Maxwell-Klein-Gordon (MKG) and Maxwell-Dirac (MD) systems physically describe the mutual interaction of moving relativistic particles with their self-generated electromagnetic field. Solving these systems in the nonrelativistic limit regime, i.e. when the speed of light $c$ formally tends to infinity, is numerically very delicate as the solution becomes highly oscillatory in time. In order to resolve the oscillations, standard time integrations schemes require severe restrictions on the time step $\tau\sim c^{-2}$ depending on the small parameter $c^{-2}$ which leads to high computational costs.

Within this thesis we propose and analyse two types of numerical integrators to efficiently integrate the MKG and MD systems in highly oscillatory nonrelativistic limit regimes to slowly oscillatory relativistic regimes.

The idea for the first type relies on asymptotically expanding the exact solution in the small parameter $c^{-1}$. This results in non-oscillatory Schrödinger-Poisson (SP) limit systems which can be solved efficiently by using classical splitting schemes. We will see that standard Strang splitting schemes, applied to the latter SP systems with step size $\tau$, allow error bounds of order $\mathcal{O}(\tau^2+c^{-N})$ for $N\in \mathbb N$ without any time step restriction. ... mehr

Within this thesis we propose and analyse two types of numerical integrators to efficiently integrate the MKG and MD systems in highly oscillatory nonrelativistic limit regimes to slowly oscillatory relativistic regimes.

The idea for the first type relies on asymptotically expanding the exact solution in the small parameter $c^{-1}$. This results in non-oscillatory Schrödinger-Poisson (SP) limit systems which can be solved efficiently by using classical splitting schemes. We will see that standard Strang splitting schemes, applied to the latter SP systems with step size $\tau$, allow error bounds of order $\mathcal{O}(\tau^2+c^{-N})$ for $N\in \mathbb N$ without any time step restriction. ... mehr

Zugehörige Institution(en) am KIT |
Institut für Angewandte und Numerische Mathematik (IANM) |

Publikationstyp |
Hochschulschrift |

Jahr |
2017 |

Sprache |
Englisch |

Identifikator |
urn:nbn:de:swb:90-764490 KITopen-ID: 1000076449 |

Verlag |
KIT, Karlsruhe |

Umfang |
V, 199 S. |

Abschlussart |
Dissertation |

Fakultät |
Fakultät für Mathematik (MATH) |

Institut |
Institut für Angewandte und Numerische Mathematik (IANM) |

Prüfungsdatum |
29.08.2017 |

Referent/Betreuer |
JProf. K. Schratz |

Projektinformation |
GRK 1294/3 (DFG, DFG KOORD, GRK 1294/3) SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015) |

Schlagworte |
Klein-Gordon, Dirac, Maxwell, Wave Equations, Schrödinger, Highly Oscillatory, Nonrelativistic Limit, Numerical Time Integration, Uniformly Accurate, Splitting |

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