Title: Solutions to the Landau-Lifshitz-Gilbert equation in the frequency space: Discretization schemes for the dynamic-matrix approach

Abstract: The dynamic-matrix method addresses the Landau-Lifshitz-Gilbert (LLG) equation in the frequency domain by transforming it into an eigenproblem. Subsequent numerical solutions are derived from the eigenvalues and eigenvectors of the dynamic-matrix. In this work we explore discretization methods needed to obtain a numerical representation of the dynamic-operator, a foundational counterpart of the dynamic-matrix. Our approach opens a new set of linear algebra tools for the dynamic-matrix method and expose the approximations and limitations intrinsic to it. We present some application examples, including a technique to obtain the dynamical matrix directly from the magnetic free energy function of an ensemble of macrospins, and an algorithmic method to calculate numerical micromagnetic kernels, including plane wave kernels. Additionally, we also show how to exploit symmetries and reduce the numerical size of micromagnetic dynamic-matrix problems by a change of basis. This work contributes to the understanding of the current magnetization dynamics methods, and could help the development and formulations of novel analytical and numerical methods for solving the LLG equation within the frequency domain.