# Numerical methods for Maxwell's equations (Summer Semester 2014)

Lecturer: | Prof. Dr. Tobias Jahnke |
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Classes: | Lecture (0155800), Problem class (0155900) |

Weekly hours: | 2+1 |

Lecture: | Wednesday 8:00-9:30 | Z 2 | Begin: 16.4.2014 |
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Problem class: | Monday 14:00-15:30 (every 2nd week) | 1C-01 Allianz building (05.20) | Begin: 28.4.2014 |

Lecturer, Problem classes | Prof. Dr. Tobias Jahnke |
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Office hours: Monday, 10 am - 11 am | |

Room 3.042 Kollegiengebäude Mathematik (20.30) | |

Email: tobias.jahnke@kit.edu | |

Problem classes | Dr. Michael Kreim |

Office hours: | |

Room Kollegiengebäude Mathematik (20.30) | |

Email: kreim@kit.edu |

The classical theory of electromagnetism is based on a system of coupled partial differential equations derived by James Clerk Maxwell about 150 years ago. These equations describe the interaction between the electric and magnetic fields, the electric and magnetic flux densities, and the electric and magnetic current densities. Solving Maxwell's equations numerically is challenging problem which appears in many different technical applications.

In this lecture numerical methods for the time-dependent Maxwell equations will be derived and analyzed. The first part will focus on finite-difference time-domain methods such as the famous Yee scheme and alternative approaches such as the ADI splitting method of Zheng, Chen, Zhang. The second part of the lecture will be devoted to space discretization by finite elements and (if time permits) discontinuus Galerkin methods.

The course consists of a lecture (Wednesday 8:00-9:30, Z 2) and problem classes (Monday 14:00-15:30, 1C-01, about every second week). Both the lecture and the problem classes will be given in English. The lecture is intended for students in mathematics, physics and other sciences with basics in ordinary and partial differential equations and the corresponding numerical methods. Some background in functional analysis (Sobolev spaces) and operator semigroup theory is helpful, but not mandatory. Previous knowledge about Maxwell's equations is not expected because a short introduction to these equations and a proof of their well-posedness will be provided in the first chapter.

# Problem classes: New room

The problem classes will take place in 1C-01 in the Allianz building, not in Z2. Problem classes will be held on 28.4., 5.5. (lecture), 19.5., 2.6., 16.6., 30.6., 14.7.

Problem set 1 (16.06.2014)

Problem set 2 (30.06.2014)

# Handwritten lecture notes

Warning: These notes are very sketchy and probably still contain a lot of typos and errors.

Lecture 1 (16.04.2014)

Lecture 2 (23.04.2014)

Lecture 3 (30.04.2014)

Lecture 4 (05.05.2014)

Lecture 5 (14.05.2014)

Lecture 6 (21.05.2014)

Lecture 7 (28.05.2014)

Lecture 8 (04.06.2014)

Lecture 9 (11.06.2014)

Lecture 10 (18.06.2014)

Lecture 11 (25.06.2014)

Lecture 12 (02.07.2014)

Lecture 13 (09.07.2014)

Lecture 14 (16.07.2014)

# Evaluation

The results of the evaluation are available here.

# Slides

Finite difference method for the wave equation (28.05.2014)

Yee grid (04.06.2014)

# Examination

Oral exam, 25 min, 28/7/2014 or 18/8/2014.

If you want to take the exam, please send a message with your preferred date to jahnke@kit.edu. Moreover, you will have to register online via the QISPOS webpage.

# References

M. Hochbruck, T. Jahnke and R. Schnaubelt: *Convergence of an ADI splitting for Maxwell's equations.*

Submitted, 2014.

J. D. Jackson: *Classical electrodynamics.*

3rd edition, New York: Wiley, 1999. (German version available online.)

P. Monk: *Finite element methods for Maxwell's equations.*

Oxford: Clarendon Press , 2006

A. Taflove and S. C. Hagness: *Computational electrodynamics: the finite-difference time-domain method.*

3. ed., Boston : Artech House, 2005.

K. Yee: *Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media.*

IEEE Transactions on Antennas and Propagation, 14 (1966), pp. 302 -- 307.

F. Zheng, Z. Chen, and J. Zhang: *Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method.*

IEEE Transactions on Microwave Theory and Techniques, 48 (2000), pp. 1550--1558.