The Landau--Lifshitz--Gilbert equation constrains the magnetization to unit length, while a fully implicit Crank--Nicolson discretization is nonlinear. This paper proposes a linear fractional-step alternative based on componentwise Gauss--Seidel updates of a Crank--Nicolson-type system. Each magnetization component is solved successively through linear elliptic subproblems, avoiding an external projection step. The construction is designed to preserve the discrete length constraint and retain the favorable stability behavior of the implicit formulation. The paper presents consistency and stability arguments and reports first-order temporal and second-order spatial accuracy. One-dimensional convergence tests and three-dimensional magnetization examples assess accuracy, norm preservation, and stability for several initial configurations. The method offers a simple structure-preserving update, but the exposition leaves a fuller convergence and stability theory, finite-element spatial discretization, and realistic micromagnetic applications as future work.
- Proposed a linear Crank--Nicolson-type fractional-step method for the Landau--Lifshitz--Gilbert equation.
- Used implicit Gauss--Seidel component updates to avoid nonlinear coupled solves.
- Preserved the magnetization-length constraint without a separate projection step.
- Presented first-order-in-time and second-order-in-space accuracy and stability analysis.
- Tested accuracy, norm preservation, and stability in one- and three-dimensional examples.