Computational Aerodynamics Assignment 6 — Flux Discretization
 Question #1
Starting from the scalar advection equation: $$\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x}=0$$ and assuming a negative wave speed: $$a<0$$ show that when limiting the second order terms such that they adhere to the rule of the positive coefficients, a second-order upwinded slope-limited scheme can be obtained as: $$u_{i+1/2}=u_{i+1}+ \phi_{i+1/2}\frac{1}{2} (u_{i+1}-u_{i+2})$$ with the limiter function: $$0\le \phi_{i+1/2}\le 2 r_i$$ and the ratio of successive gradients: $$r_i=\frac{u_i-u_{i+1}}{u_{i+1}-u_{i+2}}$$
 05.26.17
 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: $$\Lambda=\left[ \begin{array}{cc} u & 0 \\ 0 & u-a \\ \end{array} \right] ~~~~~ L=\left[ \begin{array}{cc} 1 & 2 \\ 0 & 1 \\ \end{array} \right] ~~~~~ U=\left[ \begin{array}{c} u\\ a \end{array} \right]$$ 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.
 05.30.17
 Question #3
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.
 05.17.18
 2. 2300, 0 m$^2$/s$^2$; 20190, -10080 m$^2$/s$^2$. 3. 23928.125, -11692.1875.
 $\pi$