Computational Aerodynamics Assignment 6 — Flux Discretization I
 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
Derive a 5th order accurate WENO discretization of $u_{i+1/2}$ for the advection equation with $a>0$. Specifically, do the following:
 (a) Find the three third-order accurate 2nd degree polynomials $P_1(x)$, $P_2(x)$, $P_3(x)$ obtained from $u_{i-2}$, $u_{i-1}$, $u_{i}$, $u_{i+1}$, and $u_{i+2}$. (b) Using $P_1(x)$, $P_2(x)$, $P_3(x)$ found in (a), evaluate $u_{i+\frac 12}^{(1)}$, $u_{i+\frac 12}^{(2)}$, $u_{i+\frac 12}^{(3)}$. (c) Find a fifth-order accurate 4th degree polynomial $P_4(x)$ that goes through the nodes mentioned in (a). (d) Determine the optimal weights $\gamma_1$, $\gamma_2$, $\gamma_3$. (e) Determine the coefficient of smoothness $\beta_1$ associated with $P_1(x)$. (f) Express $u_{i+1/2}$ as a function of $\beta_j$ and $u_{i+1/2}^{(j)}$ with $j=1,2,3$.
 05.17.18
 2 $u^{(1)}_{i+\frac 12} = \frac{3}{8} u_{i-2} - \frac{5}{4} u_{i-1} + \frac{15}{8} u_{i}$, $u^{(2)}_{i+\frac 12} = -\frac{1}{8} u_{i-1} + \frac{3}{4} u_{i} + \frac{3}{8} u_{i+1}$, $u^{(3)}_{i+\frac 12} = \frac{3}{8} u_{i} + \frac{3}{4} u_{i+1} - \frac{1}{8} u_{i+2}$, $\gamma_1=\frac{1}{16}$, $\gamma_2=\frac{5}{8}$, $\gamma_3=\frac{5}{16}$, $\beta_1 = \frac{1}{3} \left( 4 u_{i-2}^2-19u_{i-2}u_{i-1}+25u_{i-1}^2+11u_{i-2}u_i -31u_{i-1}u_i+10u_i^2 \right)$
 $\pi$