Assigned Wednesday, Nov. 12, 2008, due Wednesday, Nov. 17, 2008
[1] Create a program that solves the Euler equations
\[ \begin{array}{c}\dfrac{\partial \vec u}{\partial t} + (\vec u \cdot \nabla ) \vec u = -\nabla P \\ \, \\ {\rm div}(\vec u) = 0 \end{array}\]
in the rectangular region $[-4,4] \times [-4,4]$ with with an initial velocity corresponding to the vorticity distribution
of Assignment 6;
\[
\omega(x,y,0) = \left\{
\begin{array}{c|c}
( 1 - ((x-2)^2 + y^2))^3 & (x-2)^2 + y^2 \leq 1 \\
( 1 - ((x+2)^2 + y^2))^3 &(x+2)^2 + y^2 \leq 1 \\
0 & elsewhere
\end{array}
\right.
\]
and $\vec u \cdot \vec n = 0 $ on the boundary of the computational domain.
Your program should implement a projection method that uses a staggered grid velocity representation on a uniform rectangular grid with m panels in the x-direction and n panels in the y-direction. Use either 6.2.7a or 6.2.7b on page 147 of Peyret and Taylor to approximate the convection term.
I've created some of the routines that are required to construct your program. These routines are available in the StaggeredGrid repository at https://www.math.ucla.edu/~anderson/270eSupport/. There is also online documentation for these routines. Briefly, there are four functions defined in StaggeredGridUtilities.h,
[2] Verify the accuracy of convergence rates with respect to both time and spatial discretization of your program. Describe the tests you performed and turn in the data that supports your conclusion that your program is working correctly.
[3] Compute the solution to time t = 200. and create a contour plot of the vorticity distribution. Compare
your results with those obtained in Assignment 6. Which method do you think is better? How are you judging the
results?