Method of lines

The method of lines solves partial differential equations by discretizing space while leaving time continuous, producing ordinary differential equations that are integrated numerically.

Spatial discretization: The replacement of a continuous spatial domain with a finite set of points, cells, or basis functions. It converts spatial derivatives in the PDE into finite-dimensional operators.
Partial differential equation: An equation involving an unknown function and its partial derivatives with respect to multiple independent variables. The method of lines begins with this class of equations and discretizes selected variables.
Advection equation: A partial differential equation describing transport of a quantity by a velocity field. Its discretized transport operator illustrates stability and accuracy challenges in time-dependent simulations.
Fully discrete method: A numerical scheme that discretizes every independent variable, including both space and time. It contrasts with the method of lines, which leaves time continuous until an ODE integrator is chosen.
Finite difference method: A numerical method that approximates derivatives using function values at discrete points. Finite differences can supply the spatial derivative operators in a method-of-lines scheme.
Ordinary differential equation: A differential equation involving an unknown function of one independent variable and its derivatives. After space is discretized, the method produces an ODE system in time.
Heat equation: A parabolic partial differential equation describing the diffusion of heat or other quantities. Spatial discretization yields diffusion ODE systems commonly advanced by this approach.
Spectral method: A numerical method that represents a solution using global basis functions and often achieves rapid convergence for smooth solutions. Spectral approximations can provide the spatial discretization before time integration.
Semidiscretization: Discretization of some independent variables while leaving others continuous. The method of lines is a semidiscretization in which time remains continuous.
Boundary condition: A constraint specifying a solution's behavior at the boundary of a spatial or temporal domain. Boundary conditions determine how spatial operators act near the domain edges.

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