Asymptotic expansions of the global discretization error for stiff problems

Abstract

The existence of asymptotic expansions of the global discretization error for a general class of nonlinear stiff differential equations \[ y'(t) = A(t)y(t) + \varphi (t,y(t)), \] where $A(t)$ has a “stiff spectrum” characterized by a small parameter $\varepsilon $ and where $\varphi (t,y)$ is smooth, is discussed. The following methods are considered: implicit Euler, implicit midpoint, and trapezoidal rules. In strongly stiff situations ($\varepsilon $ significantly smaller than the stepsize h) the implicit Euler scheme admits a full asymptotic expansion; the same is true for the midpoint rule and for the trapezoidal rule under certain coupling conditions. In those strongly stiff cases where a full expansion does not exist for the midpoint or trapezoidal rule, the remainder term is of a reduced order but shows a regular, oscillating behavior that is described in detail. In mildly stiff situations, order reductions of the remainder term inevitably occur in any case after the start or after the change of stepsize but—as can be shown by discrete singular perturbation techniques—these order reductions are rapidly damped out as the integration proceeds. Results are illustrated by various numerical examples; in particular, numerical experience with extrapolation and defect correction is reported.