Transiant calculation for fix classes

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 \(C_{k}^{\prime j + 1,i-1}\).