A Comparison of Three Time-stepping Methods for the LLG Equation in Dynamic Micromagnetics

• Main Authors: Wredh, Simon, Kroner, Anton, Berg, Tomas
• Format: Others
• Language: English
• Published: Uppsala universitet, Avdelningen för beräkningsvetenskap 2017
• Subjects: LLG equation; Micromagnetism; Implicit midpoint; Midpoint extrapolation; Projection method; Other Computer and Information Science; Annan data- och informationsvetenskap; Computational Mathematics; Beräkningsmatematik
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-323537

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.