Optimal Error Analysis of Euler and Crank--Nicolson Projection Finite Difference Schemes for Landau--Lifshitz Equation

Abstract

The Landau--Lifshitz equation has been widely used to describe the dynamics of magnetization in a ferromagnetic material, which is highly nonlinear with the nonconvex constraint $|{m}|=1$. A crucial issue in designing efficient numerical schemes is to preserve this constraint in the discrete level. A simple and frequently used one is the projection method, which projects the numerical solution onto a unit sphere at each time step. The method has been used in many areas in the past several decades, while analysis has not been explored. In this paper, we present optimal error analysis of a backward Euler and a Crank--Nicolson semi-implicit projection finite difference scheme for the Landau--Lifshitz equation. The analysis is based on new and precise estimates of the difference between the errors of projected and unprojected solutions in both $L^2$ and $H^1$ norms. Some numerical experiments are provided to confirm our theoretical results.