Method of lines

The method of lines is a technique for solving partial differential equations in which every dimension but one is discretized. Reducing the equation to a single continuous dimension, usually time, turns it into a system of ordinary differential equations or differential-algebraic equations, which can then be handed to the many integration routines and software packages already written for those problems. In its most common form the spatial derivatives are approximated first and the time variable is left continuous, so a standard initial value solver advances the system. Because those solvers need an initial value problem, the method cannot be applied directly to purely elliptic equations such as Laplace's equation, although it has been used on them through the method of false transients, which adds an artificial time derivative, and through a semi-analytical variant that exploits the exponential of a matrix. The approach dates back to at least the early 1960s, and a long literature studies its accuracy and stability for different kinds of equation.