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

Abstract

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),quad tge 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^{mathrm{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^{mathrm{long}} grows linearly in time, it is small and it remains small in the long-time. We give a condition under which gamma _napprox gamma _n^{mathrm{long}}, i.e. frac{gamma _n}{gamma _n^{mathrm{long}}}approx 1, in the long-time. When this condition does not hold, the ratio frac{gamma _n}{gamma _n^{mathrm{long}}} is large for all time. These results describe the long-time behavior of the relative error gamma _n in the stiff situation.