A mixed mid-point Runge-Kutta like scheme for the integration of Landau-Lifshitz equation

In this paper, a novel numerical time-stepping scheme for the solution of the dynamic Landau-Lifshitz equation of micromagnetics is introduced. The equation is a non-linear problem, and is traditionally solved using semi-discretization techniques. However, these techniques corrupt some intrinsic properties of the Landau-Lifshitz (LL) dynamics such as the conservation of magnetization magnitude in time. The proposed approach uses a mixed mid-point Runge-Kutta like algorithm which satisfies LL dynamic properties for time-steps up to 1.25 ps. The accuracy of the proposed scheme has been tested and verified using μmag Standard Problem 4 against popular micromagnetic simulation software for different time-steps and cell sizes.