Symmetrized local error estimators for time-reversible one-step methods in nonlinear evolution equations
For researchers using time-reversible integrators for nonlinear evolution equations, this provides more accurate local error estimation with minimal extra computational cost.
This paper extends defect-based local error estimators to nonlinear and nonautonomous evolution equations, proving that the improved asymptotic order from the linear case holds. Numerical results for Schrödinger-type equations show successful adaptive time-stepping based on these estimators.
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 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.