RK4 (and RK3) Schemes

Equations

RK4 — four slope samples, Simpson-weighted:

\[k_1 = h f(t_n, y_n) \qquad k_2 = h f(t_n + \tfrac h2, y_n + \tfrac{k_1}2) \qquad k_3 = h f(t_n + \tfrac h2, y_n + \tfrac{k_2}2) \qquad k_4 = h f(t_n + h, y_n + k_3)\]

\[\boxed{\,y_{n+1} = y_n + \frac{k_1 + 2k_2 + 2k_3 + k_4}{6}\,}\]

RK3 — three samples:

\[k_1 = f(t_n, y_n), \quad k_2 = f(t_n + \tfrac h2, y_n + \tfrac h2 k_1), \quad k_3 = f(t_n + h, y_n + h(2k_2 - k_1))\]

\[\boxed{\,y_{n+1} = y_n + \frac{h}{6}(k_1 + 4k_2 + k_3)\,}\]

Physical meaning

Numerical quadrature applied to slopes: probe the derivative at the start, middle (twice), and end of the step, then average with Simpson-like weights so the Taylor errors cancel through 4th (3rd) order. See the Runge–Kutta concept page for the family logic.

Variables

\(h = \Delta t\) — step size · \(k_i\) — stage slopes (note RK4's \(k_i\) include the \(h\) factor as written; RK3's do not — follow one convention in code!).

Properties

	RK3	RK4
Global error	\(\mathcal{O}(h^3)\)	\(\mathcal{O}(h^4)\)
Evaluations/step	3	4
Halving \(h\) divides error by	8	16
Symplectic?	no	no

Applications & limitations

The default for smooth, non-Hamiltonian ODEs and moderate horizons: benchmark trajectories, Rutherford scattering, field-line tracing (also the streamline tracer in the fluids potential-flow widget). For long-time orbital dynamics, its non-symplectic energy drift loses to leapfrog; for stiff systems, explicit RK stability limits bite — use implicit methods.

Leapfrog update — conservation-first alternative
CFL condition — the analogous stability constraint for PDEs

Quiz

Q1 (computational). One RK4 step for \(\dot y = y\), \(y_0 = 1\), \(h = 1\): compute \(y_1\) and compare with \(e = 2.71828\).

Answer

\(k_1 = 1\), \(k_2 = 1.5\), \(k_3 = 1.75\), \(k_4 = 2.75\); \(y_1 = 1 + (1 + 3 + 3.5 + 2.75)/6 = 1 + 10.25/6 = 2.7083\). Error 0.01 with a huge step — that's 4th order for you.

Q2 (MCQ). RK4's four stages per step mean that at equal cost, comparing with forward Euler at step \(h/4\), RK4's error advantage is roughly a factor of:

(a) 4 (b) \(h\) (c) \(h^{-3}/64\)... hard to say without \(h\) (d) none