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I'm reading this paper and trying to follow everything that the author has done.

A Comparative Study of Finite Volume Method and Finite Difference Method for Convection-Diffusion Problem

But there is a problem. I cannot derive the following equation (third line in equation (6) of [1]) for two reasons:

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  1. I cannot match the expression marked by "1" in red.
  2. The signs are opposite for the part marked by "2"

Does anybody know how the author has derived this equation?

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  • $\begingroup$ @AntonMenshov Are you sure that it is finite-difference? isn't it finite-volume? $\endgroup$ – Roh Mar 17 '18 at 6:49
  • $\begingroup$ The paper has sections on both (btw, inside the paper it has both tags), and your question is pretty independent from FV and TD in particular, so I considered it fair to add an FD tag. $\endgroup$ – Anton Menshov Mar 17 '18 at 6:52
  • $\begingroup$ ....Or if you are talking about the title for the question -> it's your call. You can improve my edit, I just want it to be representative and readable. $\endgroup$ – Anton Menshov Mar 17 '18 at 6:53
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Initially, the equation (6) is derived from equation (4):

$$ F_e\Phi_e-F_w\Phi_w=D_e(\Phi_E-\Phi_P)-D_w(\Phi_P-\Phi_W) $$ under the central differencing approximation $\Phi_e=(\Phi_P+\Phi_E)/2$, $\Phi_w=(\Phi_W+\Phi_P)/2$. Naturally, (4) becomes the first line of (6):

$$ \frac{F_e}{2}(\Phi_P+\Phi_E)-\frac{\color{red}{F_w}}{2}(\Phi_W+\Phi_P) = D_e(\Phi_E-\Phi_P)-D_w(\Phi_P-\Phi_W) $$

Notice, there is probably a typo (highlighted in red) in the first line of the equation (6) as in the second term $F_w$ is mistyped with $F_e$. Otherwise, those terms just simplify and the second line of (6), where $F_w$ magically appears again does not make any sense. Nevermind, the proposed by authors continuity equation $F_e-F_w=0$ equate them anyway.

Next, naturally, we will move all the terms with $\Phi_P$ to the left-hand side, and everything else to the right-hand side and group the coefficients in front of $\Phi_P$, $\Phi_W$, and $\Phi_E$ in a convenient form. So,

$$ \frac{F_e}{2}\Phi_P-\frac{F_w}{2}\Phi_P+D_e\Phi_P+D_w\Phi_P =-\frac{F_e}{2}\Phi_E+\frac{F_w}{2}\Phi_W+ D_e\Phi_E+D_w\Phi_W $$

finally resulting in the second line of (6), where the sign seems to be wrong, as far as I understand, (highlighted in red):

$$ \left[\underbrace{\left(D_w-\frac{F_w}{2}\right)+\left(D_e\color{red}{+}\frac{F_e}{2} \right)}_{*}\right]\Phi_P =\left(D_e-\frac{F_e}{2}\right)\Phi_E+\left(D_w+\frac{F_w}{2}\right)\Phi_W $$

No, by adding $(F_w-F_e)$ to (*) (I can do it since by continuity equation the added expression is 0):

$$ \left[\left(D_w-\frac{F_w}{2}+F_w\right)+\left(D_e+\frac{F_e}{2} -F_e\right)\right]\Phi_P =\left(D_e-\frac{F_e}{2}\right)\Phi_E+\left(D_w+\frac{F_w}{2}\right)\Phi_W $$ I can arrive to: $$ \left[\left(D_w+\frac{F_w}{2}\right)+\left(D_e-\frac{F_e}{2}\right)\right]\Phi_P =\left(D_e-\frac{F_e}{2}\right)\Phi_E+\left(D_w+\frac{F_w}{2}\right)\Phi_W $$ and now by adding $F_e-F_w$ I arrive to the final third line of (6): $$ \left[\left(D_w+\frac{F_w}{2}\right)+\left(D_e-\frac{F_e}{2}\right)+(F_e-F_w)\right]\Phi_P =\left(D_e-\frac{F_e}{2}\right)\Phi_E+\left(D_w+\frac{F_w}{2}\right)\Phi_W $$

It seems like the final expression of (6) in the paper you cited is correct (provided (4) is correct, continuity equation is physical and central differencing is taken care of - which it probably is). However, the derivation that they have shown in the first two lines of equation (6) contains $\approx 1.5$ confusing bugs. 1.5 - because $F_e$ is technically equal to $F_w$.

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