Shampine_Gear_Stiffness_Paper

# Shampine_Gear_Stiffness_Paper - SIAM REVIEW Voi 21 No 1...

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SIAM REVIEW Voi. 21, No. 1, January 1979 () 1979 Society for Industrial and Applied Mathematics 0036-1445/79/2101-0001 \$01.00/0 A USER’S VIEW OF SOLVING STIFF ORDINARY DIFFERENTIAL EQUATIONS* L. F. SHAMPINE " AND C. W. GEAR Abstract. This paper aims to assist the person who needs to solve stiff ordinary differential equations. First we identify the problem area and the basic difficulty by responding to some fundamental questions: Why is it worthwhile to distinguish a special class of problems termed "stiff"? What are stiff problems? Where do they arise? How can we recognize them? Second we describe the characteristics shared by methods for the numerical solution of stiff problems. These characteristics have important implications as to the convenience and efficiency of solution of even routine problems. Understanding them is indispensable to [he assembling of codes for the very efficient solution of special problems or for solving exceptionally large problems at all. Third we shall briefly discuss what is meant by "solving" a differential equation numerically and what might be reasonably expected in the case of stiff problems. 1. Introduction. The numerical solution of ordinary differential equations is an old topic and, perhaps surprisingly, methods discovered around the turn of the century arc still the basis of the most effective, widely used codes for this purpose [23]. Great improvements in efficiency have been made, but it is probably fair to say that the most significant achievements have been in reliability, convenience, and diagnostic capabil- ities. The typical scientific problem can be solved by casual users of these codes both easily and cheaply. Nevertheless, there are several kinds of problems which classical methods do not handle very efficiently. The problems called "stiff" are too important to ignore, and are too expensive to overpower. They are too important to ignore because they occur in many physically important situations. They are too expensive to overpower because of their size and the inherent difficulty they present to classical methods, no matter how great an improvement in computer capacity becomes available. Even if one can bear the expense, classical methods of solution require so many steps that roundoff errors may invalidate the solution. It is all the more frustrating that the solutions of stiff problems look like they should be particularly easy to compute. After a few general remarks about solving differential equations, we shall use some simple examples to show where the trouble originates and what might be done about it. We shall mention a number of contexts in which stiffness was recognized and dealt with by scientists through the use of special features of the problem. Our attention here will be directed towards the phenomenon of stiffness and towards general purpose procedures for the solution of stiff differential equations.

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Shampine_Gear_Stiffness_Paper - SIAM REVIEW Voi 21 No 1...

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