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Mathematics of Computation

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On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations

Authors: A. Prothero and A. Robinson
Journal: Math. Comp. 28 (1974), 145-162
MSC: Primary 65L05
MathSciNet review: 0331793
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Abstract: The stiffness in some systems of nonlinear differential equations is shown to be characterized by single stiff equations of the form

$\displaystyle y' = g'(x) + \lambda\{y - g(x)\}.$

The stability and accuracy of numerical approximations to the solution $ y = g(x)$, obtained using implicit one-step integration methods, are studied. An S-stability property is introduced for this problem, generalizing the concept of A-stability. A set of stiffly accurate one-step methods is identified and the concept of stiff order is defined in the limit $ \operatorname{Re}(-\lambda) \to \infty$. These additional properties are enumerated for several classes of A-stable one-step methods, and are used to predict the behaviour of numerical solutions to stiff nonlinear initial-value problems obtained using such methods. A family of methods based on a compromise between accuracy and stability considerations is recommended for use on practical problems.

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Keywords: Stiff system of ordinary differential equations, implicit one-step methods, A-stability, S-stability, stiffly accurate methods, stiff order
Article copyright: © Copyright 1974 American Mathematical Society