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W. Auzinger, H. Hofstätter, O. Koch:
"Precise local error control for symmetric one-step schemes applied to nonlinear evolution equations";
Talk: SciCADE 2019, International Conference on Scientific Computation and Differential Equations, Innsbruck; 2019-07-22 - 2019-07-26.

Controlling the local error in time (and adapting the stepsize) of one-step methods for evolution equations can be performed in different ways. For a number problem classes (e.g., equations of Schrödinger
type) defect-based techniques are useful because the defect (residual) of the numerical solution can be
evaluated efficiently, resulting in an asymptotically correct local error estimator.
We recall the idea of this technique, and concentrate on the case of symmetric schemes. Here the
asymptotic quality of the estimator can be further be increased by computing a ´symmetrized´ version
of the defect. This can also be used as a corrector resulting in a higher-order scheme. We demonstrate
that this inherits symmetry, and also symplecticity, of the underlying lower-order scheme asymptotically
in a very precise way. Numerical examples are presented.