Runge–Kutta–Fehlberg 4(5)

📐 Definition

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

\begin{aligned} k_1 &= F(u^{n}, t_n),\\ k_2 &= F\!\left(u^{n} + \tfrac{1}{4}\Delta t\,k_1,\, t_n + \tfrac{1}{4}\Delta t\right),\\ k_3 &= F\!\left(u^{n} + \tfrac{3}{32}\Delta t\,k_1 + \tfrac{9}{32}\Delta t\,k_2,\, t_n + \tfrac{3}{8}\Delta t\right),\\ k_4 &= F\!\left(u^{n} + \tfrac{1932}{2197}\Delta t\,k_1 - \tfrac{7200}{2197}\Delta t\,k_2 + \tfrac{7296}{2197}\Delta t\,k_3,\, t_n + \tfrac{12}{13}\Delta t\right),\\ k_5 &= F\!\left(u^{n} + \tfrac{439}{216}\Delta t\,k_1 - 8\Delta t\,k_2 + \tfrac{3680}{513}\Delta t\,k_3 - \tfrac{845}{4104}\Delta t\,k_4,\, t_n + \Delta t\right),\\ k_6 &= F\!\left(u^{n} - \tfrac{8}{27}\Delta t\,k_1 + 2\Delta t\,k_2 - \tfrac{3544}{2565}\Delta t\,k_3 + \tfrac{1859}{4104}\Delta t\,k_4 - \tfrac{11}{40}\Delta t\,k_5,\, t_n + \tfrac{1}{2}\Delta t\right) \end{aligned}

The fifth-order solution estimate is

u^{n+1}_{(5)} = u^{n} + \Delta t\left(\tfrac{16}{135}k_1 + \tfrac{6656}{12825}k_3 + \tfrac{28561}{56430}k_4 - \tfrac{9}{50}k_5 + \tfrac{2}{55}k_6\right)

and the embedded fourth-order estimate is

u^{n+1}_{(4)} = u^{n} + \Delta t\left(\tfrac{25}{216}k_1 + \tfrac{1408}{2565}k_3 + \tfrac{2197}{4104}k_4 - \tfrac{1}{5}k_5\right)

The local error estimate is $e^{n+1} = u^{n+1}_{(5)} - u^{n+1}_{(4)}$ .

Domain and Codomain

Applies to $u^n \in \mathbb{R}^m$ (or function spaces) with sufficiently smooth $F$ to support fifth-order truncation error. Outputs $u^{n+1}_{(5)}$ and $u^{n+1}_{(4)}$ lie in the same space.

⚙️ Key Properties

Embedded Error Estimate

The difference $e^{n+1} = u^{n+1}_{(5)} - u^{n+1}_{(4)}$ provides a cheap local error proxy that enables adaptive step-size control.

Adaptive Step Control (Typical Form)

A common controller selects a new step size using a normed tolerance test and

\Delta t_{\mathrm{new}} \approx \Delta t \left(\frac{\mathrm{tol}}{\|e^{n+1}\|}\right)^{1/5}

where the exponent $1/5$ reflects the scaling $\|e^{n+1}\|=\mathcal{O}(\Delta t^5)$ of the embedded local error estimate.

Cost and Stability

RKF45 is explicit (no solves) and typically uses six drift evaluations per attempted step. Like other explicit Runge–Kutta schemes, stability restricts timesteps on stiff problems.

🎯 Special Cases and Limits

For linear $F(u,t) = Au$ , both estimates reduce to applying the stability polynomials to $z = \Delta t\,\lambda$ . As $\Delta t \to 0$ , the fourth- and fifth-order estimates coincide and converge to the exact flow.

Compared to classical RK4, RKF45 adds an embedded estimator for step-size control. Heun’s method is the two-stage analogue with lower order and no embedded pair.

Usage in Oakfield

Oakfield exposes RKF45 as an adaptive built-in explicit integrator:

Integrator name: rkf45 (Lua: sim.sim_create_context_integrator(ctx, "rkf45", {...})).
Always adaptive: Oakfield forces adaptive = true for RKF45 even if the caller requests a fixed step, since the method relies on embedded error control.
Complex + stochastic support: supports complex primary fields and optional stochastic increments after each accepted step.

Historical Foundations

📜 Embedded Runge–Kutta Pairs

Fehlberg’s embedded 4(5) construction provided a practical way to estimate local error and adapt step sizes without extra drift evaluations, helping popularize adaptive explicit solvers.

🌍 Modern Perspective

Embedded Runge–Kutta pairs (including RKF-type and Dormand–Prince variants) are now standard workhorses for adaptive nonstiff ODE integration.

📚 References

Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I
Fehlberg, Low-Order Classical Runge-Kutta Formulas with Stepsize Control (NASA TR R-315)