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Uniformly accurate exponential-type integrators for Klein-Gordon equations with asymptotic convergence to the classical NLS splitting

Authors: Simon Baumstark, Erwan Faou and Katharina Schratz
Journal: Math. Comp. 87 (2018), 1227-1254
MSC (2010): Primary 35C20, 65M12, 35L05
Published electronically: August 15, 2017
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Abstract: We introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory nonrelativistic regime without any step-size restriction under the same regularity assumptions on the initial data required for the integration of the corresponding nonlinear Schrödinger limit system. In contrast to previous works we do not employ any asymptotic/multiscale expansion of the solution. This allows us to derive uniform convergent schemes under far weaker regularity assumptions on the exact solution. In addition, the newly derived first- and second-order exponential-type integrators converge to the classical Lie, respectively, Strang splitting in the nonlinear Schrödinger limit.

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Additional Information

Simon Baumstark
Affiliation: Fakultät für Mathematik, Karlsruhe Institute of Technology, Englerstrasse 2, 76131 Karlsruhe, Germany

Erwan Faou
Affiliation: INRIA & ENS Rennes, Avenue Robert Schumann F-35170 Bruz, France

Katharina Schratz
Affiliation: Fakultät für Mathematik, Karlsruhe Institute of Technology, Englerstrasse 2, 76131 Karlsruhe, Germany

Received by editor(s): June 17, 2016
Received by editor(s) in revised form: November 1, 2016
Published electronically: August 15, 2017
Additional Notes: The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. This work was also partly supported by the ERC Starting Grant Project GEOPARDI No 279389.
Article copyright: © Copyright 2017 American Mathematical Society

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