RK4

If Runge-Kutta is the idea of sampling several slopes inside a step and averaging them, RK4 is its most famous concrete recipe — the four-slope formula that, for decades, was the default 'just solve this ODE for me' method. It hits a sweet spot: high enough accuracy that you rarely need anything fancier on a smooth problem, yet simple enough to write in a dozen lines of code.

RK4 takes a single step using four slope evaluations: k1 = f(t_n, y_n), the slope at the start; k2 = f(t_n + h/2, y_n + (h/2)*k1), a slope at the midpoint using k1 to get there; k3 = f(t_n + h/2, y_n + (h/2)*k2), a second, refined midpoint slope; and k4 = f(t_n + h, y_n + h*k3), the slope at the far end. The step is a weighted average that trusts the two midpoint slopes most: y_{n+1} = y_n + (h/6)*(k1 + 2*k2 + 2*k3 + k4). The 1-2-2-1 weighting is no accident — it is exactly the pattern that cancels the error terms through h^4, making the method fourth order, and it echoes Simpson's rule for integration (which RK4 reduces to when f depends only on t).

RK4 is fourth order: its global error is O(h^4), so halving the step cuts the error by 16. That richness is why it long served as the workhorse for non-stiff problems. The honest limits: it costs four f-evaluations per step (worth it only when you want several correct digits and f is smooth); it has a FIXED step, so it cannot adapt to easy and hard regions of the solution (modern solvers use embedded pairs for that); and like all explicit Runge-Kutta methods it is only conditionally stable, hence wrong for stiff equations, where an implicit method is required.