Micromagnetism is the study of magnetic materials on the microscopic length scale (of nano to micrometers), this scale does not take quantum mechanical effects into account, but is small enough to neglect certain macroscopic effects of magnetism in a material. The Landau-Lifshitz-Gilbert (LLG) equation is used within micromagnetism to determine the time evolution of the magnetisation vector field in a ferromagnetic solid. It is a partial differential equation with high non linearity, which makes it very difficult so solve analytically. Thus numerical methods have been developed for approximating the solution using computers. In this report we compare the performance of three different numerical methods for the LLG equation, the implicit midpoint method (IMP), the midpoint with extrapolation method (MPE), and the Gauss-Seidel Projection method (GSPM). It was found that all methods have convergence rates as expected; second order for IMP and MPE, and first order for GSPM. Energy conserving properties of the schemes were analysed and neither MPE or GSPM conserve energy. The computational time required for each method was determined to be very large for the IMP method in comparison to the other two. Suggestions for different areas of use for each method are provided.

Micromagnetics is a continuum mechanics theory of magnetic materials widely used in industry and academia. In this thesis we describe a complete numerical method, with a number of novel components, for the computational solution of dynamic micromagnetic problems by solving the Landau-Lifshitz-Gilbert (LLG) equation. In particular we focus on the use of the implicit midpoint rule (IMR), a time integration scheme which conserves several important properties of the LLG equation. We use the finite element method for spatial discretisation, and use nodal quadrature schemes to retain the conservation properties of IMR despite the weak-form approach. We introduce a novel, generally-applicable adaptive time step selection algorithm for the IMR. The resulting scheme selects error-appropriate time steps for a variety of problems, including the semi-discretised LLG equation. We also show that it retains the conservation properties of the fixed step IMR for the LLG equation. We demonstrate how hybrid FEM/BEM magnetostatic calculations can be coupled to the LLG equation in a monolithic manner. This allows the coupled solver to maintain all properties of the standard time integration scheme, in particular stability properties and the energy conservation property of IMR. We also develop a preconditioned Krylov solver for the coupled system which can efficiently solve the monolithic system provided that an effective preconditioner for the LLG sub-problem is available. Finally we investigate the effect of the spatial discretisation on the comparative effectiveness of implicit and explicit time integration schemes (i.e. the stiffness). We find that explicit methods are more efficient for simple problems, but for the fine spatial discretisations required in a number of more complex cases implicit schemes become orders of magnitude more efficient.