Crank–Nicolson

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 and 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 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.

The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time. For example, in one dimension, if the partial differential equation is

then, letting ${\displaystyle u(i\Delta x,\,n\Delta t)=u_{i}^{n}\,}$, the equation for Crank–Nicolson method is a combination of the forward Euler method at ${\displaystyle n}$ and the backward Euler method at n + 1 (note, however, that the method itself is not simply the average of those two methods, as the equation has an implicit dependence on the solution):

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