Runge-Kutta 4 (RK4)

Runge-Kutta 4th Order

Euler's method uses only the slope at the start of each step. The Runge-Kutta 4 method (RK4) samples the slope at four points within the step and takes a weighted average — achieving fourth-order accuracy.

The Formula

$k_1 = f(t,\; y)$ $k_2 = f\!\left(t + \tfrac{h}{2},\; y + \tfrac{h}{2} k_1\right)$ $k_3 = f\!\left(t + \tfrac{h}{2},\; y + \tfrac{h}{2} k_2\right)$ $k_4 = f(t + h,\; y + h\, k_3)$

$y_{\text{next}} = y + \frac{h}{6}\left(k_1 + 2k_2 + 2k_3 + k_4\right)$

• $k_1$: slope at the start
• $k_2$: slope at the midpoint using $k_1$
• $k_3$: slope at the midpoint using $k_2$ (refined)
• $k_4$: slope at the end using $k_3$

The weights $(1, 2, 2, 1)$ follow Simpson's rule for numerical integration.

Why RK4?

With the same step size, RK4 is dramatically more accurate than Euler. For $\frac{dy}{dt} = y$ with $h = 0.1$:

• Euler: error ~0.005 per step
• RK4: error ~0.000000002 per step

RK4 is the workhorse of scientific computing. It is used in physics simulations, orbital mechanics, and engineering systems everywhere Euler is too inaccurate.

Your Task

Implement rk4_step(f, t, y, h) that returns the next y value using one RK4 step.