Novel Numerical Methods for the Stochastic Landau-Lifshitz-Gilbert Equation

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 that is the Stochastic Landau-Lifshitz-Gilbert 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 ones based on Geometric 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 Stochastic Landau-Lifshitz-Gilbert 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 ordinary differential equations. 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 the simulations is to compute invariant distributions. Thus the focus of this application is to investigate how the time integration methods incorporated into a solver for the Stochastic Landau-Lifshitz-Gilbert 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.

The topic of this project lies in the intersection of stochastic and numerical analysis and its general aim was the development and analysis of numerical methods for the Stochastic Landau-Lifschitz-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 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 term in this equation needs to be taken into account. The focus of this project was to investigate time integration methods in this context and our main results are the construction of structure preserving splitting methods for the Stochastic Landau-Lifschitz-Gilbert Equation and the construction and analysis of stochastic Munthe-Kaas methods for the weak approximation of this equation. We illustrated our theoretical considerations with numerical simulations which showed that our methods realise their theoretical properties and prove to be competitive with previously used methods. We would like to emphasise that the constructed methods are also relevant for the numerical approximation of more general stochastic differential equations.