Highly Efficient Norm Preserving Numerical Schemes for Micromagnetic Energy Minimization

In this paper, two efficient and magnetization norm preserving numerical schemes based on the scalar auxiliary variable (SAV) method are developed for calculating the ground state in micromagnetic structures. The first SAV scheme is based on the original SAV method for the gradient flow model, while the second scheme features an updated scalar auxiliary variable to better align with the associated energy. To address the challenging constraint of pointwise constant magnetization length, an implicit projection method is designed, and verified by both SAV schemes. Both proposed SAV schemes are designed to ensure energy dissipation, requiring only the solution of two linear systems with constant coefficients per time step, which leads to exceptional computational efficiency. The computational efficiency is further enhanced by applying the Discrete Cosine Transform during the solving process. Numerical experiments demonstrate that our SAV schemes outperform commonly used numerical methods in terms of both efficiency and stability.