High-Order Energy-Decreasing IMEX Runge-Kutta Generalized Scalar Auxiliary Variable Methods for Landau-Lifshitz Equation

Abstract

The Landau-Lifshitz equation is a fundamental model in micromagnetics, whose nonlinear geometric structure arising from the pointwise unit-length constraint makes the construction of stable and accurate numerical schemes particularly challenging. In this work, we develop and analyze a class of extrapolated and linearized implicit-explicit Runge-Kutta (IMEX-RK) schemes within the generalized scalar auxiliary variable (GSAV) framework for the Landau-Lifshitz equation. The proposed methods require only the solution of linear systems with constant coefficients, preserve the unit-length constraint through a projection step, and exhibit a modified energy dissipation law without imposing restrictions on the damping parameter. A modified energy dissipation property is established, and rigorous error estimates are derived theoretically for second- and higher-order schemes. Numerical experiments for the second- and third-order IMEX RK-GSAV methods confirm the theoretical convergence results and demonstrate the effectiveness of the proposed approach.