Linear stability of stiff differential equation solvers (Q1062732)

Consider a linear multistep method used to solve a stiff differential equation \(y'(x)=f(y(x))\). In a typical step, the method gives an approximation \(y_ n\) to \(y(x_ n)\) and hence an approximation \(f(y_ n)\) to \(y'(x_ n)\). A number of authors, for example \textit{C. Gear} and \textit{Y. Saad} [SIAM J. Sci. Stat. Comput. 4, 583-601 (1983; Zbl 0541.65051)], recommend that in subsequent steps an approximation that exactly satisfies the corrector equation should be used instead of \(f(y_ n)\). It is shown that the resulting method, applied to the linear problem \(y'=\lambda y\), is stable if the corrector equation is stable and the residuals obtained in an iterative solution of the corrector equation are uniformly bounded.