Semi-implicit Structure Preserving Method for The Landau-Lifshitz Equation

TL;DR

This paper introduces a semi-implicit numerical scheme for the Landau-Lifshitz equation that guarantees stability, structure preservation, and norm constraint adherence, addressing theoretical gaps in existing projection methods.

Contribution

The proposed scheme combines BDF, extrapolation, and Crank-Nicolson techniques to ensure stability, structure preservation, and solution uniqueness for the Landau-Lifshitz equation.

Findings

Ensures stable computation and norm preservation.

Guarantees the uniqueness of the numerical solution.

Facilitates theoretical analysis of the normalization step.

Abstract

A critical challenge inherent to the projection method applied to the Landau-Lifshitz equation is the deficiency of rigorous theoretical justifications for the stability of its projection step. To mitigate this limitation, we introduce a semi-implicit numerical scheme, which is formulated on the basis of the first-order Backward Differentiation Formula (BDF) incorporated with one-sided extrapolation and a Crank-Nicolson-type norm-preserving procedure. This proposed scheme exhibits three fundamental characteristics: structure preservation, numerical stability, and first-order accuracy in time. In practical implementations, the scheme not only ensures stable computation and adheres to the norm constraint but also guarantees the uniqueness of the numerical solution, thereby providing substantial facilitation for the theoretical analysis of the normalizing step.