The Crank--Nicolson method applies the trapezoidal rule to a semidiscrete evolution equation. For a linear system its update is
For symmetric negative semidefinite matrices , each factor has norm at most one, since its eigenvalues are with . The same holds for , so their ordered product is a contraction for every , whether or not commute. This proves unconditional discrete L2 norm stability of the split Crank-Nicolson method for directional diffusion. Its splitting error remains first order in general, because the product's quadratic defect is .
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The Crank-Nicolson method is a numerical technique used for solving partial differential equations, particularly parabolic types (like the heat equation). It is widely utilized in computational physics and finance due to its efficacy in handling time-dependent problems. ### Key Features of the Crank-Nicolson Method: 1. **Implicit Method**: The Crank-Nicolson method is an implicit scheme, meaning that it involves solutions to equations that require solving a system of equations at each time step.