semidiscretization

Semidiscretization, or semi-discretization, is a numerical technique in which one or more independent variables are discretized while others remain continuous. In the context of partial differential equations (PDEs), semidiscretization typically means discretizing the spatial variables while keeping time continuous, a form of the method of lines.

The procedure involves choosing a spatial discretization method—such as finite differences, finite elements, or spectral methods—and constructing a discrete spatial operator that approximates the continuous differential operators. Substituting this operator into the PDE yields a system of ordinary differential equations in time: du_h/dt = F_h(u_h, t) or du/dt = A_h u + f_h. The resulting semidiscrete system evolves in time under an ODE or, in some cases, a differential-algebraic equation (DAE).

Applications of semidiscretization include diffusion-type, convection-diffusion, and wave-type equations, as well as systems from fluid dynamics and electromagnetics. The approach is widely used to study stability and to inform the choice of time-stepping schemes. It also provides a framework for error analysis that separates spatial discretization error from time discretization error once a time integrator is selected.

Analysis and considerations often focus on the properties of the semidiscrete system, such as stability and convergence. For linear problems, the semidiscrete form is a finite-dimensional ODE system whose spectral properties influence the behavior of time integrators. The choice between explicit and implicit time stepping interacts with these properties, and CFL-type conditions may emerge after full discretization.

Relation to full discretization: fully discretizing also discretizes time, yielding a complete numerical scheme. Semidiscretization + time stepping is a common approach in which the spatial discretization error is treated separately from the temporal error. Examples include finite difference discretization of the heat equation, finite element discretization of elliptic problems, and spectral discretization using Fourier modes.