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Error Analysis of Exponential Integrators for Nonlinear Wave-Type Equations

Dörich, Benjamin

Abstract:
This thesis is concerned with the time integration of certain classes of nonlinear evolution equations in Hilbert spaces by exponential integrators. We aim to prove error bounds which can be established by including only quantities given by a wellposedness result. In the first part, we consider semilinear wave equations and introduce a class of first- and second-order exponential schemes. A standard error analysis is not possible due to the lack of regularity. We have to employ appropriate filter functions as well as the integration by parts and summation by parts formulas in order to obtain optimal error bounds. In the second part, we propose two exponential integrators of first and second order applied to a class of quasilinear wave-type equations. By a detailed investigation of the differentiability of the right-hand side we derive error bounds in different norms. In the framework we can treat quasilinear Maxwell’s equations in full space and on a smooth domain as well as a class of quasilinear wave equations. In both parts, we include numerical examples to confirm our theoretical findings.

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Volltext §
DOI: 10.5445/IR/1000130187
Veröffentlicht am 10.03.2021
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Hochschulschrift
Publikationsdatum 10.03.2021
Sprache Englisch
Identifikator KITopen-ID: 1000130187
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 152 S.
Art der Arbeit Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Angewandte und Numerische Mathematik (IANM)
Prüfungsdatum 24.02.2021
Referent/Betreuer Prof. M. Hochbruck
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Schlagwörter numerical analysis, semmilinear wave equations, quasilinear wave-type equations, semilinear evolution equations, quasilinear evolution equations, operator semigroups, exponential integrators, time integration, abstract error analysis, a priori error estimates, Maxwell equations, Kerr nonlinearity
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