Heun Method

📐 Definition

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

\begin{aligned} k_1 &= F(u^{n}, t_n),\\ \tilde{u} &= u^{n} + \Delta t\,k_1,\\ k_2 &= F(\tilde{u}, t_n + \Delta t),\\ u^{n+1} &= u^{n} + \frac{\Delta t}{2}(k_1 + k_2) \end{aligned}

Domain and Codomain

Applies to $u^n \in \mathbb{R}^m$ (or function spaces) with locally Lipschitz $F$ . Output $u^{n+1}$ resides in the same space.

⚙️ Key Properties

Order and Local Error

Heun’s method is second-order accurate in time, with local truncation error $\mathcal{O}(\Delta t^3)$ .

Linear Stability (Test Equation)

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

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

Stability requires $|R(z)| < 1$ , so timesteps remain restricted on stiff problems.

Cost

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

🎯 Special Cases and Limits

As $\Delta t \to 0$ , the method converges to the exact flow under standard smoothness assumptions. Its leading term matches forward Euler, with the second stage acting as an explicit trapezoidal correction.

Interpolates between forward Euler (predictor) and RK4 in cost-versus-accuracy; shares the trapezoidal averaging idea with Crank–Nicolson but remains explicit.

Usage in Oakfield

Oakfield exposes Heun as a built-in explicit integrator:

Integrator name: heun (created via sim.sim_create_context_integrator(ctx, "heun", {...})).
Optional adaptivity: when adaptive = true, Oakfield uses the difference between the Euler predictor and Heun corrector as an error proxy for step acceptance.
Complex + stochastic support: the implementation handles real and complex fields and can apply stochastic increments after the deterministic update.

Historical Foundations

📜 Predictor–Corrector Schemes

Heun popularized this explicit second-order approach in the early 20th century as a practical predictor–corrector alternative to first-order stepping.

🌍 Modern Perspective

It is now commonly classified as an explicit Runge–Kutta method of order 2 (explicit trapezoidal / improved Euler) and is frequently used when a small accuracy upgrade over Euler is needed.

📚 References

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