MAE 766, Computational Fluid Dynamics
Spring 2017
Take Home Finals
Handed out: April 24, 2017
Due: May 2, 2017
Name:_________________________________________________________________
Instructions:
1. Write your name on the first sheet (this page) of your exam.
2. Work out the answers in the white space below each question and on the facing blank
pages of this booklet.
Problem 1
The linear advection equation can be recovered from the compressible Euler equations by
setting the following conditions: ρ=ρ(x), V(x)=constant (>0), and p(x)=constant.
1) Show that all three equations (mass, momentum, and energy) become simply an
advection equation.
2) What is the numerical flux function at the interface i+1/2 for the density (mass)
equation using the upwind scheme?
3) Assume that flow is supersonic flow. What is the numerical flux function at the
interface i+1/2 for the density (mass) equation using the van Leer flux vector splitting
scheme? Is it the same as the one from the upwind scheme?
4) Assume that flow is subsonic flow. What is the numerical flux function at the interface
i+1/2 for the density (mass) equation using the van Leer flux vector splitting scheme? Is
it the same as the one from the upwind scheme?
Problem 2.
Consider solving the classical 1D Sod-shock tube problem with the following initial
conditions:
ρ=1,
u=0, p=1
for 0≤x<0.5
ρ=0.125, u=0, p=0.1
for 0.5<x≤1
1
This question is chosen to illustrate the importance of a limiter for any higherorder reconstructed finite volume methods. Show that the second order finite
volume method with a centered slope reconstruction, where the interface values
are approximated by a linear interpolation between one upstream and one
downstream cell, will lead to the breakdown of the numerical solution.
(Hint: Prove that the reconstruction scheme will lead to negative density at the
interface.)
2
This question is chosen to illustrate the robustness of the DG methods. The DG
method is able to obtain a stable solution for this problem, although oscillations
do occur in the vicinity of the discontinuities. Obtain a second-order DG solution
to this problem at t=0.2 using a three-stage TVD Runge-Kutta time discretization
scheme with a fixed time step of 0.0004 on a uniform grid of 100 cells. You can
choose your preferred numerical flux among van Leer, Steger-Warming, Roe,
HLLC, and LDFSS. (Hint: To debug your program, you can apply your program
to solve the linear advection equation in the context of the full Euler equations by
setting the following initial conditions: ρ=1+exp(-200(x-0.5)(x-0.5)), u=1, p=10
with the periodic boundary conditions. Note also that a hard limiter for pressure is
required in order to prevent the pressure from becoming negative. This is done by
setting the pressure, p=min(p, ε), where ε is a small number (=1.0e-06, for
example).
Problem 3
Consider to solve the 1D Euler equations using van Leer flux function and one-stage
Runge-Kutta time discretization on a uniform grid of 100 cells.
a: Using DG(P0) method for the advection problem with the periodic boundary
conditions and the following two initial conditions:
Subsonic flow:
ρ=2, u=1, p=2
for 0.25<x<0.75
ρ=1, u=1, p=2
for 0.00<x<0.25 and 0.75<x<1.0
Supersonic flow:
ρ=2,
ρ=1,
u=1, p=0.2
u=1, p=0.2
for 0.25<x<0.75
for 0.00<x<0.25 and 0.75<x<1.0
Compute the solution at t=2. Plot the density profiles and compare these two numerical
solutions with the exact solution.
b: Using DG(P0) and DG(P1) methods for Lax-Harden shock tube problem with the
extrapolation boundary conditions and following initial conditions:
ρ=0.445,
u=0.698876404, p=3.52773
for 0<x<0.5
ρ=0.5,
u=0,
p= 0.571
for 0.5<x<1
Compute the solution at t=0.15. Plot the density, Mach number, and pressure profiles and
compare these two numerical solutions with the exact solution.
Problem 4
Derive explicitly the form of the mass matrix for a nodal-based DG(P2) method using the
standard finite element shape function for a 2D triangular element.
Problem 5
Solve the following elliptic problem (boundary value problem) using DG(P1) methods
⎧ d 2 u
⎪− 2
⎪
⎪ dx
⎨
⎪
⎪
⎪
⎩
= −10 in (0,1)
u x =0 = 0
du
=0
dx x =1
a. What is the exact solution ? Sketch the exact solution.
b. On a non-uniform grid of 3 points (2 cells) (i.e., x1=0, x2 = h, x3 = 1), compute the
numerical solution using BR2 scheme or the recovery method or the DDG method. You
might want to consider using the finite element shape function basis, if you decide to use
BR2 scheme.
c. Should the DG solution coincide with the exact solution at these three grid points?
Why ?