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W. Auzinger, H. Hofstätter, O. Koch:
"Symmetrized local error estimators for time-reversible one-step methods in nonlinear evolution equations";
Journal of Computational and Applied Mathematics, 356 (2019), 339 - 357.

Prior work on computable defect-based local error estimators for (linear) timereversible 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 equationsof Schrödinger 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.

nonlinear evolution equations, numerical time integration, one-step methods, time-reversible schemes, splitting methods, commutator-free Magnus-type methods, Magnus integrators, local error estimation

http://dx.doi.org/10.1016/j.cam.2019.02.011