CrankNicolson

Crank–Nicolson is a finite difference method for solving time-dependent partial differential equations, named after John Crank and Phyllis Nicolson who introduced it in 1947. It is especially suited for parabolic problems such as diffusion or heat equations and is widely used in physics, engineering, and quantitative finance.

The method is implicit and time-centered, effectively taking the average of the spatial discretization at the current and next time levels. For a one-dimensional heat equation ∂u/∂t = κ ∂^2u/∂x^2 on a grid with spacing Δx and time step Δt, the Crank–Nicolson discretization is

(u_i^{n+1} − u_i^n)/Δt = (κ/2) [ (u_{i+1}^{n+1} − 2u_i^{n+1} + u_{i−1}^{n+1})/Δx^2 + (u_{i+1}^n − 2u_i^n + u_{i−1}^n)/Δx^2 ].

This yields a linear system with a tridiagonal coefficient matrix at each time step, which can be solved efficiently. The method is second-order accurate in both time and space and, for linear diffusion problems, unconditionally stable.

Key properties include its balance between stability and accuracy and its suitability for stiff problems. While Crank–Nicolson is A-stable, it can produce nonphysical oscillations if the time step is too large relative to spatial discretization or for strongly nonlinear problems.

Extensions include applying Crank–Nicolson in multiple dimensions and using alternating direction implicit (ADI) schemes to handle higher-dimensional problems efficiently. It is also common in computational finance for solving option pricing PDEs such as the Black–Scholes equation.