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 $i-1$ 0 100 $i$ 10 110 $i+1$ 9 105 $i+2$ -10 100
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.
 05.29.17
 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 $u$ $a$ $i-1$ 0 100 $i$ 0 110 $i+1$ 0 105 $i+2$ 0 100
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 $x$ $u$ 1 0.0 4 2 0.1 3 3 0.2 3 4 0.3 4 5 0.4 5 6 0.5 8 7 0.6 11 8 0.7 12
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
 1. 2300, 0 m$^2$/s$^2$; 20190, -10080 m$^2$/s$^2$. 2. 23928.125, -11692.1875. 3. 4.5, $\frac{1465}{326}$, $10 \frac{26569}{212552}$.
 $\pi$