The Crank–Nicolson method is a numerical technique used to solve partial differential equations (PDEs), particularly the heat equation. It is an implicit finite difference method that is unconditionally stable and second-order accurate in both time and space.
It is most often applied in fields of quantitative finance for option pricing with the Black-Scholes equation, but also commonly introduced in physics problems like the Diffusion equation.
Formulation
It is based on the trapezoidal rule for numerical integration, which averages the spatial derivatives at the current time step and the next time step.
It varies in formulation by dimension, but for the one-dimensional diffusion equation, it can be expressed as follows: