# 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

 Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM) Publikationstyp Hochschulschrift Jahr 2018 Sprache Englisch Identifikator urn:nbn:de:swb:90-871804 KITopen-ID: 1000087180 Verlag KIT, Karlsruhe Umfang V, 146 S. Abschlussart Dissertation Fakultät Fakultät für Mathematik (MATH) Institut Institut für Angewandte und Numerische Mathematik (IANM) Prüfungsdatum 12.07.2018 Referent/Betreuer JProf. K. Schratz Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015)
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