Introduction to CFD Questions & Answers
 Question by Student 201427564 Professor, when you explain about 'Roe Average', you wrote like this. ${ A }_{i+ \frac {1} {2}} ({U}_{i+1} - {U}_{i}) = {F}_{i+1} - {F}_{i}$. I can not understand that how can it be possible that ${A}_{i+ \frac {1} {2}} {U}_{i+1} = {F}_{i+1}$ and ${A}_{i+ \frac {1} {2}} {U}_{i} = {F}_{i}$ . Because subscript of A and U are different each other.
 05.29.17
Hm, I don't understand the question. You need to explain better what you don't understand..
 Question by Student 201427564 Oh.. I mean How can it be possible that ${A}_{i+ \frac {1} {2}} {U}_{i+1} = {F}_{i+1}$ rather than ${F}_{i+\frac {1} {2}}$ . Are there any rules?
 05.30.17
I'm not sure what you mean. But it is not correct that $A_{i+1/2}U_{i+1}=F_{i+1}$. This is not the Roe average. The Roe average if $\Delta F= A_{i+1/2}\Delta U$. I'll give you 0.5 point for the effort.
 Question by Student 201427102 Professor, I'm confused. For finding ${F^{-}_{i+1/2}}$, you used below form. $$r^{-}_{i}=\frac {F^{-}_{i}-F^{-}_{i+1}}{F^{-}_{i+1}-F^{-}_{i+2}}$$ According to table, this form for $alpha < 0$. But $alpha > 0$ at Q#2 of assign.#8.
 05.31.17
What does “alpha” mean? I didn't use this in class. You need to rephrase your question and use the same symbols as used in class. Or, define clearly a new symbol you are introducing.
Question by Student 201427102
Professor, I'm confused. For finding ${F^{-}}_{i+1/2}$, you used below form. $$r^{−}_{i} = \frac{{F^{−}}_{i}−{F^{−}}_{i+1}}{{F^{−}}_{i+1}−{F^{−}}_{i+2}}$$ According to table, this form for a< 0. But a>0 at Q#2 of assign .
Now I see what you mean. When dealing with a system of equations, the wave speeds are not necessarily $a$ or $u$. Rather, the wave speeds are the eigenvalues. So, when determining $F^\pm_{i+1/2}$, the wave speeds are within $\Lambda^\pm$ (those play the same role as the “$a$” does for a scalar equation). 1.5 point bonus.
 Question by Student 201427564 Professor, I have a question about $\phi(r)$. In the table, $\phi(r)=max(0, min(1,r))$. But last class, you wrote like this. ${({\phi^{+}}_{i+ \frac {1} {2}})}_{1} = max(0,min(1, 2{r^{+}}_{i}))$ Why did you use $2{r^{+}}_{i}$ rather than ${r^{+}}_{i}$ ?
 06.01.17
Both $\phi=\max(0,\min(1,r))$ and $\phi=\max(0,\min(1,2r))$ respect the rule of the positive coefficients and reduce to first order at extrema and are hence valid. But $\phi=\max(0,\min(1,2r))$ is better because it is closer to the second-order stencil. 1.0 point bonus.
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