1
2\subsection{Time--discretization of the linear case~(\ref{first-DS3}) }
3Let us now proceed with the time discretization of~(\ref{eq:toto1}) with FirstOrderLinearR~(\ref{first-DS3})  by a fully implicit scheme :
4\begin{equation}
5  \begin{array}{l}
6    \label{eq:toto1-DS3}
7     M x^{\alpha+1}_{k+1} = M x_{k} +h\theta A x^{\alpha+1}_{k+1}+h(1-\theta) A x_k + h \gamma r^{\alpha+1}_{k+1}+ h(1-\gamma)r(t_k)  +hb\\[2mm]
8     y^{\alpha+1}_{k+1} =  C x^{\alpha+1}_{k+1} + D \lambda ^{\alpha+1}_{k+1} +Fz +e\\[2mm]
9     r^{\alpha+1}_{k+1} = B \lambda ^{\alpha+1}_{k+1} \\[2mm]
10  \end{array}
11\end{equation}
12
13\[R_{\free} = M(x^{\alpha}_{k+1} - x_{k}) -h\theta A x^{\alpha}_{k+1} - h(1-\theta) A x_k -hb_{k+1} \]
14\[R_{\free} = W(x^{\alpha}_{k+1} - x_{k}) -h A x_{k} -hb_{k+1} \]
15
16\paragraph{Resulting Newton step (only one step)}
17For the sake of simplicity, let us assume that $\gamma =1$
18\begin{equation}
19  \begin{array}{l}
20     (M -h\theta A)x^{\alpha+1}_{k+1} = M x_{k} +h(1-\theta) A x_k + hr^{\alpha+1}_{k+1} + hb\\[2mm]
21     y^{\alpha+1}_{k+1} =  C x^{\alpha+1}_{k+1} + D \lambda ^{\alpha+1}_{k+1} +Fz + e \\[2mm]
22     r^{\alpha+1}_{k+1} = B \lambda ^{\alpha+1}_{k+1}\\[2mm]
23  \end{array}
24\end{equation}
25that lead to with: $ (M -h\theta A) = W$
26\begin{equation}
27  \begin{array}{l}
28     x^{\alpha+1}_{k+1} = W^{-1}(M x_{k} +h(1-\theta) A x_k + r^{\alpha+1}_{k+1} +hb) = x\free + W^{-1}(r^{\alpha+1}_{k+1})\\[2mm]
29     y^{\alpha+1}_{k+1} =  ( D+hCW^{-1}B) \lambda ^{\alpha+1}_{k+1} +Fz + CW^{-1}(M
30     x_k+h(1-\theta)Ax_k + hb) +e \\[2mm]
31  \end{array}
32\end{equation}
33with $x_{\free} = x^{\alpha}_{k+1} + W^{-1}(-R_{\free})= x^{\alpha}_{k+1} - W^{-1}(W(x^{\alpha}_{k+1}
34- x_k) -hAx_k-hb_{k+1} )= W^{-1}(Mx_k +h(1-\theta)Ax_k +h b_{k+1})$
35\begin{equation}
36  \begin{array}{l}
37     y^{\alpha+1}_{k+1} =  ( D+hCW^{-1}B) \lambda ^{\alpha+1}_{k+1} +Fz + Cx_{\free}+e\\[2mm]
38     r^{\alpha+1}_{k+1} = B \lambda ^{\alpha+1}_{k+1}\\[2mm]
39  \end{array}
40\end{equation}
41
42\paragraph{Coherence with previous formulation}
43\[y_p = y^{\alpha}_{k+1} -\mathcal R^{\alpha}_{yk+1} + C^{\alpha}_{k+1}(x_p -x^{\alpha}_{k+1}) -
44D^{\alpha}_{k+1} \lambda^{\alpha}_{k+1} \]
45\[y_p = Cx_k + D \lambda _k  + C(\tilde x_{\free}) -D \lambda_k +Fz + e\]
46\[y_p = Cx_k   + C(\tilde x_{\free})  +Fz + e\]
47\[y_p = Cx_k   + C(\tilde x_{\free})  +Fz + e\]
48\[y_p = C(x_{\free})  +Fz + e\]
49
50%In the case of the system~(\ref{eq:deux}) with a affine function $f$ or $\theta =0$, the the MLCP matrix $W$ can be computed before the beginning of the time loop saving a lot of computing effort.  In the case of the system (\ref{eq:trois}) with $\theta=\gamma=0$, the MLCP matrix $W$ can be computed before the beginning of the Newton loop.
51\clearpage
52
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