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Concept
Stiff equation
Summary
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.
When integrating a differential equation numerically, one would expect the requisite step size to be relatively small in a region where the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with slope nearly zero. For some problems this is not the case. In order for a numerical method to give a reliable solution to the differential system sometimes the step size is required to be at an unacceptably small level in a region where the solution curve is very smooth. The phenomenon is known as stiffness. In some cases there may be two different problems with the same solution, yet one is not stiff and the other is. The phenomenon cannot therefore be a property of the exact solution, since this is the same for both problems, and must be a property of the differential system itself. Such systems are thus known as stiff systems.
Consider the initial value problem
The exact solution (shown in cyan) is
We seek a numerical solution that exhibits the same behavior.
The figure (right) illustrates the numerical issues for various numerical integrators applied on the equation.
One of the most prominent examples of the stiff ordinary differential equations (ODEs) is a system that describes the chemical reaction of Robertson:
If one treats this system on a short interval, for example, there is no problem in numerical integration. However, if the interval is very large (1011 say), then many standard codes fail to integrate it correctly.
Additional examples are the sets of ODEs resulting from the temporal integration of large chemical reaction mechanisms.
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