Advantages of a semi-implicit scheme over a fully implicit scheme for Landau-Lifshitz-Gilbert equation

Abstract

Magnetization dynamics in magnetic materials is modeled by the Landau-Lifshitz-Gilbert (LLG) equation, which is a nonlinear system of partial differential equations. In the LLG equation, the length of magnetization is conserved and the system energy is dissipative. Implicit and semi-implicit schemes have often been used in micromagnetics simulations due to their unconditional numerical stability. In more details, implicit schemes preserve the properties of the LLG equation, but solve a nonlinear system of equations per time step. In contrast, semi-implicit schemes only solve a linear system of equations, while additional operations are needed to preserve the length of magnetization. It still remains unclear which one shall be used if both implicit and semi-implicit schemes are available. In this work, using the implicit Crank-Nicolson (ICN) scheme as a benchmark, we propose to make this implicit scheme semi-implicit. Stability and convergence analysis, and numerical performance in terms of accuracy and efficiency are systematically studied. Based on these results, we conclude that a semi-implicit scheme is superior to its implicit analog both theoretically and numerically, and we recommend the semi-implicit scheme in micromagnetics simulations if both methods are available.