4.8.2 Numerical results

Next we report the comparison between the solution obtained using the above numerical technique and the solutions submitted by other researchers [79] to the $\mu$-mag website. The time step of the mid-point numerical algorithm is fixed and has been chosen such that $(\vert\gamma\vert M_s)^{-1} \Delta t=2.5$ ps. We observe that the time steps related to the other solutions (see Ref. [79]) are considerably smaller (less then 0.2 ps) to fulfill numerical stability requirements.

Figure: Comparison between solutions of $\mu$-mag standard problem no. 4. Plots of $<m_x>=<M_x>/M_s$ and $<m_y>=<M_y>/M_s$ versus time. The external field is applied at an angle of $170^\circ$ off the $x$-axis.

In Figs. 4.2 and 4.3 plots of $<m_y>$ ($<\cdot>$ means spatial average) as a function of time are reported. We observe that in the first case (Fig. 4.2) there is substantial agreement between the submitted solutions (see Ref. [79]) and for this reason we report, for comparison purposes, only the solution proposed by McMichael and coworkers. In Fig. 4.4 the plots of magnetization vector field when $<m_x>$ crosses zero for the first time are reported. Numerical simulations of the same problem were performed with a smaller cell edge (2.5 nm, number of cells $N=10000$). The results, reported in Fig. 4.5, show that the computed magnetization dynamics does not depend on the mesh size. As far as accuracy is concerned, the self-consistency conditions mentioned in section 4.6 have been verified by means of the computation of the values m$_$av, $\sigma _$m$^2$ and $\hat{\alpha}^{n}$. The result of this computations is reported in Figs. 4.6-4.8.3. One can observe from Fig. 4.6 that the magnetization magnitude is very well preserved, since the mean value m$_$av$\sim1\pm 10^{-16}$ and the variance $\sigma _$m$^2$ is in the order of $10^{-30}$. Moreover, one can see from Fig. 4.8.3 that the relative error $e^n_\alpha=(\hat{\alpha}^{n}-\alpha)/\alpha$ is in the order of $10^{-7}$. As far as conservative dynamics is concerned, the same problem has been simulated with $\alpha=0$. The results, shown in Fig. 4.8.3 show that the reversal of the thin-film occurs, in the sense that the average magnetization exhibits a persistent oscillation around the reversed state. This means that the precessional effects are prevalent with respect to the damping effects. The free energy is conserved as one can see from Fig. 4.8 where exchange, magnetostatic, anisotropy, Zeeman energy and the total free energy are reported as functions of time. Quantitatively speaking, the relative error $e_$g of the free energy with respect to its initial value is in the order of $10^{-8}$ as one can see from Fig. 4.9.