Finite difference method

Summary

The finite difference method solves differential equations approximately on a grid of points. It replaces each derivative by a difference quotient of the values at nearby grid points, which turns the equation into a system of algebraic equations. It is the oldest and simplest numerical method for partial differential equations. Its convergence rests on consistency and stability, as in the Lax equivalence theorem. For time-dependent problems, stability often requires a step restriction such as the Courant–Friedrichs–Lewy condition.