A Second-Order Semi-Implicit Method for the Inertial Landau-Lifshitz-Gilbert Equation  

作　　者：Panchi Li Lei Yang Jin Lan Rui Du Jingrun Chen 

机构地区：[1]School of Mathematical Sciences,Soochow University,Suzhou 215006,China [2]School of Computer Science and Engineering,Macao University of Science and Technology,Macao SAR,China [3]Center for Joint Quantum Studies and Department of Physics,School of Science,Tianjin University,92 Weijin Road,Tianjin 300072,China [4]Mathematical Center for Interdisciplinary Research,Soochow University,Suzhou 215006,China [5]Suzhou Institute for Advanced Research,University of Science and Technology of China,Suzhou,Jiangsu 215123,China [6]School of Mathematical Sciences,University of Science and Technology of China,Hefei,Anhui 230026,China

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2023年第1期182-203,共22页高等学校计算数学学报（英文版）

基　　金：P.Li is supported by the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX202711);L.Yang is supported by the Science and Technology Development Fund,Macao SAR(Grant No.0070/2019/A2);the National Natural Science Foundation of China(NSFC)(Grant No.11701598);J.Lan is supported by NSFC(Grant No.11904260);the Natural Science Foundation of Tianjin(Grant No.20JCQNJC02020);R.Du was supported by NSFC(Grant No.11501399);J.Chen is supported by NSFC(Grant No.11971021).

摘　　要：Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to nanosecond.The Landau-Lifshitz-Gilbert(LLG)equation has been widely used in micromagnetics simulations over decades.Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics,which cannot be captured intrinsically by the LLG equation.Therefore,as a generalization,the inertial LLG(iLLG)equation is proposed to model the ultrafast magnetization dynamics.Mathematically,the LLG equation is a nonlinear system of parabolic type with(possible)degeneracy.However,the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy,and exhibits more complicated structures.It behaves as a hyperbolic system at sub-picosecond timescales,while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond.Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation.In this work,we propose a second-order semiimplicit scheme to solve the iLLG equation.The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme,and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps.The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy.At each time step,the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number.Numerically,the second-order accuracy of the proposed method in both time and space is verified.At sub-picosecond timescales,the inertial effect of ferromagnetics is observed in micromagnetics simulations,in consistency with the hyperbolic property of the iLLG model;at

关 键 词：Inertial Landau-Lifshitz-Gilbert equation semi-implicit scheme second-order accuracy micromagnetics simulations 

分 类 号：O175[理学—数学]