§2 Runge- Kutta method 将改进欧拉法推广为: y+=y1+hK1+2K2 K f(xi, yu K =f(r +ph, y;+plr y(x)=,f(x, y) 首先希望能确定系数λ1 精度,即在y;=y(x)的自 =f(x,y)+∫(x R=y(x+1) d ∫(x,y)+∫1(x,y)f(x,y) sepl:将k2在(x1,y2)点作Tay K2=f(x+ph, y; phU) f(xi,y)+phf,(x;, yi+phK,(i,yi)+O(h) -y(x,)+phy(x1)+O() Sp2:将K2代入第式,得到 Va=D2 +hay'(x;)+aly'(x )+ phy(x)+O(2) y1+(1+2)hy(x;)+a2ph2y"(x;)+O(h3)§2 Runge-Kutta Method 首先希望能确定系数 1、2、p，使得到的算法格式有2阶 精度，即在 yi = y(xi ) 的前提假设下，使得 ( ) ( ) 3 Ri = y xi+1 − yi+1 = O h Step 1: 将 K2 在 ( xi , yi ) 点作 Taylor 展开 ( , ) ( , ) ( , ) ( ) ( , ) 2 1 2 1 f x y phf x y phK f x y O h K f x ph y phK i i x i i y i i i i = + + + = + + ( ) ( ) ( ) 2 = y xi + phy xi + O h 将改进欧拉法推广为： ( , ) ( , ) [ ] 2 1 1 1 1 1 2 2 K f x ph y phK K f x y y y h K K i i i i i i = + + = + = +  +  ( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( , ) f x y f x y f x y dx dy f x y f x y f x y dx d y x x y x y = + = +  = Step 2: 将 K2 代入第1式，得到   ( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( )] 2 3 1 2 2 2 1 1 2 y hy x ph y x O h y y h y x y x phy x O h i i i i i i i i = + +  +  + + = +  +  +  +     