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

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.