The transient equation for fix classes is :

$\frac{\partial C_k}{\partial t}(x,t)=E_k(x,t)$

where :

• $C_k$ is the linear mass density ;
• $E_k$ is the derivative of $C_k$ with respect to time.

The time discretization for $ j + 1 = t + \Delta t $ of this equation is
made by a semi-implicit Euler method :

$C_k^{j+1}-C_k^{j}-\frac{\Delta t}{2}\left(E_k^{j}+E_k^{j+1}\right)=0$

Resolution is then performed in the same Newton’s method iteration as drifting quality classes.

The formula for iteration i is:

$C_k^{j+1,i}=C_k^{j+1,i-1}-\frac{f\left(C_k^{j+1,i-1}\right)}{f^\prime\left(C_k^{j+1,i-1}\right)}=C_k^{j+1,i-1}-\frac{b_{i}}{a_{i}}$

where :

• $b_{i}=C_k^{j+1,i-1}-C_k^{j}-\frac{\Delta t}{2}\left(E_k^{j}+E_k^{j+1,i}\right)$
• $a_{i}=1-\frac{\Delta t}{2}E_k^{\prime j+1,i}$

$E_k^{j + 1,i}$ and $E_k^{\prime j + 1,i}$ are fonctions of $C_k^{j + 1,i-1}$ and drifting classes concentrations $C_{k}^{\prime j + 1,i-1}$.