Mathematical and numerical studies of non linear ferromagnetic materials

In this paper we are interested in the numerical modeling of absorbing ferromagnetic materials obeying the non-linear Landau-Lifchitz-Gilbert law wit h respect to the propagation and scattering of electromagnetic waves. In this work we consider the 1D problem. We first show that the corresponding Cauchy problem has a unique global solution. We then dérive a numerical scheme based on an appropriât e modification of Yee's scheme, that we show to preserve some important properties of the continuous model such as the conservation of the norm of the magnetization and the decay of the electromagnetic energy. Stability is proved under a suitable CFL condition. Some numerical results for the 1D model are presented. AMS Subject Classification. 35Q60, 35L60, 65M06. Received: September 24, 1997. Revised: April 9, 1998 and September 11, 1998. 1. I N T R O D U C T I O N 1.1. Exposit ion of t h e physical problem Without entering into the details of the theory of ferromagnetism (see for this [2,3] or [4]), we briefly describe the physics of our problem. 1.1.1. The ferromagnetic zone By a "ferromagnetic material" we mean a material that possesses a spontaneous magnetization M. (Throughout bold letters will be used for vectors.) Ferromagnetic materials such as ferrites or garnets are widely used in the microwave industry as shifters or circulât ors for instance. In this paper we study in particular the propagation of electromagnetic waves inside such ferromagnetic materials; they are said to be absorbing materials, we will see what sensé we can give to this assertion. A ferromagnetic zone is a région in which M ̂ 0, and we dénote by O such a région. This zone will be examined in this paper on a scale large enough to permit the use of a continuous magnetization vector M (x) at any time; so we do not consider a priori any domain or wall concept [1]. The magnetization M is assumed to satisfy the Landau-Lifchitz-Gilbert équation (LLG):