Computational Aerodynamics Assignment 7 — Flux Discretization II  
Question #1
Consider a system of equations $\partial U/\partial t+\partial F/\partial x=0$ with $F=AU$, $A=L^{-1}\Lambda L$ and with: $$ \Lambda=\left[ \begin{array}{cc} u & 0 \\ 0 & u-a \\ \end{array} \right] ~~~~~ L=\left[ \begin{array}{cc} 1 & 2 \\ 0 & 1 \\ \end{array} \right] ~~~~~ U=\left[ \begin{array}{c} u\\ a \end{array} \right] $$ The node properties correspond to:
Node$u$, m/s$a$, m/s
$i-1$0100
$i$10110
$i+1$9105
$i+2$-10100
Do the following:
(a)  Find $F_{i+1/2}^+$ with a minmod2 limiter 2nd-order FVS scheme.
(b)  Find $F^-_{i+1/2}$ with a minmod2 limiter 2nd-order FVS scheme.
(c)  Find $F_{i+1/2}$ with a minmod2 limiter 2nd-order FVS scheme.
05.30.17
Question #2
Consider a system of equations $\partial U/\partial t+\partial F/\partial x=0$ with $F=AU$, $A=L^{-1}\Lambda L$ and with: $$ \Lambda=\left[ \begin{array}{cc} u & 0 \\ 0 & u-a \\ \end{array} \right] ~~~~~ L=\left[ \begin{array}{cc} 1 & 2 \\ 0 & 1 \\ \end{array} \right] ~~~~~ U=\left[ \begin{array}{c} u\\ a \end{array} \right] ~~~~~ F=\left[ \begin{array}{c} u^2+2a^2\\ a(u-a) \end{array} \right] $$ The node properties correspond to:
Node$u$$a$
$i-1$0100
$i$0110
$i+1$0105
$i+2$0100
For the primitive variable vector set to: $$ Z=U=\left[ \begin{array}{c} u\\ a \end{array} \right] $$ and using a second-order-upwind slope-limited FDS scheme with the minmod2 limiter, reconstruction evolution, and arithmetic averaging, do the following:
(a)  Find the primitive variable vector on the left and right sides of the interface, $Z_{\rm L}$ and $Z_{\rm R}$.
(b)  Find the flux at the interface $F_{i+1/2}$.
Note: both $u$ and $a$ are non-dimensional.
Question #3
You wish to solve numerically the following scalar equation: $$ \frac{\partial u}{\partial t} + \frac{\partial f}{\partial x}=0 $$ with $f=\frac{1}{2}u^2$. At a certain time level, $u$ corresponds to:
Node$x$$u$
10.04
20.13
30.23
40.34
50.45
60.58
70.611
80.712
Using a WENO 2nd-3rd order interpolation of the primitive $u$ reconstructed over a FDS scheme with arithmetic averaging and with optimal weights set to $\gamma_0=\frac{1}{3}$ and $\gamma_1=\frac{2}{3}$, it is desired to find the flux at the interface between node 4 and node 5, i.e. $f_{4.5}$. For this purpose, do the following:
(a)  Find $u_{\rm L}$ between node 4 and 5 using WENO3.
(b)  Find $u_{\rm R}$ between node 4 and 5 using WENO3.
(c)  Find $f_{4.5}$ using FDS with $u_{\rm L}$ and $u_{\rm R}$ found in (a) and (b).
05.17.18
Answers
1.  2300, 0 m$^2$/s$^2$; 20190, -10080 m$^2$/s$^2$.
2.  23928.125, -11692.1875.
3.  4.5, $\frac{1465}{326}$, $10 \frac{26569}{212552}$.
Due on May 30th at 16:30. Do Questions #2 and #3 only.
05.23.19
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