8.3.1 Milshtein scheme

The Milshtein scheme () for the stochastic Landau-Lifshitz equation () is implemented as

$$M_i(t+\Delta t) = M_i(t) +$$

$$\Biggl[-\gamma' ( \mathbf{M} \times (\mathbf{H}_\mathrm{eff} + \mathbf{H}_\mathrm{th}) )$$

$$-\frac{\alpha \gamma'}{M_\mathrm{s}} \mathbf{M} \times ( \mathbf{... ...times (\mathbf{H}_\mathrm{eff} + \mathbf{H}_\mathrm{th}) ) \Biggr]_i \Delta t +$$

$$\gamma'^2 (1+\alpha^2) M_i (H_{\mathrm{th},i} \Delta t)^2 \quad,$$ (8.1)

where the thermal field (, , ) is a Gaussian process

$$H_{\mathrm{th},i}= \eta \sqrt{ 2 \frac{\alpha}{1+\alpha^2} \frac{k_B T}{\mu_0\gamma' v M_\mathrm{s}\Delta t} }$$ (8.2)

with a Gaussian random variable $\eta$. $v$ is the discretization volume of the computational cells. In the finite difference model it is the volume of a cubic computational cell and in the finite element model it is the volume of the finite element.