View source for Crank–Nicolson method

In [[numerical analysis]], the '''Crank–Nicolson method''' is a [[finite difference method]] used for numerically solving the [[heat equation]] and similar [[Partial differential equation|partial differential equations]].

== Description ==

It is a second-order method in time.  See [[Big O notation]].

It is implicit in time (see [[Explicit and implicit methods]]), and can be written as an implicit Runge–Kutta method

It is [[Numerical stability|numerically stable]].

The method was developed by [[John Crank]] and [[Phyllis Nicolson]] in the mid 20th century.

For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally stable.

However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step Δt times the thermal diffusivity to the square of space step, Δx2, is large (typically larger than 1/2 per Von Neumann stability analysis).

For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate backward Euler method is often used, which is both stable and immune to oscillations.

== See also ==

* [[Big O notation]]
* [[Explicit and implicit methods]]
* [[Numerical analysis]]
* [[Partial differential equation]]

== External links ==

* [https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method Crank–Nicolson method]

[[Category:Mathematics]]
[[Category:Physics]]