  A practical and efficient scheme for the higher order integration of the
Landau-Lifschitz-Gilbert (LLG) equation is presented. The method is based on
extrapolation of the two-step explicit midpoint rule and incorporates adaptive
time step and order selection. We make use of a piecewise time-linear stray
field approximation to reduce the necessary work per time step. The
approximation to the interpolated operator is embedded into the extrapolation
process to keep in step with the hierarchic order structure of the scheme. We
verify the approach by means of numerical experiments on a standardized NIST
problem and compare with a higher order embedded Runge-Kutta formula. The
efficiency of the presented approach increases when the stray field computation
takes a larger portion of the costs for the effective field evaluation.
