Length preserving numerical schemes for Landau-Lifshitz equation based on Lagrange multiplier approaches

TL;DR

This paper introduces two novel length-preserving numerical schemes for the Landau-Lifshitz equation using Lagrange multiplier methods, achieving high-order accuracy and efficiency, with additional energy dissipative variants validated through extensive numerical experiments.

Contribution

The paper presents new length-preserving schemes based on two Lagrange multiplier approaches, including higher-order predictor-corrector methods and energy dissipative variants, enhancing stability and computational efficiency.

Findings

Schemes effectively preserve length constraint.

Numerical experiments confirm stability and accuracy.

Performance compares favorably with existing methods.

Abstract

We develop in this paper two classes of length preserving schemes for the Landau-Lifshitz equation based on two different Lagrange multiplier approaches. In the first approach, the Lagrange multiplier $\lambda(\bx,t)$ equals to $|\nabla m(\bx,t)|^2$ at the continuous level, while in the second approach, the Lagrange multiplier $\lambda(\bx,t)$ is introduced to enforce the length constraint at the discrete level and is identically zero at the continuous level. By using a predictor-corrector approach, we construct efficient and robust length preserving higher-order schemes for the Landau-Lifshitz equation, with the computational cost dominated by the predictor step which is simply a semi-implicit scheme. Furthermore, by introducing another space-independent Lagrange multiplier, we construct energy dissipative, in addition to length preserving, schemes for the Landau-Lifshitz equation, at the expense of solving one nonlinear algebraic equation. We present ample numerical experiments to validate the stability and accuracy for the proposed schemes, and also provide a performance comparison with some existing schemes.