4.4.1.2 Energy balance and discrete Lyapunov structure

Next, we analyze the energy balance properties of the discrete dynamics. The derivation of the main equation can be carried out by scalar multiplying both sides of Eq. (4.33) by the quantity

$$\left[ \underline{\textbf{h}}_\text{eff}\left(\frac{\underline{\... ...nderline{\textbf{m}}^{n+1}\!\!-\!\underline{\textbf{m}}^{n}}{\Delta t} \right].$$ (4.36)

It is clear that, due to the antisymmetry of the matrix ${\underline{\Lambda}}(\underline{\textbf{m}})$ (which in turn is related to the antisymmetry of its $3\times3$ diagonal blocks), the dot multiplication of (4.36) and the right-hand-side of Eq. (4.33) gives zero. As far as the left-hand side is concerned, by using the expression of the discretized effective field (4.20), one obtains:

$$\frac{\underline{\textbf{m}}^{n+1}\!\!-\!\underline{\textbf{m}}^{... ...f{m}}^{n+1}\!\!-\!\underline{\textbf{m}}^{n}}{\Delta t} \right]=\mathbf{0} .$$ (4.37)

Then, by using the symmetry of the matrix ${\underline{\textrm{C}}}$ and the following mid-point approximation for the mid-point value of the applied field:

$$\underline{\textbf{h}}_a\left(t^{n}\!\!+\!\!\frac{\Delta t}{2}\ri... ...rline{\textbf{h}}_a}{dt^2} \right\vert _{t^{n}\!\!+\!\!\frac{\Delta t}{2}} ,$$ (4.38)

one can readily derive the following equation

$$\begin{split} \frac{ \underline{{\text{g}}}(\underline{\text... ...t\vert _{t^{n}\!\!+\!\!\frac{\Delta t}{2}} \quad. \end{split}$$ (4.39)

Notice that since $(\underline{\textbf{m}}^{n+1}\!-\!\underline{\textbf{m}}^{n})$ is of the order $\mathcal{O}(\Delta t)$ then the last term at the right hand side of the equation is $\mathcal{O}(\Delta t^2)$ and thus, as we expected, up to this second order term, the discrete dynamics reproduce the energy balance for semi-discretized equation (4.28). In fact, very interesting properties of the scheme can be inferred from Eq. (4.39). First we notice that, if the applied field $\underline{\textbf{h}}_a$ is piece-wise linear with respect to time, in each time interval $[t^{n},t^{n+1}]$, then the last term in right hand side vanishes and the energy balance is exactly reproduced in its mid-point time discretized version. In addition, in the case of constant applied field, the last two terms in Eq. (4.39) vanish and the energy balance reduces to the following form

$$\frac{ \underline{{\text{g}}}(\underline{\textbf{m}}^{n+1};\under... ...textbf{m}}^{n+1}\!-\!\underline{\textbf{m}}^{n}}{\Delta t}\right\vert^2 \quad.$$ (4.40)

Equation (4.40) has very important consequences. First, independently of the time step, the discretized energy $\underline{{\text{g}}}(\underline{\textbf{m}}^{n};\underline{\textbf{h}}_a)$ is decreasing. This confirms that the mid-point rule is an unconditionally stable algorithm which reproduce the relaxation behavior in LLG discrete dynamics for any choice of the time step. Notice also that the rate of variation of energy in the discrete dynamics is coherent the mid-point version of Eq. (4.29). Second, for $\alpha=0$ the energy is exactly preserved regardless of the time steps. These two properties confirms the unconditional stability of mid-point rule, but more importantly they indicate that, in the short time scale, the mid-point rule will tend to reproduce correctly the most important part in the LLG dynamics, i.e. the precessional magnetization motion.