Runge-Kutta 4th Order Method

The Runge-Kutta 4th order method (RK4) is the workhorse numerical integrator for ordinary differential equations $dy/dx = f(x, y)$. It achieves $O(h^4)$ global accuracy by sampling the slope at four points per step.

RK4 Algorithm

Given the current state $(x_n, y_n)$ and step size $h$:

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

$y_{n+1} = y_n + \frac{k_1 + 2k_2 + 2k_3 + k_4}{6}$

The weighted average gives RK4 its 4th-order accuracy: inner slopes are counted twice.

Harmonic Oscillator System

The harmonic oscillator $\ddot{y} + \omega^2 y = 0$ can be rewritten as a system of two first-order ODEs:

$\frac{dy}{dt} = v, \qquad \frac{dv}{dt} = -\omega^2 y$

RK4 is applied to both equations simultaneously at each step. With initial conditions $y(0) = 1$, $v(0) = 0$, the exact solution is $y(t) = \cos(\omega t)$.

Accuracy vs. Step Size

Your Task

Implement rk4_step(f, x, y, h) that advances the solution by one step and returns the new $y$ value. Implement rk4_solve(f, x0, y0, x_end, h) that returns a list of (x, y) tuples for the full solution. Implement harmonic_oscillator_rk4(omega, t_end, y0=1.0, v0=0.0, h=0.01) that solves the harmonic oscillator system and returns (t_values, y_values) as lists.