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 53 54%%% Local Variables: 55%%% mode: latex 56%%% TeX-master: "DevNotes" 57%%% End: 58