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W. Auzinger, I. Brezinova, H. Hofstätter, O. Koch, M. Quell:
"Practical Splitting Methods for the Adaptive Integration of Nonlinear Evolution Equations. Part II: Comparison of Local Error Estimation and Step-Selection Strategies for Nonlinear Schrödinger and Wave Equations";
Computer Physics Communications, 234 (2019), 55 - 71.

We compare the practical performance of adaptive splitting methods for the solu-tion of nonlinear Schrödinger equations. Different methods for local error estima-tion are assessed with respect to their accuracy and efficiency in conjunction with promising strategies for step-size adaptation. The numerical comparisons com-prise the cubic nonlinear Schrödinger equation with a blow-up solution, systems of coupled nonlinear Schrödinger equations, a rotational and a Gross-Pitaevskii equation under a highly oscillatory potential inducing wave chaos, and a quantum control model with a time-dependent potential. Finally, for nonlinear wave equa-tions we demonstrate the enhanced computational stability ensuing from adaptive step selection strategies close to the border mandated by the CFL condition.

Nonlinear Schrödinger equations, splitting methods, local error estimators, adaptive step-size selection, embedded methods, defect-based

http://dx.doi.org/10.1016/j.cpc.2018.08.003