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

Abstract

AbstractIn the stiff situation, we consider the long-time behavior of the relative error $$\gamma _n$$ γ n in the numerical integration of a linear ordinary differential equation $$y^{\prime }(t)=Ay(t),\quad t\ge 0$$ y ′ ( t ) = A y ( t ) , t ≥ 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 $$\gamma _n$$ γ n by comparing it to the relative error $$\gamma _n^{\mathrm{long}}$$ γ n 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 $$ \gamma _n^{\mathrm{long}}$$ γ n long grows linearly in time, it is small and it remains small in the long-time. We give a condition under which $$\gamma _n\approx \gamma _n^{\mathrm{long}}$$ γ n ≈ γ n long , i.e. $$\frac{\gamma _n}{\gamma _n^{\mathrm{long}}}\approx 1$$ γ n γ n long ≈ 1 , in the long-time. When this condition does not hold, the ratio $$\frac{\gamma _n}{\gamma _n^{\mathrm{long}}}$$ γ n γ n long is large for all time. These results describe the long-time behavior of the relative error $$\gamma _n$$ γ n in the stiff situation.