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

TL;DR

This paper introduces and rigorously analyzes a third-order semi-implicit projection method for solving the nonlinear Landau-Lifshitz-Gilbert equation, achieving high accuracy and stability with proven convergence.

Contribution

It presents a novel third-order semi-implicit scheme with convergence proof and stability analysis for the Landau-Lifshitz-Gilbert equation, including a projection step to maintain magnetization length.

Findings

Third-order accuracy in time and fourth-order in space achieved.

Unique solvability without step-size restrictions proven.

Numerical examples confirm theoretical convergence in 1D and 3D.

Abstract

The convergence analysis of a third-order scheme for the highly nonlinear Landau-Lifshitz-Gilbert equation with a non-convex constraint is considered. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz-Gilbert equation based on the third-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. We provide a rigorous convergence analysis for the fully discrete numerical solution by the introduction of two sets of approximated solutions where one set of solutions solves the Landau-Lifshitz-Gilbert equation and the other is projected onto the unit sphere. Third-order accuracy in time and fourth order accuracy in space is obtained provided that the spatial step-size is the same order as the temporal step-size and slightly large damping parameter $\alpha$ (greater than $\sqrt{2}/2$). And also, the unique solvability of the numerical solution without any assumption for the step-size in both time and space is theoretically justified, using a monotonicity analysis. All these theoretical results are guaranteed by numerical examples in both 1D and 3D spaces.