Introduction to CFD 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$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 #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$$a$
$i-1$0100
$i$0110
$i+1$0105
$i+2$0100
For the primitive variable vector set to: $$ 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.
05.17.18
Answers
2.  2300, 0 m$^2$/s$^2$; 20190, -10080 m$^2$/s$^2$.
3.  23928.125, -11692.1875.
Due on Thursday May 24th at 16:30. Do all 3 questions.
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