Temporal High-Order Accurate Numerical Scheme for the Landau-Lifshitz-Gilbert Equation

In this paper, a family of temporal high-order accurate numerical schemes for the Landau-Lifshitz-Gilbert (LLG) equation is proposed. The proposed schemes are developed utilizing the Gauss-Legendre quadrature method, enabling them to achieve arbitrary high-order time discretization. Furthermore, the geometrical properties of the LLG equation, such as the preservation of constant magnetization magnitude and the Lyapunov structure, are investigated based on the proposed discrete schemes. It is demonstrated that the magnetization magnitude remains constant with an error of (2p+3) order in time when utilizing a (2p+2)th-order discrete scheme. Additionally, the preservation of the Lyapunov structure is achieved with a second-order error in the temporal step size. Numerical experiments and simulations effectively verify the performance of our proposed algorithm and validate our theoretical analysis.