SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Plenary Lecture


Convergence of an ADI splitting for Maxwell's equations

M. Hochbruck, T. Jahnke and R. Schnaubelt

Abstract
Maxwell's equations provide the foundation for the theory of electromagnetism, and solving these equations numerically is an important task in many applications. For problems posed on a cuboid or on $\mathbb{R}^3$, the alternating direction implicit method proposed by Zheng, Chen, and Zhang (2000) is particularly attractive, because this method is unconditionally stable and computationally cheap. The main idea is, roughly speaking, to decompose the Maxwell operator in such a way that the sub-flows can be propagated in a stable and efficient way. In this talk, second-order convergence for the semi-discretization in time is shown in the framework of operator semigroup theory. The proof is based on results concerning the regularity of the Cauchy problems of the sub-flows, which then allow to apply an abstract convergence proof by Hansen and Ostermann [1]. Before the error analysis, well-posedness of Maxwell's equations on cuboids and on $\mathbb{R}^3$ will be discussed.

Bibliography
[1] E. Hansen and A. Ostermann, Dimension splitting for evolution equations, Numerische Mathematik, 108 (2008), pp. 557-570.

Organized by         Universidad de Valladolid     IMUVA