TABLE OF CONTENTS Stiffness is a subtle, difficult, and important concept in the numerical solution of ordinary differential equations. It depends on the differential equation, the initial conditions, and the numerical method. Dictionary definitions of the word "stiff" involve terms like "not easily bent," "rigid," and "stubborn." We are concerned with a computational version of these properties.
A problem is stiff if the solution being sought varies slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results.
Stiffness is an efficiency issue. If we weren't concerned with how much time a computation takes, we wouldn't be concerned about stiffness. Nonstiff methods can solve stiff problems; they just take a long time to do it.
A model of flame propagation provides an example. We learned about this example from Larry Shampine, one of the authors of the MATLAB ordinary differential equation suite. If you light a match, the ball of flame grows rapidly until it reaches a critical size. Then it remains at that size because the amount of oxygen being consumed by the combustion in the interior of the ball balances the amount available through the surface. The simple model is
The scalar variable y(t) represents the radius of the ball. The y 2 and y 3 terms come from the surface area and the volume. The critical parameter is the initial radius, ?, which is "small." We seek the solution over a length of time...
