Stiff Equations

Numerical Analysis · Axiom Academy

LESSON Stiff Differential Equations Understanding problems with multiple time scales and how to solve them efficiently A differential equation is stiff when it contains dynamics on vastly different time scales. Some components decay rapidly (fast transients), while others change slowly. The classic example is a system where one component has a time constant of microseconds while another has seconds. For a linear system y' = Ay, the stiffness ratio is: When this ratio is large (e.g., 10^6), the problem is stiff. In this example, the fast component (e^(-1000t)) decays in milliseconds, while the slow component (e^(-t)) takes seconds. After the initial transient, the solution is smooth but the fast dynamics still constrain explicit methods. Explicit methods like Forward Euler or RK4 have bounded stability regions . For stiff problems, the step size must be tiny to remain stable, even when the solution is changing slowly. This makes them impractically slow. Forward Euler gives y_(n+1) = (1 + h*lambda)*y_n. For stability, |1 + h*lambda| stability , not accuracy . The stability region of a method is the set of h*lambda values for which the numerical solution remains bounded. For stiff problems, we need methods whose stability regions include the entire left half-plane. A-stable: The stability region contains the entire left half of the complex plane (Re(h*lambda) < 0). Any stable ODE stays stable numerically.

This is the written version of the interactive lesson above. See the full Numerical Analysis course.