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: $$ r_i=\frac{u_i-u_{i+1}}{u_{i+1}-u_{i+2}} $$
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 $j=1,2,3$.
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) $
Due on Thursday May 23rd at 16:30. Do both questions.
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