Linearized, Implicit, Finite Difference Equations

The following assumptions are applied:

If $\begin{array}{l}f \cdot f >> \Delta f \cdot \Delta f\end{array}$ , then $\begin{array}{l}\Delta f \cdot \Delta f = 0\end{array}$ (Preissmann as reported by Liggett and Cunge, 1975).

If $\begin{array}{l}g=g(Q,z_s)\end{array}$ , then $\begin{array}{l}\Delta g\end{array}$ can be approximated by the first term of the Taylor Series, i.e.:

1)	$\begin{array}{l}\displaystyle \Delta g_j = \left( \frac{\partial g}{\partial Q} \right) _j \Delta Q_j + \left( \frac{\partial g}{\partial z_s} \right) _j \Delta z_j\end{array}$

If the time step, $\begin{array}{l}\Delta t\end{array}$ , is small, then certain variables can be treated explicitly; hence $\begin{array}{l}h_j^{n+1} \approx h_j^{n}\end{array}$ and $\begin{array}{l}\Delta hj \approx 0\end{array}$ .

Assumption 2 is applied to the friction slope, $\begin{array}{l}S_f\end{array}$ and the area, $\begin{array}{l}A\end{array}$ . Assumption 3 is applied to the velocity, $\begin{array}{l}V\end{array}$ , in the convective term; the velocity distribution factor, $\begin{array}{l}\beta\end{array}$ ; the equivalent flow path, $\begin{array}{l}x\end{array}$ ; and the flow distribution factor, $\begin{array}{l}\phi\end{array}$ .

The finite difference approximations are listed term by term for the continuity equation in Table 1 and for the momentum equation in Table 2. If the unknown values are grouped on the left-hand side, the following linear equations result:

2)	$\begin{array}{l}\displaystyle CQ1_j \Delta Q_j + CZ1_j \Delta z_{s,j} + CQ2_j \Delta Q_{j+1} + CZ2_j \Delta z_{s,j+1} = CB_j\end{array}$

3)	$\begin{array}{l}\displaystyle MQ1_j \Delta Q_j + MZ1_j \Delta z_{s,j} + MQ2_j \Delta Q_{j+1} + MZ2_j \Delta z_{s,j+1} = MB_j\end{array}$

Table 1 Finite Difference Approximation of the Terms in the Continuity Equation

*Term*	*Finite Difference Approximation*
$\begin{array}{l}\Delta Q\end{array}$	$\begin{array}{l}(Q_{j+1} - Q_j) + \theta (\Delta Q_{j+1} - Q_j)\end{array}$
$\begin{array}{l}\displaystyle \frac{\partial A_c}{\partial t} \Delta x_c\end{array}$	$\begin{array}{l}\displaystyle 0.5 \Delta x_{cj} \frac{\left( \frac{dA_c}{dz} \right)_j \Delta z_j + \left( \frac{dA_c}{dz} \right) _{j+1} \Delta z_{j+1}}{\Delta t}\end{array}$
$\begin{array}{l}\displaystyle \frac{\partial A_f}{\partial t} \Delta x_f\end{array}$	$\begin{array}{l}\displaystyle 0.5 \Delta x_{fj} \frac{\left( \frac{dA_f}{dz} \right)_j \Delta z_j + \left( \frac{dA_f}{dz} \right) _{j+1} \Delta z_{j+1}}{\Delta t}\end{array}$
$\begin{array}{l}\displaystyle \frac{\partial S}{\partial t} \Delta x_f\end{array}$	$\begin{array}{l}\displaystyle 0.5 \Delta x_{fj} \frac{\left( \frac{dS_f}{dz} \right)_j \Delta z_j + \left( \frac{dS_f}{dz} \right) _{j+1} \Delta z_{j+1}}{\Delta t}\end{array}$

Table 2 Finite Difference Approximation of the Terms in the Momentum Equation

*Term*	Finite Difference Approximation
$\begin{array}{l}\displaystyle \frac{\partial (Q_c \Delta x_c +Q_f \Delta x_f )}{\partial t \Delta x_e}\end{array}$	$\begin{array}{l}\displaystyle \frac{0.5}{\Delta x_e \delta t} (\delta Q_{cj} \Delta x_{cj} + \delta Q_{fj} \Delta x_{fj} +\delta Q_{cj+1} \Delta x_{cj} + \delta Q_{fj+1} \Delta x_{fj})\end{array}$
$\begin{array}{l}\displaystyle \frac{\Delta \beta VQ}{\Delta x_ej}\end{array}$	$\begin{array}{l}\displaystyle \frac{1}{\Delta x_{ej}} \left[ (\beta VQ)_{j+1} - (\beta VQ)_j \right] + \frac{\theta}{\Delta x_{ej}} \left[ (\beta VQ)_{j+1} - (\beta VQ)_j \right]\end{array}$
$\begin{array}{l}\displaystyle g \overline{A} \frac{\Delta z_s}{\Delta x_e}\end{array}$	$\begin{array}{l}\displaystyle g \overline{A} \left[ \frac{z_{s,j+1} - z_{s,j}}{\Delta x_{ej}} + \frac{\theta}{\Delta x_{ej}} (\Delta z_{s,j+1} - \Delta z_{s,j}) \right] + \theta g \Delta \overline{A} \frac{(z_{s,j+1} -z_{s,j})}{\Delta x_{ej}}\end{array}$
$\begin{array}{l}g \overline{A} \left( \overline{S}_f + \overline{S}_h \right)\end{array}$	$\begin{array}{l}g \overline{A} \left( \overline{S}_f + \overline{S}_h \right) + 0.5 \theta g \overline{A} \left[ (\Delta S_{fj+1} + \Delta S_{fj}) + (\Delta S_{hj+1} + \Delta S_{hj}) \right] + 0.5 \theta g (\overline{S}_f + \overline{S}_h ) ( \Delta A_j + \Delta A_{j+1} )\end{array}$
$\begin{array}{l}\overline{A}\end{array}$	$\begin{array}{l}0.5 (A_{j+1} + A_j)\end{array}$
$\begin{array}{l}\overline{S}_f\end{array}$	$\begin{array}{l}0.5 (S_{fj+1} + S_fj)\end{array}$
$\begin{array}{l}\partial A_j\end{array}$	$\begin{array}{l}\displaystyle \left( \frac{dA}{dZ} \right) _j \Delta z_j\end{array}$
$\begin{array}{l}\partial S_{fj}\end{array}$	$\begin{array}{l}\displaystyle \left( \frac{-2S_f}{K} \frac{dK}{dz} \right)_j \Delta z_j + \left( \frac{2s_f}{Q} \right)_j \Delta Q_j\end{array}$
$\begin{array}{l}\partial \overline{A}\end{array}$	$\begin{array}{l}0.5 (\Delta A_j + \Delta A_{j+1} )\end{array}$

The values of the coefficients are defined in Tables 3 and 4.

Table 3 Coefficients for the Continuity Equation

Coefficient	Value
$\begin{array}{l}CQ1_j\end{array}$	$\begin{array}{l}\displaystyle \frac{-\theta}{\Delta x_{ej}}\end{array}$
$\begin{array}{l}CZ1_j\end{array}$	$\begin{array}{l}\displaystyle \frac{0.5}{\Delta t \Delta x_{ej}} \left[ \left( \frac{dA_c}{dz} \right)_j \Delta x_{cj} + \left( \frac{dA_f}{dz} + \frac{dS}{dz} \right) _j \Delta x_{fj} \right]\end{array}$
$\begin{array}{l}CQ2_j\end{array}$	$\begin{array}{l}\displaystyle \frac{\theta}{\Delta x_{ej}}\end{array}$
$\begin{array}{l}CZ2_j\end{array}$	$\begin{array}{l}\displaystyle \frac{0.5}{\Delta t \Delta x_{ej}} \left[ \left( \frac{dA_c}{dz} \right)_{j+1} \Delta x_{cj} + \left( \frac{dA_f}{dz} + \frac{dS}{dz} \right) _{j+1} \Delta x_{fj} \right]\end{array}$
$\begin{array}{l}CB_j\end{array}$	$\begin{array}{l}\displaystyle - \frac{Q_{j+1} - Q_j}{\Delta x_{ej}} + \frac{Q_j}{\Delta x_{ej}}\end{array}$

Table 4 Coefficients of the Momentum Equation

Term	Value
$\begin{array}{l}MQ1_j\end{array}$	$\begin{array}{l}\displaystyle 0.5 \frac{\Delta x_{cj} \phi _j +\Delta x_{fj} (1-\phi _j)}{\Delta x_{ej} \Delta t} - \frac{\beta _j V_j \theta}{\Delta x_{ej}} + \theta g\overline{A} \frac{(S_{fj} +S_{hj})}{Q_j}\end{array}$
$\begin{array}{l}MZ1_j\end{array}$	$\begin{array}{l}\displaystyle \frac{-gA\theta}{\Delta x_{ej}} + 0.5g(z_{j+1} -z_j) \left( \frac{dA}{dz} \right)_j \left( \frac{\theta}{\Delta x_{ej}} \right) - g\theta \overline{A} \right[ \left( \frac{dK}{dz} \right)_j \left( \frac{S_{fj}}{K_j} \right) + \left( \frac{dA}{dz} \right)_j \left( \frac{S_{hj}}{A_j} \right) \right] + 0.5\theta g \left( \frac{dA}{dz} \right)_j (\overline{S}_f + \overline{S}_h)\end{array}$
$\begin{array}{l}MQ2_j\end{array}$	$\begin{array}{l}\displaystyle \displaystyle 0.5 \left[ \Delta x_{cj} \phi _j+1} + \Delta x_{fj} (1-\phi _{j+1}) \right] \left( \frac{1}{\Delta x_{ej} \Delta t} \right) + \beta _{j+1} V_{j+1} \left( \frac{\theta}{\Delta x_{ej}} \right) + \frac{\theta gA}{Q_{j+1}} (S_{fj+1} + S_{hj+1} )\end{array}$
$\begin{array}{l}MZ2_j\end{array}$	$\begin{array}{l}\displaystyle \frac{g \overline{A} \theta}{\Delta x_{ej}} + 0.5g ( z_{j+1} -z_j ) \left( \frac{dA}{dz} \right) _{j+1} \left( \frac{\theta}{\Delta x_{ej}} \right) - \theta g \overline{A} \left[ \left( \frac{dK}{dz} \right) _{j+1} \left( \frac{S_{fj+1}}{K_{j+1}} \right) + \left( \frac{dA}{dz} \right) _{j+1} + \left( \frac{S_{j+1}}{A_{j+1}} \right) \right] + 0.5 \theta g \left( \frac{dA}{dz} \right) _{j+1} (\overline{S} _f + \overline{S} _h )\end{array}$
$\begin{array}{l}MB_j\end{array}$	$\begin{array}{l}\displaystyle - \left[ \left( \beta _{j+1} V_{j+1} Q_{j+1} - \beta _j V_j Q_j \right) \left( \frac{1}{\Delta x_{ej}} \right) + \left( \frac{g \overline{A}}{\Delta x_{ej}} \right) ( z_{j+1} - z_j ) + g \overline{A} \left( \overline{S} _f + \overline{S} _h \right) \right]\end{array}$