The Crank-Nicolson Method
Crank-Nicolson averages explicit and implicit evaluations to reach second-order accuracy in time with unconditional linear stability.
The trapezoidal rule in time
The Crank-Nicolson method integrates a time-dependent PDE by averaging the right-hand side at the current and next time levels: u_{n+1} = u_n + (dt/2)[f(u_n) + f(u_{n+1})]. It is the trapezoidal rule applied in time and, for the heat equation and similar diffusion problems, combines two attractive properties: second-order accuracy in the time step and unconditional stability for linear problems.
Because it involves f at the new level, Crank-Nicolson is implicit and requires a linear (or nonlinear) solve each step, but the payoff is that the step size is set by accuracy rather than by a stability limit.
A-stable but not L-stable
Crank-Nicolson is A-stable: no linear mode grows for any step size. It is not L-stable, however, so very fast decaying modes are damped only weakly and can produce slowly decaying oscillations when the initial data or forcing is not smooth. When such oscillations are troublesome, a few backward-Euler steps at the start (Rannacher smoothing) or a shift toward a more strongly damping scheme restores clean behavior.
Use and variants
The method is a standard choice for parabolic problems needing better accuracy than backward Euler without the step limit of explicit schemes. The theta-method generalizes it: theta = 0 gives forward Euler, theta = 1 backward Euler, and theta = 1/2 Crank-Nicolson, letting a single parameter trade accuracy for damping.
- Average of current and next right-hand sides (trapezoidal)
- Second-order accurate and A-stable for linear problems
- Not L-stable: weak damping of the fastest modes
- Theta-method generalization spans Euler to Crank-Nicolson
Crank-Nicolson and its theta-method relatives appear throughout diffusion-dominated transport modeling where second-order temporal accuracy without a restrictive step limit is desired.