Discretization-Induced Stiffness in Micromagnetic Simulations

Abstract

In the numerical integration of the Landau-Lifshitz-Gilbert (LLG) equation, stiffness (stability restrictions on the time step size for explicit methods) is known to be a problem in some cases. We examine the relationship between stiffness and spatial discretization size for the LLG with exchange and magnetostatic effective fields. A maximum stable time step is found for the reversal of a single-domain spherical nanoparticle with a variety of magnetic parameters and numerical methods. From the lack of stiffness when solving a physically equivalent ODE problem, we conclude that any stability restrictions in the partial differential equation case arise from the spatial discretization rather than the underlying physics. We find that the discretization-induced stiffness increases as the mesh is refined and the damping parameter is decreased. In addition, we find that the use of the FEM/BEM method for magnetostatic calculations increases the stiffness. Finally, we observe that the use of explicit magnetostatic calculations within an otherwise implicit time integration scheme (i.e. a semi-implicit scheme) does not cause stability issues.