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

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

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

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

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

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

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

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

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.