Runge-Kutta method

Definition

We wish to approximate a solution to the first order differential equation

$$\frac{dy(t)}{dt}=y^{\prime }(t)=f(y(t),t)\text{ given }y(t_0)=y_0$$

Then 4th order approximation would be

\begin{align*} k_{1} &= f(y_{0}, t_{0}) \\ k_{2} &= f\left( y_{0}+k_{1}\frac{dt}{2}, t_{0} + \frac{dt}{2} \right) \\ k_{3} &= f\left( y_{0}+k_{2}\frac{dt}{2}, t_{0} + \frac{dt}{2} \right) \\ k_{4} &= f\left( y_{0}+k_{3}dt, t_{0} + dt \right) \\

\end{align*}

$$Finallywegetaweightedaverageoftheseslopes.>$$

y(t_{0}+dt) = y(t_{0}) + \frac{k_{1} + 2k_{2} + 2k_{3} + k_{4}}{6}dt

Source

Swarthmore Edu