4.4 Mid-point LLG discrete dynamics

We now proceed to derive the full discretization of LLG equation by applying the mid-point rule (see Appendix C) to the spatially semi-discretized system of ODEs given by Eq. (4.22). In the following, we will denote the value of physical quantities at the n-th time step with the suffix $n$. The mid-point rule consists in the following time-stepping scheme, written for the for the generic $l-$th cell:

$$\frac{\textbf{{m}}_l^{n+1}\!\!-\!\textbf{{m}}_l^{n}}{\Delta t}= ... ...ha \frac{\textbf{{m}}_l^{n+1}\!\!-\!\textbf{{m}}_l^{n}}{\Delta t} \right] ,$$ (4.32)

where $\Delta t$ is the time step which, for the sake of simplicity, is assumed here constant. Nevertheless, due to the single-step nature of mid-point rule, most considerations in the following can be generalized to nonconstant time steps. Equation (4.32) can be rewritten in terms of mesh vectors as follows

$$\frac{\underline{\textbf{m}}^{n+1}\!\!-\!\underline{\textbf{m}}^{... ...line{\textbf{m}}^{n+1}\!\!-\!\underline{\textbf{m}}^{n}}{\Delta t} \right] .$$ (4.33)

Equation (4.33) defines $\underline{\textbf{m}}^{n+1}$ in terms of $\underline{\textbf{m}}^n$ implicitly. By solving this equation for $\underline{\textbf{m}}^{n+1}$, we generate a map

$$\underline{\textbf{m}}^{n+1} = \Phi(\underline{\textbf{m}}^{n},\Delta t)$$ (4.34)

which describe the LLG discrete dynamics. We will discuss the technique for solving the implicit equation (4.33) in the following section. In this section, instead, we will focus on the properties of the map (4.34) defined implicitly by Eq. (4.32) or equivalently by Eq. (4.33). As a preliminary consideration, we notice that, in most LLG numerical investigations, numerical discretization of LLG equation is carried out starting from the Landau-Lifshitz form (4.2) which has the advantage of explicitly expressing the time derivative of $\textbf{{m}}$. Conversely, in the approach we propose, we directly discretized the original LLG equation in which the time derivative is implicitly contained. In fact, since the mid-point scheme is already implicit, the implicit nature of LLG equation does not introduce any further complication, but rather it drastically simplifies the treatment and the algorithm.