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Markus Ableidinger

Dipl.-Ing. Markus Ableidinger

Dipl.-Ing. Markus Ableidinger (Zum Team)

S2 0619
Tel.: +43 732 2468 4167

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FWF-Projekt P26314  (Neues Fenster)

The topic of this project lies in the intersection of stochastic and numerical analysis and its general aim is the development and analysis of numerical methods for the stochastic Landau-Lifshitz-Gilbert equation modelling micromagnetic phenomena. In 1963, W. F. Brown motivated a stochastic partial differential equation modelling the uniform vector magnetisation of a fine ferromagnetic particle, where the magnitude of the vector is essentially constant, but its direction is subject to thermal fluctuations. An important requirement for a successful numerical treatment of this equation is that the numerical method respects the qualitative behaviour of its solution, where the most prominent constraints on the evolution of the magnetisation vector are
a) that the length of the vector is constant in time at each spatial location and,
b) conditions with respect to the so-called Landau energy of the system are satisfied.
In addition, the interplay of these geometric aspects and the Stratonovich-type, multiplicative noise forcing in the nonlinear stochastic partial differential equation needs to be taken into account for the construction of convergent and reliable numerical algorithms. A fundamental role is played by the time integration scheme in this construction, and in the deterministic setting a number of approaches, in particular based on Geometrical Numerical Integration techniques, have been proposed. In the literature on stochastic numerics, neither the specific structure of the Stratonovich stochastic ordinary differential equations arising in the spatial discretisation of the stochastic Landau-Lifshitz-Gilbert equation, nor the inherent geometrical structure of the problem have been addressed in any systematic way. Further aspects of importance for dealing with the SLLG equation and its space discretised stochastic ordinary differential equation are concerned with taking advantage of the "smallness of the noise" for the efficiency of the method and the stability of the numerical methods for the space discretised stochastic ODEs. The latter aspect concerns on the one hand the efficiency of the method in terms of the choice of explicit or implicit time integrators, time step-sizes or solvers for nonlinear systems, and on the other hand, the reliability of long.time simulations, e.g., when the goal of simulations is to compute invariant distributions. Thus the focus of this project is to investigate how the time integration methods incorporated into a solver for the SLLG equation can be designed and/ or improved for reliability of the dynamics and efficiency, where we propose methods originating from Geometric Numerical Integration theory, and to study stability properties of the schemes.