Direct solution of Landau-Lifshitz-Gilbert equation for domain walls in thin Permalloy films

The Gilbert equation expressed in conventional polar angle notation is solved for one-dimensional Neel and Bloch walls in a thin Permalloy film, using different numerical methods to compare the extent of stability. The methods examined include the Euler, so-called modified Dufort-Frankel, and backward Euler methods. The backward Euler method is found to be stable under arbitrary magnitude of time difference for the case of Neel walls if the contribution of the magnetization of the nearest neighboring cells to the demagnetizing field is treated implicitly. The method is also stable in the case of Bloch walls, though the stability limit for the time step is only 10 times as large as that of the Euler method. The Dufort-Frankel method is quite unstable. Two-dimensional computation is feasible using the backward Euler method. The principle and the results of the calculations are given