Nonstiffness

Nonstiffness is a characteristic used in the analysis of differential equations and dynamical systems to describe problems that do not require extremely disparate time scales to be resolved. In nonstiff problems, the dynamics evolve on comparable time scales, so the eigenvalues of the system’s linearization (for example, the Jacobian) have similar magnitudes and decay rates. This allows explicit time-stepping methods to advance the solution with time steps governed mainly by accuracy, not by stability constraints.

Stiffness is the contrast, referring to problems that contain both fast and slow dynamics. In stiff systems, there exist rapid transients that force explicit solvers to take very small time steps for stability, even if the overall interest lies in slower behavior. This often manifests as a large disparity among the eigenvalues of the system, with some having large negative real parts and others near zero. The resulting stability restrictions make many implicit methods more efficient.

A common practical criterion is the stiffness ratio, roughly the quotient between the fastest and the slowest relevant time scales. If this ratio is small (near unity), the problem is typically nonstiff. If the ratio is large (for example, several orders of magnitude), the problem is typically stiff. This division is problem-dependent and informal, but it guides the choice of numerical methods.

Numerical implications follow naturally. Nonstiff problems are usually well suited to explicit methods, such as standard Runge–Kutta schemes, with step sizes determined by accuracy. Stiff problems, by contrast, are often better treated with implicit methods or specialized stiff solvers that provide stability for large time steps.