Micromagnetics is a continuum mechanics theory of magnetic materials widelyused in industry and academia. In this thesis we describe a complete numericalmethod, with a number of novel components, for the computational solution ofdynamic micromagnetic problems by solving the Landau-Lifshitz-Gilbert (LLG)equation. In particular we focus on the use of the implicit midpoint rule (IMR), atime integration scheme which conserves several important properties of the LLGequation. We use the finite element method for spatial discretisation, and usenodal quadrature schemes to retain the conservation properties of IMR despitethe weak-form approach.We introduce a novel, generally-applicable adaptive time step selection algorithmfor the IMR. The resulting scheme selects error-appropriate time steps for a vari-ety of problems, including the semi-discretised LLG equation. We also show thatit retains the conservation properties of the fixed step IMR for the LLG equation.We demonstrate how hybrid FEM/BEM magnetostatic calculations can be coupledto the LLG equation in a monolithic manner. This allows the coupled solver tomaintain all properties of the standard time integration scheme, in particularstability properties and the energy conservation property of IMR. We also de-velop a preconditioned Krylov solver for the coupled system which can efficientlysolve the monolithic system provided that an effective preconditioner for the LLGsub-problem is available.Finally we investigate the effect of the spatial discretisation on the comparativeeffectiveness of implicit and explicit time integration schemes (i.e. the stiffness).We find that explicit methods are more efficient for simple problems, but for thefine spatial discretisations required in a number of more complex cases implicitschemes become orders of magnitude more efficient.