Runge–Kutta 4

📐 Definition

For $u_t = F(u,t)$ with timestep $\Delta t$ , define

\begin{aligned} k_1 &= F(u^{n}, t_n),\\ k_2 &= F\!\left(u^{n} + \tfrac{\Delta t}{2} k_1,\, t_n + \tfrac{\Delta t}{2}\right),\\ k_3 &= F\!\left(u^{n} + \tfrac{\Delta t}{2} k_2,\, t_n + \tfrac{\Delta t}{2}\right),\\ k_4 &= F\!\left(u^{n} + \Delta t\, k_3,\, t_n + \Delta t\right), \end{aligned}

then update

u^{n+1} = u^{n} + \frac{\Delta t}{6}(k_1 + 2k_2 + 2k_3 + k_4)

Domain and Codomain

Applies to $u^n \in \mathbb{R}^m$ (or function spaces) with sufficiently smooth $F$ to support fourth-order truncation error. Output $u^{n+1}$ remains in the same space.

⚙️ Key Properties

Order and Local Error

Classical RK4 is fourth-order accurate in time, with local truncation error $\mathcal{O}(\Delta t^5)$ .

Linear Stability (Test Equation)

For the linear test equation $u_t = \lambda u$ , the stability polynomial is

R(z) = 1 + z + \tfrac{1}{2}z^2 + \tfrac{1}{6}z^3 + \tfrac{1}{24}z^4, \qquad z = \lambda \Delta t

RK4 is explicit, so stability imposes CFL-like timestep restrictions for many PDE semi-discretizations.

Cost

Four drift evaluations per step; no linear or nonlinear solves.

🎯 Special Cases and Limits

As $\Delta t \to 0$ , RK4 converges to the exact solution under standard smoothness assumptions. For autonomous linear systems $u_t = Au$ , the update corresponds to applying the degree-4 stability polynomial to $z=\Delta t\,\lambda$ for each eigenvalue $\lambda$ .

Compared to Crank–Nicolson, RK4 avoids implicit solves but requires smaller timesteps for stiff problems.

Usage in Oakfield

Oakfield exposes RK4 as a built-in explicit integrator:

Integrator name: rk4 (Lua: sim.sim_create_context_integrator(ctx, "rk4", {...})).
Optional adaptivity: when adaptive = true, Oakfield compares an RK4 update against a cheaper Heun estimate to build an error norm for step acceptance.
Complex + stochastic support: works for real/complex primary fields and supports optional stochastic increments.

Historical Foundations

📜 Development of Runge–Kutta Methods

Runge and Kutta developed families of multi-stage explicit schemes in the late 19th and early 20th centuries, providing systematic ways to build higher-order integrators without higher derivatives.

🌍 Modern Perspective

RK4 remains a widely used default explicit integrator due to its simplicity, accuracy per step, and predictable error behavior on smooth nonstiff dynamics.

📚 References

Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I
Butcher, Numerical Methods for Ordinary Differential Equations