_{0})+{\frac {1}{2}}h^{2}y''(t_{0})+O(h^{3}).}$

${\displaystyle \mathrm {LTE} =y(t_{0}+h)-y_{1}={\frac {1}{2}}h^{2}y''(t_{0})+O(h^{3}).}$

${\displaystyle y}$拥有三阶有界导数时，这个结果是成立的。[2]

## 全局截尾误差

${\displaystyle |{\text{GTE}}|\leq {\frac {hM}{2L}}(e^{L(t-t_{0})}-1)\qquad \qquad }$

## 註腳

1. ^ 命名自它的发明者萊昂哈德·歐拉
2. ^ 初值問題
3. ^ 即每步误差
4. ^ 欧拉方法经常应用于作为构建一些更复杂方法的基础，例如预估-校正方法

## 参考资料

1. ^
2. ^ Butcher 2003，第60頁
3. ^ Atkinson 1989，第344頁
4. ^ Butcher 2003，第49頁
5. ^ Atkinson 1989，第346頁；Lakoba 2012，公式 (1.16)
6. ^ Iserles 1996，第7頁
7. ^ Butcher 2003，第63頁