📂Numerical Analysis

Stiff Equation

Terminology

When solving ordinary differential equations numerically, using numerical methods, an equation that requires very small mesh sizes is called a stiff equation.

Explanation

In Japanese it seems to be translated as 硬い方程式, but that wording doesn’t convey the nuance well, so I simply use “stiff equation.”

The phrase that a problem is stiff is often used in the context of solving a Cauchy problem, but it’s actually difficult to give a precise definition. Simply put, one can regard it as a problem that becomes hard to solve due to numerical issues; even the very simple Dahlquist problem $y ' = - \lambda y$ becomes stiff when $\lambda > 0$ is large. Among partial differential equations, even the relatively simple case of solving the heat equation behaves this way.

Usually implicit methods are recommended over explicit methods to overcome such problems (implicit methods preferred over explicit methods), but this is not guaranteed to be a solution — even if you gain in allowable mesh size, other computational costs may increase.