Semi-implicit Structure Preserving Method for The Landau-Lifshitz Equation

A critical challenge inherent to the projection method applied to the Landau-Lifshitz equation is the deficiency of rigorous theoretical justifications for the stability of its projection step. To mitigate this limitation, we introduce a semi-implicit numerical scheme, which is formulated on the basis of the first-order Backward Differentiation Formula (BDF) incorporated with one-sided extrapolation and a Crank-Nicolson-type norm-preserving procedure. This proposed scheme exhibits three fundamental characteristics: structure preservation, numerical stability, and first-order accuracy in time. In practical implementations, the scheme not only ensures stable computation and adheres to the norm constraint but also guarantees the uniqueness of the numerical solution, thereby providing substantial facilitation for the theoretical analysis of the normalizing step.