##  [Method of Lines](/method-lines-0) 

 Definition

A semi‑discretization procedure that discretizes all independent variables except one—commonly space is discretized while time remains continuous—yielding a system of ordinary differential equations (ODEs) in the remaining variable to be integrated with ODE solvers.

 

 

 

 

 

 





## Principle

Principle

Separate spatial discretization from temporal integration: construct spatial difference, finite volume or spectral approximations to reduce the PDE to an ODE system whose temporal evolution can be handled by established ODE integrators (explicit, implicit, stiff solvers).

 

 

 

 

 





## Demonstration

Demonstration

For a reaction‑diffusion PDE, apply finite differences in space to produce a large coupled system of ODEs for nodal values; then use an implicit stiff ODE solver to advance the solution in time respecting diffusion‑induced stiffness.

 

 

 

 

## Misapplication

Misapplication

Using an ODE solver that does not account for stiffness introduced by the spatial discretization (e.g., explicit integrator on a stiff semi‑discrete system) leading to impractically small time steps or instability.

 

 

 

 

 





## Consequence

Consequence

Allows reuse of sophisticated time integrators and clear separation of concerns: spatial accuracy and temporal integrator choice can be tuned independently, and adaptive time stepping can be applied to the semi‑discrete system.

 

 

 

 

## Reversal

Reversal

Discretize time first (Rothe's method) or use fully discrete space‑time methods that treat space and time simultaneously, rather than leaving one variable continuous to exploit ODE solvers.

 

 

 

 

 





## Boundary

Boundary

Applies when one independent variable can be left continuous and efficiently handled by ODE techniques; less natural for methods that require intrinsic coupling of space and time (e.g., certain space‑time variational formulations).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Close to operator splitting (which manipulates temporal evolution operators) and to fully discrete schemes; tension arises in choosing whether to treat temporal coupling via ODE integrators or via combined discretizations.

 

 

 

 

 





## Synthesis

Synthesis

A pragmatic route from PDE to ODE: discretize spatial operators to produce a semi‑discrete system and then apply ODE solver technology for time evolution, enabling modular solver design and targeted treatment of stiffness and adaptivity.