Changjian Xie

Changjian Xie

Research Fellow, Department of Mathematics, NUS

Research Talks

2023 $13^{th} CSCM$

July 15, 2023

Talk, CSCM, Nanjing, China

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 talk, I will introduce 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 timestep 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 one- and three- dimensional spaces.

2019 CSIAM Contributed paper

September 21, 2019

Talk, CSIAM, foshan, China

In ferromagnets, the intrinsic magnetic order, known as magnetization, makes these materials ideal for information storage and manipulation. From the modeling perspective, magnetization dynamics is described by the Landau-Lifshitz equation with pointwise length constraint. From the numerical perspective, typically, second-order in time schemes are either explicit with strong stability restriction on the stepsize due to the high nonlinearity or implicit with a nonlinear system of equations to be solved at each step. In the talk, we will introduce several second-order semi-implicit schemes based on the second-order backward-differentiation-formula and the onesided interpolation from former steps with a projection step. For these schemes, we are able to prove the uniqueness of the numerical solution to the linear system of equations at each step. For one of these schemes, we then prove its second-order accuracy under the mild condition that the stepsize in time is proportional to the gridsize in space. Examples in 1D and 3D are given to verify the analysis results. A benchmark problem from National Institute of Standards and Technology is also tested to verify the applicability of these schemes.

2018 JSIAM Graduate International Symposium

December 08, 2018

Talk, JSIAM, Nanjing, China

In this talk, we will introduce a second-order semi-implicit projection methods for Landau-Lifschitz model in micromagnetics and give some main theoretical results, which are unconditional unique solvability and optimal rate convergence analysis. Here, we will present some numerical examples to verify the second-order convergence rate both in time and space.