4.6 Accuracy tests for LLG discrete dynamics

We have shown that mid-point rule time-stepping preserves magnetization magnitude conservation and Lyapunov structure of LLG equation. Nevertheless, since the time-stepping equations (4.33) are solved through an iterative procedure, the properties of mid-point rule we have demonstrated in Sec. 4.4 are fulfilled only within a certain accuracy related to the tolerance which we imposed on the quasi-NR technique. In this respect, it is important to test the accuracy of the preservation of LLG properties during the computation. To this end, as far as magnetization magnitude conservation, we will check the accuracy with the following quantities:

$$_ =\frac{1}{N}\sum_{l=1}^N \vert\textbf{{m}}_l\vert \quad,\quad \sigma_ ^2=\frac{1}{N}\sum_{l=1}^N (\vert\textbf{{m}}_l\vert- _ )^2 \quad,$$ (4.51)

which are, respectively, mean value and variance of the magnetization magnitude over the cells of the mesh. As far as the energy balance equation is concerned, we use the self-consistency criterion proposed by Albuquerque and coworkers [94]. This criterion is based on Eq. (4.28) rewritten in the following form

$$\alpha= \left( \frac{d }{d t}\underline{{\text{g}}}(t) \right) \Big/ \left\vert \frac{d \underline{\textbf{m}}}{d t}\right\vert^2 .$$ (4.52)

which is valid for constant applied field. It is quite natural, to test the preservation of energy dynamics in numerical computation, to compute

$$\hat{\alpha}^{n}=- \left( \frac{ \underline{{\text{g}}}(\underlin... ...\textbf{m}}^{n+1}\!-\!\underline{\textbf{m}}^{n}}{\Delta t}\right\vert^2 \quad.$$ (4.53)

at each time step, and compare the value of the sequence $\hat{\alpha}^{n}$ with the constant $\alpha$. In this respect, we observe that if we could exactly invert the nonlinear system of equations (4.43), then the sequence $\hat{\alpha}^{n}$ would be constant coincident with $\alpha$. This can be immediately derived from the property (4.40) of the mid-point rule. However, since we determine the new time step by an iterative procedure, the sequence will be in fact nonconstant and it will usually exhibit an oscillatory behavior. It has been shown in Ref. [94] that numerical instabilities can be detected from the observation of the behavior of the values $\hat{\alpha}^{n}$. For the case of conservative dynamics, the discretized energy is conserved according to Eq. (4.29) for $\alpha=0$:

$$\underline{{\text{g}}}(t)=\underline{{\text{g}}}(t_0) \quad\forall  t \ge t_0\quad.$$ (4.54)

With mid-point rule time discretization, this property becomes (see Eq. (4.40)):

$$\underline{{\text{g}}}(\underline{\textbf{m}}^{n+1};\underline{\t... ...derline{{\text{g}}}(\underline{\textbf{m}}^{n};\underline{\textbf{h}}_a) \quad,$$ (4.55)

which holds regardless of the time step. One can test the accuracy of the scheme by recording the deviation of the total energy from its initial value. Again, one cannot expect that this property will be exactly fulfilled as far as we solve the time stepping algorithm with an iterative procedure. In this respect, we will verify a-posteriori that the energy conservation is guaranteed with sufficient precision by computing the relative error $e_$g of $\underline{{\text{g}}}(\underline{\textbf{m}}^n;\underline{\textbf{h}}_a)$ with respect to the initial energy $\underline{{\text{g}}}(\underline{\textbf{m}}^0;\underline{\textbf{h}}_a)$:

$$e_ ^n=\frac{\underline{{\text{g}}}(\underline{\textbf{m}}^0;\underli... ...ine{{\text{g}}}(\underline{\textbf{m}}^0;\underline{\textbf{h}}_a)} \quad ,$$ (4.56)

and checking that the sequence $e_$g$^n$ remains within prescribed tolerance.