Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation

Published in Submitted to Applied Numerical Mathematics (2019), 2019

Jingrun Chen, Cheng Wang, Changjian Xie. submitted to Applied Numerical Mathematics. 2019.



The numerical approximation for the Landau-Lifshitz equation, the dynamics of magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting challenges in developing numerical methods. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz equation based on the second-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. Subsequently, we provide a rigorous convergence analysis for the fully discrete numerical solution by introducing two sets of approximated solutions to preceed estimation alternatively, with unconditional stability and second-order accuracy in both time and space, provided that the spatial step-size is the same order as the temporal step-size, which remarkably relax restrictions of temporal step-size compared to the implicit schemes. And also, the unique solvability of the numerical solution without any assumptions for the step size in both time and space is theoretically justified, which turns out to be the first such result for the micromagnetics model. All these theoretical properties are verified by numerical examples in both oneand three- dimensional spaces.

Best Paper Award Honorable Mention