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

Abstract:

The stiffness in some systems of nonlinear differential equations is shown to be characterized by single stiff equations of the form \[ y’ = g’(x) + \lambda \{y - g(x)\}.\] 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.