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

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