What is meant by numerical stiffness in differential equations?

Numerical stiffness in the context of differential equations refers to a situation where the solutions exhibit rapid variations, often leading to difficulties in finding accurate numerical solutions. This typically occurs in problems where there are vastly different time scales present in the solution, meaning some components of the solution evolve much faster than others. As a result, standard numerical methods may struggle to maintain accuracy because they can require impractically small step sizes to resolve the rapidly changing components without introducing significant numerical error.

Consequently, when dealing with stiff equations, a naive approach might lead to oscillations or divergences in the numerical solution, highlighting the inherent challenges in managing these types of problems. This stiffness necessitates the use of specialized numerical methods, such as implicit solvers, that can handle the stability issues associated with these rapid changes effectively.

In summary, the focus on rapid variation in solutions defines numerical stiffness, making it essential for practitioners studying differential equations to recognize and employ appropriate methods to mitigate these challenges.