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B. Stiftner:
"Second-order in time numerical integration of the Landau-Lifshitz-Gilbert equation";
Supervisor, Reviewer: D. Praetorius, J. Kraus, A. Prohl; Institute for Analysis and Scientific Computing, 2018; oral examination: 2018-06-21.

In computational micromagnetism, the Landau-Lifshitz-Gilbert equation (LLG) is the fundamental mathematical model for the understanding and simulation of time-dependent micromagnetic phenomena. The non-linear nature of the equation, a non-convex side constraint, and the non-uniqueness of solutions aggravate the development of efficient numerical algorithms. The (second-order) tangent plane scheme from [Alouges et al. (Numer. Math., 128, 2014)] and the midpoint scheme from [Bartels and Prohl (2006) (SIAM J. Numer. Anal., 44)] provide us with two finite-element-based algorithms, which are both (formally) second-order in time and unconditionally convergent. The particular structure of both algorithms suggests the numerically expensive implicit treatment of possible lower-order terms and of coupled systems like, e.g., the computation of the stray field or, more general, the coupling of LLG with the full Maxwell system. To avoid this and to conserve the second-order in time convergence, we employ an implicit-explicit second-order in time Adams-Bashforth-type approach, where we treat the lower-order terms explicitly in time. For couplings with other equations, this decouples the approximate computation of the magnetization (i.e., the solution of LLG), and of the coupled equation (e.g., electrical and magnetic field of the coupling of LLG with the full Maxwell system). The resulting algorithms are formally second-order in time. For the coupling with eddy currents, this yields a decoupled second-order in time tangent plane scheme. For the coupling with the spin diffusion equation, this yields a decoupled second-order in time midpoint scheme. Moreover, we provide certain assumptions in a unified framework, which covers, in particular, physically relevant dissipative effects. We extend the existing convergence analysis and prove unconditional convergence of our extended algorithms. Moreover, we discuss the efficient solution of the corresponding (linear and non-linear) variational problems. Numerical experiments with our extensions confirm the preservation of the second-order in time convergence, reduced computational costs and the applicability to physically relevant examples of our algorithms.

http://www.asc.tuwien.ac.at/~praetorius/download/thesis/phd/stiftner2018.pdf