The Effective Stability of Adaptive Timestepping ODE Solvers

Abstract

We consider the behavior of certain adaptive timestepping methods, based upon embedded explicit Runge--Kutta pairs, when applied to dissipative ODEs. It has been observed numerically that the standard local error controls can impart desirable stability properties, but this has been rigorously verified only for very special, low-order, Runge--Kutta pairs. The rooted-tree expansion of a certain quadratic form, central to the stability theory of Runge--Kutta methods, is derived. This, together with key assumptions on the sequence of accepted time-steps and the local error estimate, provides a general explanation for the observed stability of such algorithms on dissipative problems. Under these assumptions, which are expected to hold for typical numerical trajectories, two different results are proved. First, for a large class of embedded Runge--Kutta pairs of order $(1,2)$, controlled on an error-per-unit-step basis, all such numerical trajectories will eventually enter a particular bounded set. This occurs for sufficiently small tolerances independent of the initial conditions. Second, for pairs of arbitrary orders $(p-1,p)$, operating under either error-per-step or error-per-unit-step control, similar results are obtained when an additional structural assumption (that should be valid for many cases of interest) is imposed on the dissipative vector field. Numerical results support both the analysis and the assumptions made.