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3.3 Algorithm (Runge-Kutta) Method of order 4

There are generalization of R-K Method of order 2 to higher order methods. Without getting into analytical details , we state the R-K method of order 4. It is a widely used algorithm. For the initial value problems (1.1), set $x_i=a+ih$, i=i=0,1,2..n and $y_0=y_0$ for k=0,1,2...n-1, define $y_{k+1}$ by

$$y_{k+1}=y_k+\frac{1}{6}(k_1+2k_2+2k_3+k_4)$$

where

$$k_1=hf(x_k,y_k)$$

$$k_2=hf(x_k+\frac{h}{2},y_k+\frac{k_1}{2})$$

Flowchart of R-K Method of order 4

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