Modelling of two phase flow in packed bed using conservation equations

[Chestermiller]

Hi Chet, is the claim that the numerical dispersion from the upwind scheme is exactly the same as the physical dispersion in the central differencing scheme, or is the claim only that they are of the same order of magnitude?

To terms of 2nd order accuracy, they are the same when $\Delta x$ and l are related in the way we have identified.

 

[casualguitar]

To terms of 2nd order accuracy, they are the same when $\Delta x$ and l are related in the way we have identified.

Thanks Chet. Have I identified this relationship correctly in my post #450 above?

 

[casualguitar]

Hi Chet, my apologies please ignore the above as its not correct. I will revert later this morning on this

 

[casualguitar]

Hi Chet, am I correct in saying -
"If the truncation error on the advection term in the upwind scheme is the same as the dispersion term in the central differencing scheme (when l = delta x/2), then the numerical dispersion associated with the upwind scheme will be equal to the physical dispersion in the central differencing scheme"

I'm having a bit of trouble proving that they are the same mathematically. Working on it though. Just checking that I have the right idea.

Heres what I've attempted so far:

Central Differencing Scheme (CDS) Dispersion Term:
\begin{equation*}
l \frac{\phi_{x+\Delta x/2} (h_{x+\Delta x} - h_x) - \phi_{x-\Delta x/2} (h_x - h_{x-\Delta x})}{\Delta x^2}
\end{equation*}

Upwind Scheme (US) Advection Term:
\begin{equation*}
\frac{\phi_{x-\Delta x/2} h_{x-\Delta x} - \phi_{x+\Delta x/2} h_x}{\Delta x}
\end{equation*}

To determine the truncation error for the advection term in the Upwind Scheme, we can use the Taylor Series expansion around x:

Expanding (h_{x-\Delta x}) using the Taylor series:
\begin{equation*}
h_{x-\Delta x} = h_x - \Delta x \frac{\partial h}{\partial x} + \frac{\Delta x^2}{2} \frac{\partial^2 h}{\partial x^2} - \dots
\end{equation*}

Substituting this into the US advection term and simplifying I think should give the truncation error term:
\begin{equation*}
\frac{\Delta x}{2} \frac{\partial^2 h}{\partial x^2}
\end{equation*}

However I havent been able to get this yet. If I did, then I'd equate this truncation error with the dispersion term in CDS when ($l = \frac{\Delta x}{2}$).

With ($l = \frac{\Delta x}{2}$), the dispersion term in CDS becomes:
\begin{equation*}
\frac{\Delta x}{2} \frac{\partial^2 h}{\partial x^2}
\end{equation*}

This term is the same as the truncation error from the US. This indicates that when $(l = \frac{\Delta x}{2})$, the truncation error in the upwind scheme is equivalent to the physical dispersion modeled by the CDS.

Is this the right track?

 

[casualguitar]

Hi Chet, am I correct in saying -
"If the truncation error on the advection term in the upwind scheme is the same as the dispersion term in the central differencing scheme (when l = delta x/2), then the numerical dispersion associated with the upwind scheme will be equal to the physical dispersion in the central differencing scheme"

I'm having a bit of trouble proving that they are the same mathematically. Working on it though. Just checking that I have the right idea.

Heres what I've attempted so far:

Central Differencing Scheme (CDS) Dispersion Term:
\begin{equation*}
l \frac{\phi_{x+\Delta x/2} (h_{x+\Delta x} - h_x) - \phi_{x-\Delta x/2} (h_x - h_{x-\Delta x})}{\Delta x^2}
\end{equation*}

Upwind Scheme (US) Advection Term:
\begin{equation*}
\frac{\phi_{x-\Delta x/2} h_{x-\Delta x} - \phi_{x+\Delta x/2} h_x}{\Delta x}
\end{equation*}

To determine the truncation error for the advection term in the Upwind Scheme, we can use the Taylor Series expansion around x:

Expanding (h_{x-\Delta x}) using the Taylor series:
\begin{equation*}
h_{x-\Delta x} = h_x - \Delta x \frac{\partial h}{\partial x} + \frac{\Delta x^2}{2} \frac{\partial^2 h}{\partial x^2} - \dots
\end{equation*}

Substituting this into the US advection term and simplifying I think should give the truncation error term:
\begin{equation*}
\frac{\Delta x}{2} \frac{\partial^2 h}{\partial x^2}
\end{equation*}

However I havent been able to get this yet. If I did, then I'd equate this truncation error with the dispersion term in CDS when ($l = \frac{\Delta x}{2}$).

With ($l = \frac{\Delta x}{2}$), the dispersion term in CDS becomes:
\begin{equation*}
\frac{\Delta x}{2} \frac{\partial^2 h}{\partial x^2}
\end{equation*}

This term is the same as the truncation error from the US. This indicates that when $(l = \frac{\Delta x}{2})$, the truncation error in the upwind scheme is equivalent to the physical dispersion modeled by the CDS.

Is this the right track?

I havent yet been able to work this out. Is this a dead end by any chance? I'm not 100% sure that it is the truncation error I should be considering. My apologies for all of the questions on this