Differential Equations and Linear Algebra, 4th Edition

2.6 The Runge–Kutta Method

We now discuss a method for approximating the solution $y=y(x)$ of the initial value problem

$${\displaystyle \frac{dy}{dx}}=f(x,y),y(x_0)=y_0$$ (1)

that is considerably more accurate than the improved Euler method and is more widely used in practice than any of the numerical methods discussed in Sections 2.4 and 2.5. It is called the Runge–Kutta method, after the German mathematicians who developed it, Carl Runge (1856–1927) and Wilhelm Kutta (1867–1944).

With the usual notation, suppose that we have computed the approximations $y_1,y_2,y_3,\dots ,y_n$ to the actual values $y(x_1),y(x_2),y(x_3),\dots ,y(x_n)$ and now want to compute $y_{n+1}\approx y(x_{n+1})\text{.}$ Then

$$y(x_{n+1})-y(x_n)={\displaystyle {\int }_{x_n}^{x_{n+1}}y\text{′}(x)dx}={\displaystyle {\int }_{x_n}^{x_{n+h}}y\text{′}(x)dx}$$ (2)

by the fundamental theorem of calculus. Next, Simpson’s rule ...