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

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 $\hat{I}\pm$ (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.

This work is essentially different with arXiv:2510.25172 since this numerical method with linear system of variable coefficients is proposed, that is highly depends on the numerical solver. The analysis for the gyromagnetic term is different. The constraint for the main result is different