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I am reading the paper, http://math.mit.edu/~gs/cse/codes/mit18086_navierstokes.pdf

In the paper, the nonlinear term is treated as mix of central central difference and upwind difference using a transition parameter $\gamma \in [0,1]$, and it's defined as $$\frac{U^*-U}{\Delta t} = ((\bar{U}^h)^2 - \gamma \lvert \bar{U}^h \rvert \tilde{U}^h )_x - ((\bar{U}^v \bar{V}^h) - \gamma \lvert \bar{V}^h \rvert \tilde{U}^v )_y $$ The variables $\bar{U}^h$ and $\tilde{U}^h$ are defiend as, $$ (\bar{U}^h)_{i+\frac{1}{2},j} = \frac{U_{i+1,j}+U_{i,j}}{2}\\ (\bar{U}^v)_{i,j+\frac{1}{2}} = \frac{U_{i,j+1}+U_{i,j}}{2}\\ (\tilde{U}^h)_{i+\frac{1}{2},j} = \frac{U_{i+1,j}-U_{i,j}}{2}\\ (\tilde{U}^v)_{i,j+\frac{1}{2}} = \frac{U_{i,j+1}-U_{i,j}}{2}\\ $$

$U^2$ term is $$ ((\bar{U}^h)^2 - \gamma \lvert \bar{U}^h \rvert \tilde{U}^h )_{i+\frac{1}{2}, j} = \bar{U}^h \begin{cases} \left(\frac{1-\gamma}{2}\right) U_{i+1,j} + \left(\frac{1+\gamma}{2}\right) U_{i,j} \;\;\; \text{if} \;\; \bar{U}^h\ge0\\ \left(\frac{1+\gamma}{2}\right) U_{i+1,j} + \left(\frac{1-\gamma}{2}\right) U_{i,j} \;\;\; \text{if} \;\; \bar{U}^h\lt 0\\ \end{cases} $$

And it says, "One can easily see that this becomes averaged central differencing for $\gamma=0$ and conservative upwinding for $\gamma=1$."

But I cannot see this becomes the conservative upwind difference when $\gamma =1$. I thought the conservative upwinding is $$ \begin{cases} (U_{i,j}^2-U_{i-1,j}^2)/\Delta x \;\;\; \text{if}\;\;U_{i,j}\ge0 \\ (U_{i+1,j}^2-U_{i,j}^2)/\Delta x \;\;\; \text{if}\;\;U_{i,j}\lt0 \\ \end{cases} $$

If I plug $U_{i,j}$ in $\bar{U}$ and $\tilde{U}$ with the condition $\bar{U}^h \ge 0$ and $\gamma=1$, the first term becomes $$ ((\bar{U}^h_{i+\frac{1}{2}} U_{i,j}) - (\bar{U}^h_{i-\frac{1}{2}} U_{i-1,j}))/{\Delta x}\\ =((\frac{U_{i+1,j}+U_{i,j}}{2}U_{i,j}) - (\frac{U_{i,j}+U_{i-1,j}}{2} U_{i-1,j}))/{\Delta x} $$

This is not the conservative upwinding difference I thought. I am confused because it depends on $\bar{U}^h$, not $U_{i,j}$. Can someone explain and show the right conservative upwinding difference form?

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Your last term $$ ((\frac{U_{i+1,j}+U_{i,j}}{2}U_{i,j}) - (\frac{U_{i,j}+U_{i-1,j}}{2} U_{i-1,j}))/{\Delta x} $$ is a conservative approximation.

The idea of conservative upwind scheme for $(u^2)_x$ is to consider it as a difference of fluxes $u u$ at $i+1/2$ and $i-1/2$. The arithmetic averages in this formula are velocities at these points, say $U_{i+1/2,j}$ and $U_{i-1/2,j}$. The values $U_{i,j}$ and $U_{i-1,j}$ are upwinded values of $u$ at these points. In fact this formula shall be used when the averaged velocities are positive, i.e. $U_{i+1/2,j}>0$ and $U_{i-1/2,j}>0$.

P.S. here I add the derivation of such conservative upwind approximation (where I skip the index $j$): $$ (u u)_x(x_i) \approx \frac{(u u)(x_{i+1/2})-(u u)(x_{i-1/2})}{\Delta x} \approx \frac{U_{i+1/2} U_i -U_{i-1/2}U_{i-1}}{\Delta x} $$ if $$ U_{i+1/2}=0.5 (U_i+U_{i+1}) > 0 \hbox{ and } U_{i-1/2}=0.5 (U_{i-1}+U_{i}) > 0 $$

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