structural:summarization, linguistic:voice_change
Crank–Nicolson for nonlinear problems

Because the Crank–Nicolson method is implicit, it is generally impossible to solve exactly. Instead, an iterative technique should be used to converge to the solution. One option is to use Newton's method to converge on the prediction, but this requires the computation of the Jacobian. For a high-dimensional system like those in computational fluid dynamics or numerical relativity, it may be infeasible to compute this Jacobian.

A Jacobian-free alternative is fixed-point iteration. If f {\displaystyle f} is the velocity of the system, then the Crank–Nicolson prediction will be a fixed point of the map Φ (x) = x 0 + h 2. {\displaystyle \Phi (x)=x_{0}+{\frac {h}{2}}\left.} If the map iteration x (i + 1) = Φ (x (i)) {\displaystyle x^{(i+1)}=\Phi (x^{(i)})} does not converge, the parameterized map Θ (x, α) = α x + (1 − α) Φ (x) {\displaystyle \Theta (x,\alpha)=\alpha x+(1-\alpha)\Phi (x)}, with α ∈ (0, 1) {\displaystyle \alpha \in (0,1)}, may be better behaved. In expanded form, the update formula is

where x i {\displaystyle x^{i}} is the current guess and x i − 1 {\displaystyle x_{i-1}} is the previous time-step.

Even for high-dimensional systems, iteration of this map can converge surprisingly quickly.