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Uniformly Accurate Methods for Klein-Gordon type Equations

Baumstark, Simon

Abstract (englisch):

The main contribution of this thesis is the development of a novel class of uniformly accurate methods for Klein-Gordon type equations.
Klein-Gordon type equations in the non-relativistic limit regime, i.e., $c\gg 1$, are numerically very challenging to treat, since the solutions are highly oscillatory in time. Standard Gautschi-type methods suffer from severe time step restrictions as they require a CFL-condition $c^2\tau<1$ with time step size $\tau$ to resolve the oscillations. Within this thesis we overcome this difficulty by introducing limit integrators, which allows us to reduce the highly oscillatory problem to the integration of a non-oscillatory limit system. This procedure allows error bounds of order $\mathcal{O}(c^{-2}+\tau^2)$ without any step size restrictions. Thus, these integrators are very efficient in the regime $c\gg 1$. However, limit integrators fail for small values of $c$.
In order to derive numerical schemes that work well for small as well as for large $c$, we use the ansatz of "twisted variables", which allows us to develop uniformly accurate methods with respect to $c$. In particular, we introduce efficient and robust uniformly accurate exponential-type integrators which resolve the solution in the relativistic regime as well as in the highly oscillatory non-relativistic regime without any step size restriction. ... mehr

Volltext §
DOI: 10.5445/IR/1000087180
Veröffentlicht am 06.11.2018
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Publikationstyp Hochschulschrift
Publikationsjahr 2018
Sprache Englisch
Identifikator urn:nbn:de:swb:90-871804
KITopen-ID: 1000087180
Verlag Karlsruher Institut für Technologie (KIT)
Umfang V, 146 S.
Art der Arbeit Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Angewandte und Numerische Mathematik (IANM)
Prüfungsdatum 12.07.2018
Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015)
Referent/Betreuer Schratz, J. Prof. K.
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