4.4.1.1 Point-wise magnitude conservation

The first important property of the discrete dynamics can be readily obtained from Eq. (4.32) by scalar multiplying both sides of the equation by $(\textbf{{m}}_l^{n+1}+\textbf{{m}}_l^{n})$. This leads immediately to

$$({\textbf{{m}}_l^{n+1}\!\!-\!\textbf{{m}}_l^{n}})\cdot (\textbf{... ...}}_l^{n+1}\vert^2-\vert\textbf{{m}}_l^{n}\vert^2=0 , l=1\ldots N ,$$ (4.35)

which means that the magnitude of the magnetization vector remains constant in each cell. Thus, the mid-point rule preserves exactly the magnitude preservation property (4.27), regardless of the time step.