Crank–Nicolson method

The Crank–Nicolson method advances numerical solutions of differential equations by averaging the forward- and backward-Euler evaluations. It is second-order accurate in time and unconditionally stable for linear diffusion equations.

Trapezoidal rule: A numerical integration rule that approximates an integral by averaging function values at interval endpoints. The method applies this time-integration rule to the differential equation.
Heat equation: A partial differential equation describing the diffusion of heat or another scalar quantity. It is the standard diffusion equation used to illustrate the method’s properties.
Diffusion equation: A partial differential equation that models the spread of a quantity from regions of high concentration to low concentration. The method is especially associated with linear diffusion problems.
Runge–Kutta method: A family of time-integration methods that combines derivative evaluations within each step. Explicit Runge–Kutta schemes avoid an implicit solve but face step-size stability limits.
Forward Euler method: An explicit time-stepping method that advances a solution using the derivative at the current time. Crank–Nicolson averages this method’s derivative evaluation with a future-time evaluation.
Finite difference method: A numerical technique that approximates derivatives using values on a discrete grid. Spatial finite differences commonly turn diffusion equations into systems advanced by this method.
Finite element method: A numerical method that approximates differential equations by combining simple functions over a mesh. Finite-element spatial discretizations can use Crank–Nicolson for their time evolution.
BDF2: A second-order backward differentiation formula using the current and previous solution values. It is another second-order implicit option, but unlike Crank–Nicolson it is not time-reversible.
Backward Euler method: An implicit time-stepping method that evaluates the derivative at the end of each time step. Its future-time evaluation supplies the other half of Crank–Nicolson’s average.
Method of lines: A technique that discretizes a differential equation in space while leaving time continuous. It reduces a diffusion equation to an ordinary differential system for time stepping.

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