Crank-Nicolson method

The Crank-Nicolson method is a finite difference method for numerically solving the heat equation and similar partial differential equations. It was developed by John Crank and Phyllis Nicolson in the 1940s. The method is second order in time, implicit in time and numerically stable, and it rests on the trapezoidal rule, which gives its second order convergence. For diffusion equations it is unconditionally stable, although the approximate solution can still contain decaying spurious oscillations when the time step multiplied by the thermal diffusivity, relative to the square of the space step, is large. When large time steps or high spatial resolution are needed, the less accurate backward Euler method is often used instead because it is both stable and immune to such oscillations.