Computational Aerodynamics Questions & Answers  
Question by Prasanna
Professor, I am a bit confused about question #5 of Assignment 4. I have to find $$\frac{\partial F_3}{\partial U_4}=\frac{\partial (\rho u^2 +P)}{\partial U_4}=\frac{\partial (\rho u^2)}{\partial U_4}+\frac{\partial P}{\partial U_4}$$ and the difficult part seems to be in determining the $\frac{\partial P}{\partial U_4}$ term. The alternative method you taught involves using the chain rule, for example, $$\frac{\partial F_3}{\partial U_4}=\frac{\partial F_3}{\partial \rho_1} \frac{\partial \rho_1}{\partial U_4}+\frac{\partial F_3}{\partial \rho_2} \frac{\partial \rho_2}{\partial U_4}+\frac{\partial F_3}{\partial u} \frac{\partial u}{\partial U_4} +\frac{\partial F_3}{\partial \phi} \frac{\partial \phi}{\partial U_4} $$ where $F_3=F_3(\rho_1,\rho_2,u,\phi)$ and $\phi$ is some variable. For this problem, $\phi$ has to be a function of $U_1,U_2,U_3,U_4$ and also a function of P such that I can evaluate $\frac{\partial \phi}{\partial U_4}$. But then if I could express P in terms of $\phi$ which is a function of $U_1,U_2,U_3,U_4$, I would use the first method which you taught to evaluate the flux jacobian terms instead. I would like your comment regarding this.
You don't necessarily need to express $\phi$ as a function of $U$ to determine $\partial \phi/\partial U$ in the same way as you don't need to express $F$ as a function of $U$ to obtain $\partial F/\partial U$.
Question by Van Tien
Professor, in Assignment 5, Question #3, for the extrapolation, I am confused to use the 1D-Lagrange interpolation or 2D-Lagrange interpolation. In the case of 1D-Lagrange interpolation, I am not sure the polynomial function is based on x or y coordinate. In the case of 2D-Lagrange interpolation, I think I need more information from the other nodes.
Use a 1D extrapolation polynomial. 2D is too time consuming to compute.
Question by Student 201983196
Professor, in Assignment7, Question#2, How do I calculate this equation, $\frac {\mid{A}\mid (Z_L,Z_R)}{2}(U(Z_R)-U(Z_L))$? Is $\frac {\mid{A}\mid (Z_L,Z_R)}{2}$ 2X1 matrix? But $(U(Z_R)-U(Z_L))$ also is 2X1 matrix? I don't know how to calculate $F_(i+\frac{1}{2})=\frac{F(Z_L)+F(Z_R)}{2}+\frac {\mid{A}\mid (Z_L,Z_R)}{2}(U(Z_R)-U(Z_L))$
No, $|A|$ is a $2 \times 2$ matrix determined from an average state function of $Z_L$ and $Z_R$.
Question by Prasanna
Professor, for Assignment #7, Question #3, are the answers posted in the following order: $u_L$,$u_R$ in decimal digits, $u_R$ in fraction, $f(u_L)$ in decimal digits, $f(u_L)$ in fraction respectively?
I updated the answers to make them more clear.
Question by Student 201983196
Professor, In Assignment#7, Question#2 (b), I use 2nd order polynomial about node(4,5,6) and node(5,6,7). and then using optimal weight, calculate $u_R$. but my solution is wrong. I don't know how to solve this Question#2(b).
Hm, I see a problem in your approach. You shouldn't be finding a polynomial when determining the flux with a TVD minmod2 limiter.
Question by Student 201627128
Professor, in class when you explained how to find WENO3, you found a highest degree polynomial through the data points. Using a similar approach I was able to find $u_L$ equal to 4.5 as in the solutions, however, when I apply the same strategy to find $u_R$, I get 4.25 instead, which does not match the solution. I tried to do it in reconstruction evolution and again found 4.5 for $u_L$ but this time $u_R$ becomes 5. Is there a separate approach to find $u_R$?
I don't understand why using reconstruction-evolution would give you a different answer. You need to find $u_{\rm R}$ by interpolating $u$. Once $u$ is interpolated and $u_{\rm L}$ and $u_{\rm R}$ are found, then apply reconstruction evolution.
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