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\title{{\Huge {\sl Alamode}:} A LAyered MOdel Development Environment for
simulation of impurity diffusion in semiconductors}
\author{D.W. Yergeau and R.W. Dutton\\ 
Stanford University}


\newcommand {\ALAMODE}{{\sl Alamode}}

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\samepage \begin{table}[htf] 
\begin{center}
\begin{tabular}{|c|l|} \hline
Material number & {\sl Alamode}\ name \\  \hline
0 & Gas \\ 
1 & Oxide \\  
2 & Nitride \\ 
3 & Silicon \\ 
4 & PolySilicon \\ 
5 & OxiNitride \\ 
6 & Aluminum \\ 
7 & Photoresist \\ 
8 & GaAs \\  \hline
\end{tabular}

\label{table:suprem-materials}
\end{center}
\end{table}
}

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\samepage \begin{table}[htf] 
\begin{center}
\begin{tabular}{|c|l||c|l|} \hline
Impurity & {\sl Alamode}\ name & Impurity & {\sl Alamode}\ name \\  \hline
0 & Vacancy & 25 & Gallium \\ 
1 & Interstitial & 26 & DeviceCKT \\ 
2 & Arsenic & 27 & Impurity27 \\ 
3 & Phosphorus & 28 & Impurity28 \\ 
4 & Antimony & 29 & Impurity29 \\ 
5 & Boron & 30 & Impurity30 \\ 
6 & Electrons & 31 & Beryllium \\ 
7 & Holes & 32 & ElectricallyActiveBeryllium \\ 
8 & Xvelocity & 33 & Magnesium \\ 
9 & Yvelocity & 34 & ElectricallyActiveMagnesium \\ 
10 & Dry02 & 35 & Selenium \\ 
11 & Wet02 & 36 & ElectricallyActiveSelenium \\ 
12 & Trap & 37 & Silicon \\ 
13 & Gold & 38 & ElectricallyActiveSilicon \\ 
14 & Potential & 39 & Tin \\ 
15 & Xstress & 40 & ElectricallyActiveTin \\ 
16 & Ystress & 41 & Germanium \\ 
17 & MixedCompOfStress & 42 & ElectricallyActiveGermanium \\ 
18 & Cesium & 43 & Zinc \\ 
19 & ChangeInInterfaceArea & 44 & ElectricallyActiveZinc \\ 
20 & ElectricallyActiveArsenic & 45 & Carbon \\ 
21 & ElectricallyActivePhosphorus & 46 & ElectricallyActiveCarbon \\ 
22 & ElectricallyActiveAntimony & 47 & Generic \\ 
23 & ElectricallyActiveBoron & 48 & ElectricallyActiveGeneric \\ 
24 & PolysiliconGrainSize & 49 & DummyImpurity \\  \hline
\end{tabular}

\label{table:suprem-impurities} 
\end{center}
\end{table}
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Gamma_g$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3254: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Gamma_h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3256: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C\left({\vec{x}}\right) = g\left({\vec{x}}\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3260: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${w}\left({\vec{x}}\right) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3264: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\int_\Omega {w} C_{,t} {d \Omega} =
\int_\Omega {w} {\nabla\cdot}{\left(D{\nabla}{C}\right)} {d \Omega}
&=& \int_\Omega {w} \left(D C_{,i}\right)_{,i} {d \Omega} \nonumber \\ 
&=& -\int_\Omega \w_{,i} D C_{,i} {d \Omega} + 
    \int_\Omega \left({w} D C_{,i}\right)_{,i} {d \Omega} \\ 
&=& -\int_\Omega \w_{,i} D C_{,i} {d \Omega} +
    \int_{\Gamma} n_i {w} D C_{,i} {d \Gamma} \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_{,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3268: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C^0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3270: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\cal{S}} &=& \left\{ C\left({\vec{x}},t\right) \in C^0\left(\Omega \times
]0,T[\right) |\; 
C\left({\vec{x}},t\right) = g\left({\vec{x}},t\right) \mbox{ on }
\Gamma_g\right\}  \\ 
{\cal{V}} &=& \left\{ {w}\left({\vec{x}}\right) \in C^0\left(\Omega\right) |\;
{w}\left({\vec{x}}\right) = 0 \mbox{ on } \Gamma_g\right\}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C\left({\vec{x}},t\right) \in {\cal{S}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3272: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${w}\left({\vec{x}}\right)
\in {\cal{V}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3274: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int_\Omega {w} C_{,t} {d \Omega} = -\int_\Omega \w_{,i} D C_{,i}
{d \Omega} + \int_{\Gamma_h} {w} h\left({\vec{x}},t,C\right) {d \Gamma}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\epsilon}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3276: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\eta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3278: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bar{\Omega} = \cup_{e\in{\epsilon}} \Omega^e$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3280: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bar{\Omega}^a\cap\bar{\Omega}^b = \emptyset$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3282: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left\{a,b\in{\epsilon}
\mbox{ and } a\neq b\right\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3284: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C\left(
{\vec{x}},t\right)\in{{\cal{S}} }^h \subset {\cal{S}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3286: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${w}\left({\vec{x}}\right)\in{{\cal{V}} }^h
\subset {\cal{V}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3288: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${{\cal{S}} }^h \subset {\cal{S}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3290: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${{\cal{S}} }^h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3292: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\cal{S}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3294: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C^h\left({\vec{x}},t\right) \in {{\cal{S}} }^h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3296: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\w^h\left({\vec{x}}\right)
\in {{\cal{V}} }^h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3298: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int_\Omega \w^h C_{,t}^h {d \Omega} = -\int_\Omega \w^h_{,i}
D\left(C^h\right ) C_{,i}^h
{d \Omega} + \int_{\Gamma_h} \w^h h\left({\vec{x}},t,C^h\right) {d \Gamma}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N_A\left(\X_A\right) = 1, A\in{\eta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3300: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N_A\left(\X_B\right) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3302: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A\neq B$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3304: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\w^h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3308: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${{\cal{V}} }^h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3310: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\w^h\left({\vec{x}}\right) = \sum_A w_A N_A\left({\vec{x}}\right)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w_A$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3312: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w_A = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3316: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A\in\nodes_{g}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3318: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}C^h\left({\vec{x}},t\right) = \sum_A c_A\left(t\right) N_A\left({\vec{x}}\right)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_A\left(t\right) = g\left(\X_A,t\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3320: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_{,t}^h$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3322: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}C_{,t}^h\left({\vec{x}},t\right) = \sum_A \dot{c}_A\left(t\right)
N_A\left({\vec{x}}\right) 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\int_\Omega \sum_A w_A N_A 
\sum_B \dot{c}_B N_B 
{d \Omega} =
-\int_\Omega \sum_A w_A {N_A}_{,i} D\left(C^h\right) \sum_B c_B
{N_B}_{,i} {d \Omega} + \int_{\Gamma_h} \sum_A w_A N_A
h\left({\vec{x}},t,C^h\right) {d \Gamma}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\sum_A w_A \int_\Omega N_A \sum_B \dot{c}_B N_B {d \Omega} =
-\sum_A w_A \int_\Omega {N_A}_{,i} D\left(C^h\right) \sum_B c_B
{N_B}_{,i} {d \Omega} + \sum_A w_A \int_{\Gamma_h} N_A
h\left({\vec{x}},t,C^h\right){d \Gamma}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\sum_A w_A \underbrace{\left(\int_\Omega N_A \sum_B \dot{c}_B N_B
{d \Omega} - R_A \right)}_{G_A} = 0
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}R_A = - \int_\Omega {N_A}_{,i} D\left(C^h\right) \sum_B c_B
{N_B}_{,i} {d \Omega} + \int_{\Gamma_h} N_A h\left({\vec{x}},t,C^h\right)
{d \Gamma} 
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A\in {\eta} - \nodes_g$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3326: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$G_A$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3328: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}G_A = \int_\Omega N_A \sum_B \dot{c}_B N_B {d \Omega} + \int_\Omega
{N_A}_{,i} D\left(C^h\right) \sum_B c_B {N_B}_{,i} {d \Omega} -
\int_{\Gamma_h} N_A h\left({\vec{x}},t,C^h\right) {d \Gamma} = 0
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:nonlinear-system}
G_A = \int_\Omega N_A \sum_B \dot{c}_B N_B {d \Omega} + \int_\Omega
{\nabla}{N_A}\cdot D\left(C^h\right) {\nabla}{C^h} {d \Omega} -
\int_{\Gamma_h} N_A h\left({\vec{x}},t,C^h\right) {d \Gamma} = 0
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf c} = \left\{c_B\right\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3332: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf R} =
\left\{R_A\right\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3334: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\label{eq:ode-system}
{\bf M}{\bf \dot{c}} &=& -\int_\Omega
{\nabla}{N_A} \cdot D\left(C^h\right) {\nabla}{C^h} {d \Omega} +
\int_{\Gamma_h} N_A h\left({\vec{x}},t,C^h\right) {d \Gamma} \\ 
&=& {\bf R}\left({\bf c}\right) \nonumber
\end{eqnarray}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf c}\left(t=0\right) =
{\bf c}_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3336: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_0\left({\vec{x}},t=0\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3338: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\Delta t_n} =
t_{n+1} - t_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3340: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray}{\bf M}{\bf v}_{n+1} &=& {\bf R}\left({\bf c}_{n+1}\right)
\label{eq:gen-trap-ode}\\ 
{\bf c}_{n+1} &=& {\bf c}_n + {\Delta t_n} {\bf v}_{n+\alpha} 
\label{eq:gen-trap-a} \\ 
{\bf v}_{n+\alpha} &=& (1-\alpha) {\bf v}_n + \alpha
{\bf v}_{n+1} \label{eq:gen-trap-b}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf c}_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3342: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf v}_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3344: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf c}\left(t_n\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3346: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf \dot{c}}\left(t_n\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3348: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3350: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf c}_{n+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3354: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf \tilde{c}}_{n+1} = {\bf c}_n + (1-\alpha) {\Delta t_n}
{\bf v}_n
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf c}_{n+1} = {\bf \tilde{c}}_{n+1} + \alpha {\Delta t_n}
{\bf v}_{n+1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf v}_{n+1} = \frac{{\bf c}_{n+1}-{\bf \tilde{c}}_{n+1}}{
\alpha {\Delta t_n}}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{\alpha {\Delta t_n}}{\bf M}
\left({\bf c}_{n+1}-{\bf \tilde{c}}_{n+1}\right) =
 {\bf R}\left({\bf c}_{n+1}\right)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:gentrap-conc-form}
{\bf R}'\left({\bf c}_{n+1}\right) = {\bf R}\left(
{\bf c}_{n+1}\right) - \frac{1}{\alpha {\Delta t_n}} {\bf M}
{\bf c}_{n+1} + \frac{1}{\alpha {\Delta t_n}} {\bf M}
{\bf \tilde{c}_{n+1}} = {\bf 0}
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}\label{eq:gentrap-rate-form}
{\bf R}'\left({\bf v}_{n+t}\right) = {\bf R}\left(
{\bf \tilde{c}}_{n+1}+\alpha{\Delta t_n} {\bf v}_{n+1}\right) -
{\bf M}{\bf v}_{n+1} = {\bf 0}
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha < \frac{1}{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3366: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha \geq \frac{1}{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3380: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$0 < \gamma \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3384: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf c}_{n+\gamma}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3386: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t_n + \gamma {\Delta t_n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3388: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3390: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:trbdf-tr}
{\bf R}'\left({\bf c}_{n+\gamma}\right)= {\bf R}
\left({\bf c}_{n+\gamma}\right)- \frac{2}{\gamma
{\Delta t_n}}{\bf M}{\bf c}_{n+\gamma}+\frac{2}{\gamma
{\Delta t_n}}{\bf M}{\bf c}_{n} + {\bf R}\left({\bf c}_{n}\right)
= {\bf 0} 
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t_{n+\gamma}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3394: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t_{n+1} = t_n + {\Delta t_n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3396: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:trbdf-bdf}
{\bf R}'\left({\bf c}_{n+1}\right)=
{\bf R}\left({\bf c}_{n+1}\right)- 
\frac{\left(2-\gamma\right)}{\left(1-\gamma\right)
{\Delta t_n}}{\bf M}{\bf c}_{n+1} + 
\frac{1}{\gamma\left(1-\gamma\right)
{\Delta t_n}}{\bf M}{\bf c}_{n+\gamma} 
-\frac{\left(1-\gamma\right)^2}{\gamma\left(1-\gamma\right){\Delta t_n}} 
 {\bf M}{\bf c}_{n} = {\bf 0}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma = 2 - \sqrt{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3398: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:divided-difference}
{\bf \tau}_{n+1} = 2 C {\Delta t_n} \left|
\gamma^{-1}{\bf R}\left({\bf c}_{n}\right) -
\gamma^{-1}\left(1-\gamma\right)^{-1}{\bf R}\left(
{\bf c}_{n+\gamma}\right)
+\left(1-\gamma\right)^{-1}{\bf R}\left({\bf c}_{n+1}\right) \right|
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}C = \frac{-3\gamma^2+4\gamma-2}{12(2-\gamma)}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf q}_{n+1} = {\bf M} {\bf c}_{n+1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}e_{{n+1},i} = \epsilon_R |q_{{n+1},i}| + \epsilon_A
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:timestep-error}
r^2=\frac{1}{N}\sum_{i=1}^{N}
\left(\frac{\tau_{{n+1},i}}{e_{{n+1},i}}\right)^2 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\dto^{*} = {\Delta t_n} r^{-1/3}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\min\left(\dto^{*},2{\Delta t_n}\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3404: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\Delta t_n} = 0.7\dto^{*}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3408: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t = t_n
+\gamma {\Delta t_n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3410: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t_{n+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3412: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf R}'\left({\bf c}\right)={\bf 0}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3414: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\Delta{\bf c}^{k+1} =
-\left({\bf T}\left({\bf c}^k\right)\right)^{-1}
{\bf R}'\left({\bf c}^k\right)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf T}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3416: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}T_{AB} = \frac{\partial {R'}_A}{\partial c_B}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf R}\left({\bf c}\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3418: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}T_{AB} =  \frac{\partial R_A}{\partial c_B} + s M_{AB}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:newton-rel-converge}
\frac{\left|\left|{\bf R}\left({\bf c}^{k+1}\right)\right|\right|}{
\left|\left|{\bf R}\left({\bf c}^{0}\right)\right|\right|}< \epsilon_R
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:newton-abs-converge}
\left|\left|{\bf R}\left({\bf c}^{k+1}\right)\right|\right|
< \epsilon_A
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:newton-update-converge}
\left(\frac{1}{N}\sum_{i=1}^{N}\left(
\frac{\Delta{\bf c}_{,i}^{k+1}}{
\epsilon_R {\bf c}_{,i}^{k+1}+\epsilon_A}\right)^2
\right)^{\frac{1}{2}} < 1
\end{equation}
}

\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial C}{\partial t} = -\lambda C
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda > 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3420: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$z=\Delta t \lambda$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3422: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}C_{n+\gamma} = K_{\mbox{TR}}(\gamma,z) C_{n} 
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K_{\mbox{TR}}(\gamma,z) = (2 - \gamma z) / (2 + \gamma z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3424: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-1 <
K_{\mbox{TR}}(\gamma,z) < 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3426: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}C_{n+1} = K_{\mbox{BDF2}}(\gamma,z) (C_{n+\gamma} - (1-\gamma)^2 C_{n})
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K_{\mbox{BDF2}}(\gamma,z) > 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3428: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$((1-\gamma)^2
C_{n} > C_{n+\gamma})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3430: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-10^{10}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3434: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial C}{\partial t} = \left(K_f I - K_r\right) C
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$10^5$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3440: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial O}{\partial t} = \left(K_f I - K_r\right) C = 
\left(K_f I - K_r\right) (O - o) 
\end{equation}
}

\appendix
\stepcounter{chapter}
\stepcounter{chapter}
\stepcounter{chapter}

\end{document}
