semidiscretized

Semidiscretization, sometimes described as semidiscretized, is a numerical approach in which a mathematical model is discretized with respect to some but not all of its variables. In the context of partial differential equations, semidiscretization typically means discretizing the spatial variables while keeping time continuous. This converts a PDE into a system of ordinary differential equations in time, forming the basis of the method of lines approach.

Common spatial discretization methods include finite difference, finite element, spectral, and finite volume techniques. The resulting semi-discrete system often has a form such as M du/dt = F(u,t) or du/dt = A u + f(t), where M is a mass matrix and A is a discretized operator that represents the spatial effects. The choice of discretization determines consistency, stability, and accuracy, with error typically depending on the spatial mesh size or basis order.

Stability and efficiency are central considerations. Since the semi-discrete system is an ODE system, the time integration method must be chosen accordingly. Explicit time integrators are simple but may require very small time steps for stiff problems, while implicit methods handle stiffness better at the cost of solving algebraic systems at each step. Structure-preserving discretizations may aim to retain physical properties such as mass conservation or energy dissipation.

Applications are widespread and include diffusion, heat conduction, wave propagation, fluid dynamics, electromagnetism, and chemical kinetics. Semidiscretization is often an intermediate step before full discretization, where both space and time are discretized. It provides a flexible framework for analyzing spatial discretization effects independently from temporal integration.