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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. Sometimes the step size is forced down to an unacceptably small level in a region where the solution curve is very smooth. The phenomenon being exhibited here is known as stiffness. In some cases we may have two different problems with the same solution, yet problem one is ''not'' stiff and problem two ''is'' stiff. Clearly the phenomenon cannot 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. It is thus appropriate to speak of ''stiff systems''. == Motivating example == 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. Applying this method instead of Euler's method gives a much better result (blue). the numerical results decrease monotonically to zero, just as the exact solution does. }} One of the most prominent examples of the stiff ODEs is a system that describes the chemical reaction of Robertson: : If one treats this system on a short interval, e.g. 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. Here, the stiffness arises from the coexistence of very slow and very fast reactions. To solve them, the software packages KPP and Autochem can be used. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stiff equation」の詳細全文を読む スポンサード リンク
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