Crank–Nicolson Integration

📐 Definition

For $u_t = F(u,t)$ and timestep $\Delta t$ , the Crank–Nicolson update solves

u^{n+1} = u^{n} + \frac{\Delta t}{2}\Big(F(u^{n}, t_n) + F(u^{n+1}, t_{n+1})\Big)

implicitly for $u^{n+1}$ .

Domain and Codomain

Applies to time-dependent states $u^n$ in $\mathbb{R}^m$ (or function spaces) with Lipschitz $F$ . Output $u^{n+1}$ is defined by solving a nonlinear or linear system each step.

⚙️ Key Properties

Order and Symmetry

Crank–Nicolson is second-order accurate in time for smooth $F$ . For linear problems it is symmetric (time-reversible) and can preserve qualitative structure better than first-order implicit stepping.

Linear Stability (Test Equation)

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

u^{n+1} = \frac{1 + \tfrac{1}{2}z}{1 - \tfrac{1}{2}z}\,u^n, \qquad z = \lambda \Delta t

The method is A-stable but not strongly damping for $\Re(z)\ll 0$ , so very stiff dissipative modes may decay slowly and can exhibit weak oscillations in some settings.

Implicit Solve Requirement

Each step requires a linear or nonlinear solve for $u^{n+1}$ ; fixed-point iteration and Newton-type methods are common.

🎯 Special Cases and Limits

Reduces to the trapezoidal rule for scalar ODEs. For linear $F(u,t) = Au + g(t)$ , the update solves $(I - \tfrac{\Delta t}{2}A)u^{n+1} = (I + \tfrac{\Delta t}{2}A)u^{n} + \tfrac{\Delta t}{2}(g^{n} + g^{n+1})$ .

Compared with explicit Runge–Kutta methods (e.g., RK4), Crank–Nicolson trades nonlinear solves for unconditional stability on diffusion operators.

Usage in Oakfield

Oakfield exposes Crank–Nicolson as an implicit built-in integrator implemented via fixed-point iteration:

Integrator name: crank_nicolson (Lua: sim.sim_create_context_integrator(ctx, "crank_nicolson", {...})).
Solve strategy: iterates on the implicit midpoint/trapezoid equation using a Picard-style update until the guess/update residual drops below the integrator tolerance (no Jacobians/Newton).
Adaptive fallback: when adaptive = true, Oakfield reduces dt across a small number of attempts if convergence is not achieved.
Complex + stochastic support: handles complex primary fields and supports optional stochastic increments after the implicit update.

Historical Foundations

📜 Diffusion Computation

Crank–Nicolson arose from mid-20th-century work on numerically solving diffusion equations with finite differences, combining the trapezoidal rule in time with practical linear solves.

🌍 Modern Perspective

It remains a standard second-order implicit method in PDE time stepping, especially where reduced numerical damping is desirable.

📚 References

Crank, The Mathematics of Diffusion
Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations II