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8.3.2 Heun scheme

The Heun scheme ([*]) requires the calculation of the predictor

$\displaystyle \bar M_i(t+\Delta t)$ $\textstyle =$ $\displaystyle M_i(t) +$  
    $\displaystyle \Biggl[-\gamma' ( \mathbf{M} \times (\mathbf{H}_\mathrm{eff} + \mathbf{H}_\mathrm{eff}) )$  
    $\displaystyle -\frac{\alpha \gamma'}{M_\mathrm{s}} \mathbf{M} \times ( \mathbf{... ...s (\mathbf{H}_\mathrm{eff}+\mathbf{H}_\mathrm{eff}) ) \Biggr]_i \Delta t \quad.$ (8.3)

Then the magnetization is updated as
$\displaystyle M_i(t+\Delta t)$ $\textstyle =$ $\displaystyle M_i(t) +$  
    $\displaystyle \frac{1}{2} \Biggl[-\gamma' \left( (\mathbf{M} + \bar \mathbf{M}) \times (\mathbf{H}_\mathrm{eff} + \mathbf{H}_\mathrm{th}) \right)$  
    $\displaystyle -\frac{\alpha \gamma'}{M_\mathrm{s}} (\mathbf{M} + \bar \mathbf{M... ...s (\mathbf{H}_\mathrm{eff}+\mathbf{H}_\mathrm{th}) \right) \Biggr]_i \Delta t .$ (8.4)


Werner Scholz 2000-05-16