Novel numerical methods for the Landau-Lifshitz-Gilbert equation

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. Further aspects of importance for dealing with the Stochastic Landau-Lifschitz-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. The focus of this application is to investigate how the time integration methods incorporated into a solver for the Stochastic Landau-Lifschitz-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.