Relative error stability and instability of matrix exponential approximations for stiff numerical integration of long-time solutions

We study the relative error in the numerical integration of the long-time solution of a linear ordinary differential equation y′(t)=Ay(t),t≥0, where A is a normal matrix. The numerical long-time solution is obtained by using at any step an approximation of the matrix exponential. This paper analyzes the relative error in the stiff situation and it shows that, in this situation, some A-stable approximants exhibit instability with respect to perturbations in the initial value of the long-time solution.