What exactly makes an ODE stiff?

Stiffness arises when the system has components that decay much faster than the solution of interest evolves. There is no single sharp definition, but the practical signature is that explicit methods are forced to use very small steps for stability even when accuracy would permit large ones.

Why do stiff problems require implicit methods?

Implicit methods can have stability regions covering the entire left half-plane (A-stability), so they remain stable at large step sizes for rapidly decaying modes. Explicit methods have bounded stability regions, which forces tiny steps and makes them impractical for stiff problems.

Stiff ODEs and Stability

Stiff differential equations contain processes evolving on widely separated time scales, so explicit methods are forced to take impractically tiny steps for stability; their efficient solution requires implicit methods with strong stability properties.

Definition

A differential equation is called stiff when it admits solution components that decay on very different time scales, so that numerical stability rather than accuracy dictates the step size; stability theory analyses which methods can take large steps without error growth.

Scope

This topic covers the phenomenon and informal definition of stiffness, the linear test equation and region of absolute stability, the concepts of A-stability, A(alpha)-stability, and L-stability, why explicit methods fail on stiff problems, and the implicit methods — implicit Runge-Kutta and backward differentiation formulas — that solve them.

Core questions

What makes a problem stiff, and why does it defeat explicit methods?
How is the region of absolute stability defined through the linear test equation?
What do A-stability and L-stability require, and why are they important for stiff problems?
Which methods provide the stability needed for stiff and differential-algebraic systems?

Key theories

Absolute stability and the test equation: Applying a method to the scalar linear test equation produces an amplification factor; the set of step-size-times-eigenvalue products for which this factor has magnitude at most one is the method's region of absolute stability, which must contain the problem's stiff eigenvalues to allow large steps.
A-stability and L-stability: A method is A-stable if its stability region contains the entire left half-plane, so it is stable for all decaying modes regardless of step size, and L-stable if it additionally damps very stiff modes completely; these properties single out the implicit methods suited to stiff problems.

Mechanisms

On a stiff problem the fastest-decaying mode has a large negative eigenvalue; an explicit method's bounded stability region forces the step size to resolve that mode even long after it has physically died out, making the computation hopelessly slow. Implicit methods such as the backward Euler method, implicit Runge-Kutta schemes, and backward differentiation formulas have stability regions covering the left half-plane (or most of it), so they remain stable at large steps and let the step size be chosen by accuracy alone. Each step then requires solving a (generally nonlinear) algebraic system, typically by a Newton iteration using the Jacobian.

Clinical relevance

Stiffness is pervasive in chemical reaction networks, combustion, electrical circuits, control systems, and method-of-lines discretizations of parabolic partial differential equations; recognizing stiffness and selecting an appropriately stable implicit solver is essential to obtain results in feasible time, and most production ODE software includes automatic stiffness detection and switching.

History

The notion of stiffness was identified by Curtiss and Hirschfelder in 1952, and the supporting stability theory — A-stability and the order barriers — was developed by Dahlquist; Gear's backward differentiation formula codes and later high-order implicit Runge-Kutta methods established the practical toolkit for stiff and differential-algebraic problems.

Key figures

Germund Dahlquist
C. William Gear
Ernst Hairer
Gerhard Wanner

Seminal works

hairer1996
iserles2008

Frequently asked questions

What exactly makes an ODE stiff?: Stiffness arises when the system has components that decay much faster than the solution of interest evolves. There is no single sharp definition, but the practical signature is that explicit methods are forced to use very small steps for stability even when accuracy would permit large ones.
Why do stiff problems require implicit methods?: Implicit methods can have stability regions covering the entire left half-plane (A-stability), so they remain stable at large step sizes for rapidly decaying modes. Explicit methods have bounded stability regions, which forces tiny steps and makes them impractical for stiff problems.