nonstiff

In numerical analysis, nonstiff refers to ordinary differential equations or dynamical systems for which explicit time-stepping methods can be used efficiently without suffering prohibitive stability constraints. The term contrasts with stiff problems, where fast transient modes force very small time steps or require implicit methods to maintain stability. The classification of a problem as nonstiff is not absolute; it depends on the numerical method and the desired accuracy, and some problems may be nonstiff for certain solvers and stiff for others.

Nonstiffness is commonly associated with systems whose linearization has eigenvalues with comparable magnitudes and moderate negative real parts, i.e., no large disparity in time scales. In linear test equations like y' = Ay, a problem is typically considered nonstiff if the eigenvalues do not require extremely small explicit steps for stability. Conversely, stiffness emerges when there is a wide separation of time scales, such as eigenvalues with large negative real parts in magnitude relative to others.

Implications for numerical methods: For nonstiff problems, explicit integrators such as classical Runge-Kutta schemes can take reasonably large steps dictated by accuracy rather than stability. In stiff problems, explicit methods become impractical due to severe stability limits; implicit methods (backward Euler, BDF) or semi-implicit schemes are preferred, sometimes with specialized solvers for the resulting algebraic equations.

Diagnosis and examples: There is no universal threshold for nonstiffness; practitioners assess stiffness using the Jacobian spectrum, stiffness ratio, or practical solver performance. Many ordinary differential equations arising in model reduction, certain population dynamics, and some chemical kinetics are nonstiff, though fast transients can still occur. The concept remains a practical guide for selecting numerical methods and understanding step-size requirements.