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 equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the 1940s.

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

Δ

t

{\displaystyle \Delta t}

times the thermal diffusivity to the square of space step,

Δ

x

2

{\displaystyle \Delta x^{2}}

, 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.

Principle

The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time. For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method—the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator. For example, in one dimension, suppose the partial differential equation is

∂

u

∂

t

=

F

(

u

,

x

,

t

,

∂

u

∂

x

,

∂

2

u

∂

x

2

)

.