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$\displaystyle \frac{ \phi_{i, k(w)} V_{i, k(w)} - \phi_{i, k-1(w)} V_{i, k-1(w)} }
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$\displaystyle \frac{9}{8} \frac{\phi_{i+1,k} - \phi_{i,k}}{\Delta x}
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$\displaystyle \frac{9}{8} \frac{\phi_{i,k+1} - \phi_{i,k}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{H_{i(u), k} - H_{i-1(u), k}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{H_{i(u), k+1} - H_{i(u), k}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{V_{i+1, k(w)} - V_{i, k(w)}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{V_{i, k(w)} - V_{i, k-1(w)}}{\Delta x} - \Dinv{24}
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$\displaystyle \frac{9}{8} \frac{\psi_{i(u), k(w)} - \psi_{i-1(u), k(w)} }{\Delta
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$\displaystyle \frac{9}{8} \frac{\psi_{i(u), k(w)} - \psi_{i(u), k-1(w)} }{\Delta
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2822}%
$\displaystyle \frac{9}{8} \frac{\phi_{i, k(w)} - \phi_{i, k-1(w)}}{\Delta x} - \Dinv{24}
\frac{\phi_{i, k+1(w)} - \phi_{i, k-2(w)}}{\Delta z},
\Deqlab{RisanZ}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2828}%
$\displaystyle \frac{9}{8}
\frac{ \phi_{i(u), k} H_{i(u), k}
- \phi_{i-1(u), k} H_{i-1(u), k}}{\Delta x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2832}%
$\displaystyle - \Dinv{24}
\frac{\phi_{i+1(u), k} H_{i+1(u), k}
- \phi_{i-2(u), k} H_{i-2(u), k} }{\Delta z},
\Deqlab{RisanZ-2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2838}%
$\displaystyle \frac{9}{8}
\frac{\phi_{i, k(w)} V_{i, k(w)}
- \phi_{i, k-1(w)} V_{i, k-1(w)} }{\Delta x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2842}%
$\displaystyle - \Dinv{24}
\frac{\phi_{i, k+1(w)} V_{i, k+1(w)}
- \phi_{i, k-2(w)} V_{i, k-2(w)}}{\Delta z}.
\Deqlab{RisanZ-3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2849}%
$\displaystyle \DP{u_{i(u), k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2853}%
$\displaystyle - u_{i(u), k} \left( \DP{u}{x} \right)_{i(u), k}
- w_{i(u), k} \left( \DP{u}{z} \right)_{i(u), k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2857}%
$\displaystyle - c_{pv} \overline{\theta}_{i(u), k}
\left( \DP{\Pi}{x} \right)_{i(u), k}
+ \left[ D_u \right]_{i(u), k}.
\Deqlab{RisanAA}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2860}%
$\displaystyle \DP{w_{i, k(w)}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2864}%
$\displaystyle - u_{i, k(w)} \left( \DP{w}{x} \right)_{i, k(w)}
- w_{i, k(w)} \left( \DP{w}{z} \right)_{i, k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2868}%
$\displaystyle - c_{pv} \overline{\theta}_{i, k(w)}
\left( \DP{\Pi}{z} \right)_{i, k(w)}
+ \left[ D_w \right]_{i, k(w)}
+ g
\left(
\frac{\theta^{\prime}}{\overline{\theta}}
- \frac{R}{p_0}
\frac{\overline{\theta}}{\overline{\Pi}^{c_{vv} / R}}
\rho_s
\right)_{i, k(w)}.
\Deqlab{RisanAB}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2871}%
$\displaystyle \DP{\Pi_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2875}%
$\displaystyle - \frac{\overline{c}^2_{i,k}}{c_{pv} \overline{\rho}_{i,k}
\overline{\theta}^2_{i,k}}
\left[ \left( \DP{}{x} (\overline{\rho} \overline{\theta} u)
\right)_{i,k} +
\left( \DP{}{z} (\overline{\rho} \overline{\theta} w)
\right)_{i,k} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2877}%
$\displaystyle + \frac{\overline{c}^2_{i,k}}{c_{pv} \overline{\rho}_{i,k}
\overline{\theta}_{i,k}}
\left( \frac{L}{c_{pv} \overline{T}_{i,k} } -1  \right)
\left[ M_{cond} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2881}%
$\displaystyle + \frac{\overline{c}^2_{i,k}}{c_{pv} \overline{\theta}_{i,k}^2
\overline{\Pi}_{i,k}}
\left( [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k} \right).
\Deqlab{RisanAD}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2884}%
$\displaystyle \DP{{\rho_s}_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2888}%
$\displaystyle - \left[ \DP{}{x} \left( \rho_s u \right) \right]_{i,k}
- \left[ \DP{}{z} \left( \rho_s w \right) \right]_{i,k}
+ \left[ M_{cond} \right]_{i,k}
+ \left[ M_{fall} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2892}%
$\displaystyle + \left[ D_{\rho_s} \right]_{i,k}.
\Deqlab{RisanAE}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2897}%
$\displaystyle \left[ M_{cond} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2901}%
$\displaystyle \frac{ 4 \pi {r_d}_{i,k} \overline{\rho}_{i,k} N^{*} }
{{R_h}_{i,k}} (S_{i,k} - 1),
\Deqlab{TrisanS2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2905}%
$\displaystyle {R_h}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2909}%
$\displaystyle \frac{L^2}{k R [\theta^2]_{i,k} [\Pi^2]_{i,k}},
\Deqlab{TrisanS3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2913}%
$\displaystyle S_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2917}%
$\displaystyle \frac{{p_0}^{\frac{c_{pv}}{R}} \Pi_{i,k}}
{\left[ p_{\ast} \right]_{i,k}},
\Deqlab{TrisanS4}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2921}%
$\displaystyle \left[ M_{fall} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2925}%
$\displaystyle \left[
\DP{}{z} \left( \rho_s V_{term} \right)
\right]_{i,k}
\Deqlab{TrisanV2},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2929}%
$\displaystyle {V_{term}}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2933}%
$\displaystyle \left[ C_{sc} \right]_{i,k}
\frac{2 \left[ r_d^2 \right]_{i,k} g \rho_I}{9 \eta_{i,k} }
\Deqlab{TrisanV3},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2937}%
$\displaystyle {r_d}_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2941}%
$\displaystyle \left(
r_{aero}^3 + \frac{3 {\rho_s}_{i,k}}
{4 \rho_I \pi \overline{\rho}_{i,k} N^{*}}
\right)^{1/3}
\Deqlab{TrisanV3-2},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2945}%
$\displaystyle \left[ C_{sc} \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2949}%
$\displaystyle 1 + \frac{4}{3} \left[ K_n \right]_{i,k},
\Deqlab{TrisanV4}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2953}%
$\displaystyle \left[ K_n \right]_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2957}%
$\displaystyle \frac{\lambda_{i,k}}{{r_d}_{i,k}},
\Deqlab{TrisanV4-2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2961}%
$\displaystyle \lambda_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2965}%
$\displaystyle \frac{k_B \overline{\theta}_{i,k}}
{\sqrt{2} \pi \sigma^2 p_0 \overline{\Pi}^{c_{vv}/R}_{i,k}},
\Deqlab{TrisanV5}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2969}%
$\displaystyle \eta_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2973}%
$\displaystyle \eta_{ref}
\left(
\frac{T_{ref} + C_{CO_2}}
{\overline{\theta}_{i,k} \overline{\Pi}_{i,k} + C_{CO_2}}
\right)
\left(
\frac{\overline{\theta}_{i,k} \overline{\Pi}_{i,k} }{T_{ref}}
\right)^{3/2}
\Deqlab{TrisanV6}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2976}%
$\displaystyle \DP{\theta_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2980}%
$\displaystyle - u_{i,k} \left( \DP{\theta}{x} \right)_{i,k}
- w_{i,k} \left( \DP{\theta}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2984}%
$\displaystyle + \Dinv{\overline{\Pi}_{i,k}} \frac{L
[M_{cond}]_{i,k}}{\overline{\rho}_{i,k} c_{pv}}
+ \Dinv{\overline{\Pi}_{i,k}}
\left( [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k} \right)
+ \left[ D_{\theta} \right]_{i,k}.
\Deqlab{RisanAC}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2987}%
$\displaystyle \DP{{K_m}_{i,k}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2991}%
$\displaystyle - u_{i,k} \left( \DP{{K_m}}{x} \right)_{i,k}
- w_{i,k} \left( \DP{{K_m}}{z} \right)_{i,k}
- \frac{3 g {C_m}^2 l^2}{2 \overline{\theta}_{i,k}}
\left( \DP{\theta^{\prime}}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2993}%
$\displaystyle + {C_m}^2 l^2 \left[ \left( \DP{u}{x} \right)^2_{i,k}
+ \left( \DP{w}{z} \right)^2_{i,k} \right]
+ \frac{{C_m}^2 l^2}{2} \left( \DP{u}{z} + \DP{w}{x} \right)^2_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2995}%
$\displaystyle - \frac{{K_m}_{i,k}}{3} \left( \DP{u}{x} + \DP{w}{z} \right)_{i,k}
+ \Dinv{2} \left[
\left( \DP[2]{{K_m}^2}{x} \right)_{i,k}
+ \left( \DP[2]{{K_m}^2}{z} \right)_{i,k} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2999}%
$\displaystyle + \left( \DP{{K_m}}{x} \right)^2_{i,k}
+ \left( \DP{{K_m}}{z} \right)^2_{i,k}
- \frac{C_{\varepsilon}}{2 C_m l^2} {K_m}^2_{i,k}.
\Deqlab{RisanAF}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3010}%
$\displaystyle \DP{\phi}{t} + U \DP{\phi}{x} = 0 \Deqlab{TrisanA}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3014}%
$\displaystyle \frac{U \Delta t}{\Delta x} \leq 1 \Deqlab{TrisanB}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3016}%
$ U$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure647}%
\begin{figure} \begin{center}
\Depsf[78mm]{ps/timesplit.ps}
  
 \end{center}
  \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3031}%
$ \Pi^{\prime}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3035}%
$\displaystyle \Pi^{\prime} - \alpha {\rm Div}  \Deqlab{SoundDecay1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3037}%
$ {\rm Div} \equiv (\partial u / \partial x) + (\partial w / \partial z)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3039}%
$ \alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3043}%
$\displaystyle \alpha = 5.0 \times 10^{-7} \times \frac{{\rm min} [(\Delta x)^2, (\Delta  z)^2]}{\Delta \tau} \Deqlab{SoundDecay2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3053}%
$\displaystyle \nu_H \DP[2]{\phi}{x} + \nu_V \DP[2]{\phi}{z} \Deqlab{NumDisp1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3055}%
$ \nu_H$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3057}%
$ \nu_V$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3061}%
$\displaystyle \nu_H = \nu_V = 1.0 \times 10^{-4} \times \frac{{\rm min} [(\Delta x)^2,  (\Delta z)^2]}{\Delta t} \Deqlab{NumDisp2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3071}%
$ u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3073}%
$ w$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3077}%
$ \phi^{t + \Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3079}%
$ \phi^{\tau +
\Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3081}%
$ \phi^{t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3083}%
$ \phi^{\tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3085}%
$ \phi^{t -
\Delta t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3087}%
$ \phi^{\tau - \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3091}%
$\displaystyle u_{i(u),k}^{\tau + \Delta \tau} = u_{i(u),k}^{\tau}    - \left( c_{pv} \overline{\theta} \DP{[\Pi^{\prime}]^{\tau}}{x}     \right)_{i,u(k)} \Delta \tau   + {F_u}_{i(u),k}^{t} \Delta \tau \Deqlab{TrisanC}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3093}%
$ F_u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3097}%
$\displaystyle {F_u}_{i(u),k}^{t} = - \left( u^t \DP{u^t}{x} \right)_{i(u), k}   - \left( w^t \DP{u^t}{z} \right)_{i(u), k}   + \left[ D_u \right]_{i(u),k}^{t - \Delta t} \Deqlab{TrisanD}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3099}%
$ t - \Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3102}%
$\displaystyle w_{i,k(w)}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3106}%
$\displaystyle w_{i,k(w)}^{\tau} - c_{pv} \overline{\theta}_{i,k(w)}
\left[ \beta \DP{[\Pi^{\prime}]^{\tau + \Delta \tau}}{z} + (1 -
\beta) \DP{ [\Pi^{\prime}]^{\tau} }{z} \right]_{i,k(w)} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3110}%
$\displaystyle + {F_w}_{i,k(w)}^{t} \Delta \tau, \Deqlab{TrisanE}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3113}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3117}%
$\displaystyle \left[ \Pi^{\prime} \right]_{i,k}^{\tau}
- \left[ \frac{ \overline{c}^2 }{c_{pv} \overline{\rho}
\overline{\theta}^2 } \DP{}{x} (\overline{\rho} \overline{\theta}
u^{\tau + \Delta \tau})\right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3119}%
$\displaystyle - \beta \left[ \frac{ \overline{c}^2 }{c_{pv} \overline{\rho}
\overline{\theta}^2 } \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau + \Delta \tau})\right]_{i,k} \Delta \tau
- (1 - \beta) \left[ \frac{ \overline{c}^2 }{c_{pv} \overline{\rho}
\overline{\theta}^2 } \DP{}{z} (\overline{\rho} \overline{\theta}
w^{\tau})\right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3121}%
$\displaystyle + \left[ \frac{\overline{c}^2}{c_{pv} \overline{\theta}}
\left( \frac{L}{c_{pv} \overline{T}} - 1 \right)
\frac{M_{cond}^{\tau}}{\overline{\rho}} \right]_{i,k} \Delta \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3125}%
$\displaystyle + \left[ \frac{ \overline{c}^2 }{c_{pv} \overline{\theta}^2
\overline{\Pi} } ( Q_{rad} + Q_{dis} )
\right]_{i,k} \Delta \tau
\Deqlab{TrisanF}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3127}%
$ \beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3129}%
$ \beta = 0.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3131}%
$ F_w$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3134}%
$\displaystyle {F_w}_{i,k(w)}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3138}%
$\displaystyle - \left( u^t \DP{w^t}{x} \right)_{i, k(w)}
- \left( w^t \DP{w^t}{z} \right)_{i, k(w)}
+ g
\frac{ [\theta^{\prime}]_{i,k(w)}^{t}}{\overline{\theta}_{i,k(w)}}
- g
\frac{R}{p_0}
\frac{\overline{\theta}_{i, k(w)}}
{ \overline{\Pi}^{c_{vv} / R}_{i, k(w)} }
{\rho_s}_{i, k(w)}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3142}%
$\displaystyle + \left[ D_w \right]_{i,k(w)}^{t - \Delta t}
\Deqlab{TrisanG}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3144}%
$ u^{\tau + \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3146}%
$ w^{\tau + \Delta \tau}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3149}%
$\displaystyle \begin{pmatrix}
A_1 & B_2 & \cdots & 0 & 0 \\
C_1 & A_2 & \cdots & 0 & 0 \\
\colon & \colon &  & \colon & \colon \\
0 & 0 & \cdots & A_{km - 1} & B_{km} \\
0 & 0 & \cdots & C_{km - 1} & A_{km}
\end{pmatrix}
\begin{pmatrix}
\Pi^{\prime}_{1,1} & \Pi^{\prime}_{2,1} & \cdots & \Pi^{\prime}_{km-1,1} &
\Pi^{\prime}_{km,1} \\
\Pi^{\prime}_{1,2} & \Pi^{\prime}_{2,2} & \cdots & \Pi^{\prime}_{km-1,2} &
\Pi^{\prime}_{km,2} \\
\colon & \colon & & \colon & \colon \\
\Pi^{\prime}_{1,km-1} & \Pi^{\prime}_{2,km-1} & \cdots &
\Pi^{\prime}_{km-1,km-1} & \Pi^{\prime}_{km,km-1} \\
\Pi^{\prime}_{1,km} & \Pi^{\prime}_{2,km} & \cdots &
\Pi^{\prime}_{km-1,km} & \Pi^{\prime}_{km,km}
\end{pmatrix} ^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3154}%
$\displaystyle =
\begin{pmatrix}
D_{1,1} & D_{2,1} & \cdots & D_{km-1,1} & D_{km,1} \\
D_{1,2} & D_{2,2} & \cdots & D_{km-1,2} & D_{km,2} \\
\colon & \colon & & \colon & \colon \\
D_{1,km-1} & D_{2,km-1} & \cdots & D_{km-1,km-1} & D_{km,km-1} \\
D_{1,km} & D_{2,km} & \cdots & D_{km-1,km} & D_{km,km}
\end{pmatrix} ^{\tau}
\Deqlab{TrisanH}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3156}%
$ km$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3161}%
$\displaystyle A_1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3165}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_1
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,1(w)},
\Deqlab{TrisanI}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3167}%
$\displaystyle A_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3171}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_k
\Dinv{(\Delta z)^2}
[ ( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k(w)}
+ ( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k-1(w)}
],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3175}%
$\displaystyle (k = 2, 3, \dots , km -1 ) \Deqlab{TrisanJ}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3179}%
$\displaystyle A_{km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3183}%
$\displaystyle 1 + \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{km}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,km-1(w)},
\Deqlab{TrisanK}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3187}%
$\displaystyle B_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3191}%
$\displaystyle - \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k-1}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k-1(w)},
\Deqlab{TrisanL}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3195}%
$\displaystyle C_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3199}%
$\displaystyle - \beta^2 \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k+1}
\Dinv{(\Delta z)^2}
( c_{pv} \overline{\rho} \overline{\theta}^2)_{i,k(w)},
\Deqlab{TrisanM}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3201}%
$\displaystyle D_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3205}%
$\displaystyle [ \Pi^{\prime} ]_{i,1}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c}^2 \Delta \tau}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{1}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,1}
- \left[ \frac{\overline{c}^2 \Delta \tau}{c_{pv}
\overline{\theta} } \right]_{1}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,1}
+ F_{i,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3209}%
$\displaystyle - \beta \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{1}
\Dinv{\Delta z} (c_{pv} \overline{\rho} \overline{\theta}^2)_{i,0(w)}
E_{i,0(w)}, \Deqlab{TrisanN}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3212}%
$\displaystyle D_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3216}%
$\displaystyle [ \Pi^{\prime} ]_{i,k}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c}^2 \Delta \tau}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{k}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,k}
- \left[ \frac{\overline{c}^2 \Delta \tau}{c_{pv}
\overline{\theta} } \right]_{k}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,k}
+ F_{i,k},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3220}%
$\displaystyle (k = 2, 3, \dots , km -1 )
\Deqlab{TrisanO}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3222}%
$\displaystyle D_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3226}%
$\displaystyle [ \Pi^{\prime} ]_{i,km}^{\tau}
- (1 - \beta) \left[ \frac{\overline{c}^2 \Delta \tau}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{km}
\left[ \DP{(\overline{\rho} \overline{\theta} w^{\tau})}{z}
\right]_{i,km}
- \left[ \frac{\overline{c}^2 \Delta \tau}{c_{pv}
\overline{\theta} } \right]_{km}
\left[ \DP{u^{\tau + \Delta \tau}}{x} \right]_{i,km}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3230}%
$\displaystyle + F_{i,km}
+ \beta \left[ \frac{\overline{c}^2 (\Delta \tau)^2}{c_{pv}
\overline{\rho} \overline{\theta}^2} \right]_{km}
\Dinv{\Delta z} (c_{pv} \overline{\rho} \overline{\theta}^2)_{i,km(w)}
E_{i,km(w)}. \Deqlab{TrisanP}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3235}%
$\displaystyle E_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3239}%
$\displaystyle \left[ \DP{\left( \alpha {\rm Div} \right)^{\tau}}{z}
\right]_{i,k(w)}
- (1 - \beta )
\left[ \DP{[\Pi^{\prime}]^{\tau}}{z} \right]_{i,k(w)}
+ \left[ \frac{F^{t}_{w}}{c_{pv} \overline{\theta}}
\right]_{i,k(w)},
\Deqlab{TrisanQ}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3241}%
$\displaystyle F_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3245}%
$\displaystyle - \beta
\left(
\frac{\overline{c}^2 \Delta \tau}
{c_{pv} \overline{\rho} \overline{\theta}}
\right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3247}%
$\displaystyle \Big[
\DP{}{z}
\Big\{ \overline{\rho} \overline{\theta}
\Big( w^{\tau} - c_{pv} \overline{\theta} \Delta \tau
\left\{
(1 - \beta) \DP{[\Pi^{\prime}]^{\tau}}{z}
- \DP{\left( \alpha {\rm Div} \right)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau \Big) \Big\}_{i,k(w)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3251}%
$\displaystyle + \frac{\overline{c}^2}{c_{pv} \overline{\theta}}
\left(\frac{L}{c_{pv} \overline{T}} -1 \right)
\frac{M_{cond}^{\tau}}{\overline{\rho}}
\Delta \tau
+ \frac{ \overline{c}^2 }{c_{pv} \overline{\theta}^2
\overline{\Pi} } ( Q_{rad} + Q_{dis} ) \Delta \tau
\Big]_{i,k}
\Deqlab{TrisanR}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3253}%
$ \tau + \Delta \tau$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3261}%
$\displaystyle {\rho_s}_{i,k}^{*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3265}%
$\displaystyle {\rho_s}_{i,k}^{\tau}
+ \left[ M_{fall} \right]_{i,k}^{\tau} \Delta \tau
+ \left[ F_{\rho_s} \right]_{i,k}^{t} \Delta \tau,
\Deqlab{TrisanU-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3269}%
$\displaystyle {\rho_s}_{i,k}^{**}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3273}%
$\displaystyle {\rm FillNeg.} [{\rho_s}_{i,k}^{*}],
\Deqlab{TrisanU-2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3277}%
$\displaystyle {\rho_s}_{i,k}^{\tau + \Delta \tau}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3281}%
$\displaystyle {\rho_s}_{i,k}^{**}
+ \left[ M_{cond} \right]_{i,k}^{**} \Delta \tau.
\Deqlab{TrisanU-3}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3283}%
$ {\rm FillNeg.} [{\rho_s}_{i,k}^{*}]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3285}%
$ {\rho_s}_{i,k}^{**}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3287}%
$ F_{\rho_s}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3292}%
$\displaystyle \left[ F_{\rho_s} \right]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3296}%
$\displaystyle - \left[ \DP{}{x} ({\rho_s}^t u^t) \right]_{i,k}
- \left[ \DP{}{z} ({\rho_s}^t w^t) \right]_{i,k}
+ {D_{\rho_s}}_{i,k}^{t}.
\Deqlab{TrisanV}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3300}%
$\displaystyle {\theta^{\prime}}_{i,k}^{\tau + \Delta \tau}  = {\theta^{\prime}}_{i,k}^{\tau} + {F_{\theta}}_{i,k}^{t} \Delta \tau +     \Dinv{\overline{\Pi}_{i,k}}     \frac{L \left[ M_{cond} \right]^{**}_{i,k}}    {c_{pv} \overline{\rho}_{i,k}} \Delta \tau   \Deqlab{TrisanS}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3302}%
$ F_{\theta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3305}%
$\displaystyle {F_{\theta}}_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3309}%
$\displaystyle - u_{i,k}^{t} \left( \DP{{\theta^{\prime}}^t}{x} \right)_{i,k}
- w_{i,k}^{t} \left( \DP{{\theta^{\prime}}^t}{z} \right)_{i,k}
- w_{i,k}^{t} \left( \DP{\overline{\theta}^t}{z} \right)_{i,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3313}%
$\displaystyle + \Dinv{\overline{\Pi}_{i,k}}
( Q_{rad} + Q_{dis})_{i,k}^{t} + {D_{\theta}}_{i,k}^{t - \Delta t}
\Deqlab{TrisanT}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3317}%
$\displaystyle {K_m}_{i,k}^{t+ \Delta t} = {K_m}_{i,k}^{t- \Delta t} + 2 \Delta t [F_{K_m}]_{i,k}^{t} \Deqlab{TrisanW}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3320}%
$\displaystyle [F_{K_m}]_{i,k}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3324}%
$\displaystyle - \left[ u_{i(u),k} \left( \DP{K_m}{x} \right)_{i(u),k} \right]_{i,k}^t
- \left[ w_{i,k(w)} \left( \DP{K_m}{z} \right)_{i,k(w)} \right]_{i,k}^t$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3326}%
$\displaystyle - \left[
\frac{3 g {C_m}^2 l^2}{2 \overline{\theta}}
\left( \DP{\theta^{\prime}}{z} \right)_{i,k(w)}
\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3328}%
$\displaystyle + {C_m}^2 l^2 \left[ \left( \DP{u}{x} \right)^2 \right]_{i,k}^{t -
\Delta t}
+ {C_m}^2 l^2 \left[ \left( \DP{w}{z} \right)^2 \right]_{i,k}^{t -
\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3330}%
$\displaystyle + \frac{{C_m}^2 l^2}{2} \left[
\left\{ \left( \DP{u}{z} \right)_{i(u),k(w)}
\right\}_{i,k}^{t - \Delta t}
+ \left\{ \left( \DP{w}{x} \right)_{i(u),k(w)}
\right\}_{i,k}^{t - \Delta t}
\right]^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3332}%
$\displaystyle - \Dinv{3} [K_m]_{i,k}^{t - \Delta t}
\left[ \left( \DP{u}{x} \right)_{i,k}^{t - \Delta t}
+ \left( \DP{w}{z} \right)_{i,k}^{t - \Delta t}
\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3334}%
$\displaystyle + \Dinv{2} \left[
\left\{ \DP{}{x} \left( \DP{K_m^2}{x} \right)_{i(u),k}
\right\}_{i,k}^{t - \Delta t}
+ \left\{ \DP{}{z} \left( \DP{K_m^2}{z} \right)_{i,k(w)}
\right\}_{i,k}^{t - \Delta t} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3336}%
$\displaystyle + \left[ \left\{ \left( \DP{K_m}{x} \right)^2 \right\}_{i(u),k}
\right]_{i,k}^{t - \Delta t}
+ \left[ \left\{ \left( \DP{K_m}{z} \right)^2 \right\}_{i,k(w)}
\right]_{i,k}^{t - \Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3340}%
$\displaystyle - \frac{C_{\varepsilon}}{2 C_m l^2} [K_m^2]_{i,k}^{t - \Delta t}
\Deqlab{TrisanX}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3342}%
$ t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3352}%
$ t + \Delta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3358}%
$ \theta^{\prime}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3362}%
$ \rho_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3364}%
$ K_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3368}%
$ \overline{\phi}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3372}%
$\displaystyle \overline{\phi^t} = \phi^t + \mu_a (\overline{\phi^{t - \Delta t}} - 2  \phi^t + \phi^{t + \Delta t})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3374}%
$ \mu_a$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3376}%
$ \mu_a = 0.1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3380}%
$ \bf {17}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3382}%
$ \bf {100}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3384}%
$ \bf {29}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3386}%
$ \bf {35}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3388}%
$ \bf {12}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3390}%
$ \bf {50}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3392}%
$ \bf {44}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3394}%
$ \bf {120}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\setcounter{equation}{0}
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\renewcommand{\thechapter}{\Alph{chapter}}

\renewcommand{\theequation}{\thechapter .\arabic{equation}}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3398}%
$\displaystyle QSUMPN(i,k)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3402}%
$\displaystyle 1.0 /$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3404}%
$\displaystyle \{
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3406}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i+1,k) )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3408}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k-1) )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3410}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k+1) )
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3412}%
$\displaystyle + 0.25 \ast [
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3414}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i+2,k) )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3416}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k-2) )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3420}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k+2) )
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\Deqlab{A}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3422}%
$\displaystyle DQFILL(i,k)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3426}%
$\displaystyle - {\rm min}( 0.0, \rho_s(i,k) )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3428}%
$\displaystyle + {\rm max}( 0.0, \rho_s(i,k) )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3430}%
$\displaystyle * \{
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3432}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i+1,k) ) * QSUMPN(i+1,k)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3434}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i,k-1) ) * QSUMPN(i,k-1)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3436}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i,k+1) ) * QSUMPN(i,k+1)
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3438}%
$\displaystyle + 0.25 * [
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3440}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i+2,k) ) * QSUMPN(i+2,k)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3442}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i,k-2) ) * QSUMPN(i,k-2)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3444}%
$\displaystyle + {\rm min}( 0.0, \rho_s(i,k+2) ) * QSUMPN(i,k+2)
]
\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3448}%
$\displaystyle \Deqlab{B}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3452}%
$\displaystyle \hat{\rho_s} (i,k)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3456}%
$\displaystyle \rho_s (i,k) + DQFILL (i,k)
\Deqlab{C}.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3460}%
$ \hat{\rho_s}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3462}%
$ DQFILL(i,k)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3464}%
$ 3/16$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3466}%
$ 1/16$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3468}%
$ (i,k) = (M,N)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3472}%
$ (i,k) = (M-2,N), (M-1,N), (M,N), (M+1,N), (M+2,N), (M,N-2), (M,N-1),
(M,N+1), (M,N+2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3474}%
$ DQFILL(i,k) = 0$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3479}%
$\displaystyle DQFILL (M-2,N)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3483}%
$\displaystyle 0.25 * \rho_s(M-2,N) \rho_s(M,N) * QSUMPN(M,N),$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3487}%
$\displaystyle \Deqlab{D}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3489}%
$\displaystyle DQFILL (M-1,N)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3493}%
$\displaystyle 0.75 * \rho_s(M-1,N) \rho_s(M,N) * QSUMPN(M,N),$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3497}%
$\displaystyle \Deqlab{E}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3499}%
$\displaystyle DQFILL (M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3503}%
$\displaystyle - \rho_s (M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3507}%
$\displaystyle \Deqlab{F}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3509}%
$\displaystyle DQFILL (M+1,N)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3513}%
$\displaystyle 0.75 * \rho_s(M+1,N) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3517}%
$\displaystyle \Deqlab{G}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3519}%
$\displaystyle DQFILL (M+2,N)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3523}%
$\displaystyle 0.25 * \rho_s(M+2,N) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3527}%
$\displaystyle \Deqlab{H}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3529}%
$\displaystyle DQFILL (M,N-2)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3533}%
$\displaystyle 0.25 * \rho_s(M,N-2) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3537}%
$\displaystyle \Deqlab{I}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3539}%
$\displaystyle DQFILL (M,N-1)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3543}%
$\displaystyle 0.75 * \rho_s(M,N-1) \rho_s(M,N) * QSUMPN(M,N),$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3547}%
$\displaystyle \Deqlab{J}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3549}%
$\displaystyle DQFILL (M,N+1)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3553}%
$\displaystyle 0.75 * \rho_s(M,N+1) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3557}%
$\displaystyle \Deqlab{K}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3559}%
$\displaystyle DQFILL (M,N+2)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3563}%
$\displaystyle 0.25 * \rho_s(M,N+2) \rho_s(M,N) * QSUMPN(M,N),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3567}%
$\displaystyle \Deqlab{L}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3570}%
$\displaystyle QSUMPN (M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3574}%
$\displaystyle \{
0.75 \ast [
\rho_s(M-1,N)
+ \rho_s(M+1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3576}%
$\displaystyle + \rho_s(M,N-1)
+ \rho_s(M,N+1)
]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3578}%
$\displaystyle + 0.25 \ast [
\rho_s(M-2,N)
+ \rho_s(M+2,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3582}%
$\displaystyle + \rho_s(M,N-2)
+ \rho_s(M,N+2)
]
\}^{-1}
\Deqlab{M}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3585}%
$\displaystyle \sum_{i} \sum_{k} \hat{\rho_s} (i, k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3589}%
$\displaystyle \sum_{i} \sum_{k}
\left[ \rho_s (i,k) + DQFILL (i,k) \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3593}%
$\displaystyle \sum_{i} \sum_{k} \rho_s (i,k)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3595}%
$\displaystyle + DQFILL (M-2,N)
+ DQFILL (M-1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3597}%
$\displaystyle + DQFILL (M,N)
+ DQFILL (M+1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3599}%
$\displaystyle + DQFILL (M+2,N)
+ DQFILL (M,N-2)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3601}%
$\displaystyle + DQFILL (M,N-1)
+ DQFILL (M,N+1)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3603}%
$\displaystyle + DQFILL (M,N+2)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3607}%
$\displaystyle \sum_{i} \sum_{k} \rho_s (i,k)
- \rho_s (M,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3609}%
$\displaystyle + \rho_s (M,N) * QSUMPN(M,N)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3611}%
$\displaystyle \{
0.75
* [ \rho_s(M-1,N) + \rho_s(M+1,N)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3613}%
$\displaystyle + \rho_s(M,N+1) + \rho_s(M,N+1) ]$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3615}%
$\displaystyle + 0.25
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3617}%
$\displaystyle + \rho_s(M,N+2) + \rho_s(M,N+2) ]
\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3623}%
$\displaystyle + \rho_s (M,N) * QSUMPN(M,N) * QSUMPN(M,N)^{-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3629}%
$\displaystyle \sum_{i} \sum_{k} \rho_s (i,k)
\Deqlab{N}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}


\end{document}
