Analysis And Applications Of Numerical Methods And Modeling For The Landau-Lifshitz Equation In Micromagnetic Materials

Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz(LL)equation,which is nonlinear,nonlocal,and has a non-convex const.raint,posing challenges in developing numerical methods.In this paper,we propose two second-order semi-implicit projection methods based on the second-order backward differentiation formula and the second-order interpolation formula using the information at first two temporal steps.We prove the unconditional solvability and verify the secondorder accuracy in both one-and three-dimensions for these methods.We compare the efficiency of these methods with that of other two popular methods which indicates that semi-implicit method is the most efficient one given the same accuracy of the error.In order to get the robustness,we employ this method to the benchmark problem dedicated to magnetic materials.Based on the original semi-implicit scheme,we propose a fully discrete semiimplicit approach,which is constructed to solve LL equation by usage of the second-order backward differentiation formula and one-sided extrapolation(using previous time-step numerical values).A projection step is further used to preserve the length of the magnetization.We conduct the rigorous convergence analysis by introducing two sets of approximated solutions for the fully discrete numerical solutions,where one set of solution solves the LL equation and the other is projected onto the unit sphere.The second-order accuracy in both time and space can be obtained as long as the temporal step and spatial grid-size stay at the same level.Also,the unique solvability is proved using the monotone analysis without any assumption for temporal and spatial step-size.On the other hand,the numerical examples in both 1D and 3D are consistent with the theoretical part.We propose two kinds of Gauss-Seidel projection methods with unconditional stability by solving heat equations 5 times and 3 times,respectively,which compared to that of the original version with solving heat equation 7 times at each time step,to improve the efficiency.We propose a second-order linear method for the LL equation with large damping parameters with treating linear diffusion term implicitly and two nonlinear terms to be explicit,then acting on a projection onto the unit sphere point-wisely.The second-order accuracy in both time and space is verified by 1D and 3D numerical examples.We obtain the error estimates of linear second-order method and apply this method to the domain wall motion of the ferromagnetic thin film.In summary:We propose the second-order semi-implicit projection methods with unconditional stability and prove the optimal rate convergence and unique solvability with the reduction of the constraint on the temporal step for the LL equation;·We propose two Gauss-Seidel projection methods with unconditional stability and the improvement of efficiency for the LL equation;·In terms of the LL equation with dissipation dominating,we construct the linear second-order method for the dissipation-dominated LL equation with solving the linear system of equations of constant coefficient and the extremely improvement of efficiency and proving the convergence of the method rigorously.