# ADI schemes for the time integration of Maxwell equations

Zerulla, Konstantin

##### Abstract:
This thesis is concerned with the analysis and construction of alternating direction implicit (ADI) splitting schemes for the time integration of linear isotropic Maxwell equations on cuboids.
The work is organized in two major parts.

The first part deals with time discrete approximations to exponentially stable Maxwell equations.
By means of a divergence cleaning technique and artificial damping, we obtain an ADI scheme with approximations that also decay exponentially in time.
The decay rate is here uniform with respect to the time discretization.
One of the main ingredients in the proof of the decay behavior is an observability estimate for the numerical approximations.
This inequality is obtained by means of a discrete multiplier technique.
We also provide a rigorous error analysis for the uniformly exponentially stable ADI scheme, yielding convergence of order one in a space similar to $H^{-1}$.
The error result makes only assumptions on the initial data and the model parameters.

In the second part, we analyze time discrete approximations to linear isotropic Maxwell equations on a heterogeneous cuboid.
In this setting, the domain consists of two different homogeneous subcuboids.
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 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Hochschulschrift Publikationsdatum 25.01.2021 Sprache Englisch Identifikator KITopen-ID: 1000128718 Verlag Karlsruher Institut für Technologie (KIT) Umfang 220 S. Art der Arbeit Dissertation Fakultät Fakultät für Mathematik (MATH) Institut Institut für Analysis (IANA) Prüfungsdatum 16.12.2020 Referent/Betreuer Prof. R. Schnaubelt Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019) Schlagwörter Maxwell equations, ADI schemes, regularity analysis, exponential stability, time integration, splitting method, observability, error estimates
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