Introduction to CFD Assignment 7 — 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.