Would you help with a question about differential equations?

Question

Salut

I was wondering if you could tell me what the difference between a "stiff" and a "nonstiff" differential equation is? I am not too lazy to research the answer myself (this question is in fact part of said research), I am just new to the discipline and, never having had a formal course, seek information where I can find it.

Thanks for reading my senseless blithering,

Max

Answer 1

Like many things in life, the best source is Wikipedia [1]. To make a long story short: a stiff equation is one that is unstable with respect to numerical schemes, where a nonstiff equation is the opposite. To see the picture better, we first consider what it means to solve a differential equation numerically. Basically, since the computer is a digital device, it is not advisable to try to model the entire continuum; rather, a numerical scheme to solve a differential equation starts by chosing a grid on which to solve the equation. Once we have a grid, we can reduce the differential equation to relations between values on a finite number of points.

Now, depending on the spacing between grid elements that one choses, one may sometimes end up with a result that hides smaller scale features (in general there will always be errors, since the numerical scheme is an approximation at best). A heuristic consideration is the following: suppose you look at a differential equation with grid size roughly equals 1. Then by the Nyquist theorem, you will cut off effects with frequencies higher than 1. For a purely linear equation it is usually no biggie. But for a highly nonlinear equation, the nonlinear feedback may work in such a way that high frequency signals can mix to produce a low frequency one. Whereas an analytic solution to the differential equation will capture this effect, a numerical scheme with large grid spacing will completely ignore this effect, producing a solution much different from that arrived from the analytical method.

For a lot of these kinds of situations, the situation can be made more amenable once one choses small enough grid points to capture the oscillation of the solution.

The stiffness of the equation is not only dependent on the equation, however. It is also dependant on the numerical method one uses to solve it. Certain numerical methods may be more stable then others (more generous about giving up errors from approximations) when applied to certain equations. One of the major tasks when tackling a problem numerically is to first find a scheme that is stable.