Introduction to CFD Assignment 8 — 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: $$\phi_{i+1/2}=\max(0,~\min(1,2r_i))$$ 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 slope-limited 2nd-order FVS scheme. (b) Find $F^-_{i+1/2}$ with a slope-limited 2nd-order FVS scheme. (c) Find $F_{i+1/2}$ with a slope-limited 2nd-order FVS scheme.
 05.30.17
 Due on Thursday June 1st at 16:30
 $\pi$