Symmetrized local error estimators for time-reversible one-step methods in nonlinear evolution equations

TL;DR

This paper extends defect-based local error estimators to nonlinear, nonautonomous evolution equations, demonstrating improved asymptotic order and efficiency for time-reversible integrators, with successful adaptive time-stepping in numerical experiments.

Contribution

It introduces symmetrized local error estimators for nonlinear evolution equations, maintaining linear case asymptotic properties and enabling efficient adaptive schemes.

Findings

Estimators achieve improved asymptotic order as step size decreases.

Computational effort is comparable or lower than traditional estimators.

Numerical experiments confirm effectiveness for Schrödinger-type equations.

Abstract

Prior work on computable defect-based local error estimators for (linear) time-reversible integrators is extended to nonlinear and nonautonomous evolution equations. We prove that the asymptotic results from the linear case [W. Auzinger and O. Koch, An improved local error estimator for symmetric time-stepping schemes, Appl.Math.Lett. 82 (2018), pp. 106-110] remain valid, i.e., the modified estimators yield an improved asymptotic order as the step size goes to zero. Typically, the computational effort is only slightly higher than for conventional defect-based estimators, and it may even be lower in some cases. We illustrate this by some examples and present numerical results for evolution equations of Schr\"odinger type, solved by either time-splitting or Magnus-type integrators. Finally, we demonstrate that adaptive time-stepping schemes can be successfully based on our local error estimators.