Relative error long-time behavior in matrix exponential approximations for numerical integration: the stiff situation

In the stiff situation, we consider the long-time behavior of the relative error $\gamma_n$ in the numerical integration of a linear ordinary differential equation $y^\prime(t)=Ay(t),\ t\ge 0$, where $A$ is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g. a polynomial or a rational approximation. We study the long-time behavior of $\gamma_n$ by comparing it to the relative error $\gamma_n^{\rm long}$ in the numerical integration of the long-time solution, i.e. the projection of the solution on the eigenspace of the rightmost eigenvalues. The error $ \gamma_n^{\rm long}$ grows linearly in time, it is small and it remains small in the long-time. We give a condition under which $\gamma_n\approx \gamma_n^{\rm long}$, i.e. $\frac{\gamma_n}{\gamma_n^{\rm long}}\approx 1$, in the long-time. When this condition does not hold, the ratio $\frac{\gamma_n}{\gamma_n^{\rm long}}$ is large for all time. These results describe the long-time behavior of the relative error $\gamma_n$ in the stiff situation.