What is crank nicolson method?

Crank-Nicolson is a finite difference numerical method to approximate solutions of partial differential equations in two or more dimensions. It is an implicit method that takes into account the averaging of both time and space. Crank-Nicolson is particularly suited to solving parabolic equations, where the solution evolves over time, as it has a second-order accuracy in both space and time.

The method combines the advantages of the forward Euler method and the backward Euler method. At each time step, the method solves a linear system of equations to obtain the solution at the next time step. The resulting equations can be solved using various techniques, such as direct or iterative methods. Crank-Nicolson is unconditionally stable, which means the method can handle stiff problems without requiring extremely small time steps.

Crank-Nicolson has applications in various fields, including fluid dynamics, heat transfer, and electromagnetism. However, the method can be computationally expensive for large systems, as it requires the solution of a linear system of equations at each time step. Hence, other numerical methods may be more suitable for specific problems.