Common
Crank–Nicolson method
Finite difference method for numerically solving parabolic differential equations
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...
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