Practical, structure-preserving methods for integrating classical Heisenberg spin systems are discussed. Two new integrators are derived and compared, including (1) a symmetric energy and spin-length preserving integrator based on a Red-Black splitting of the spin sites combined with a staggered timestepping scheme and (2) a (Lie-Poisson) symplectic integrator based on Hamiltonian splitting. The methods are applied to both 1D and 2D lattice models and are compared with the commonly used explicit Runge–Kutta, projected Runge–Kutta, and implicit midpoint schemes on the bases of accuracy, conservation of invariants and computational expense. It is shown that while any of the symmetry-preserving schemes improves the integration of the dynamics of solitons or vortex pairs compared to Runge-Kutta or projected Runge Kutta methods, the staggered Red-Black scheme is far more efficient than the other alternatives.
Academic Press
Journal of Computational Physics

Frank, J.E, Huang, W, & Leimkuhler, B.J. (1997). Geometric Integrators for Classical Spin Systems. Journal of Computational Physics, 133, 160–172. doi:10.1006/jcph.1997.5672