EML 5725 Computational Fluid Dynamics Spring 2003
Groups Project 2

1.

Fix all problems with respect of neatness, comments, error evaluation, exact solution drawing, etc with your previous program.

2.

Copy it into a new file cn.f, cn.cpp, or whatever, and copy ftcs.in into cn.in.

3.

Change the cn files to use the Crank-Nicholson finite difference formula instead of the forward-time, central-space (simple explicit) scheme. See the supporting material page for helpful subroutines. From now on, restrict yourself to the case that V₁=1.

4.

You will need to replace the specification of by specification of a . Figure out the value of p for which the CN is scheme is the most accurate for a given number of points. Then test various values of C to figure out for which value of C the scheme is most accurate for the same number of computed points.

5.

Copy the cn files into mf files, and change the Crank-Nicholson formula into the Mitchell-Fairweather one. (To get the MF scheme, replace the second order accurate approximations of the second order x-derivatives by fourth order accurate compact ones. Then multiply the entire finite difference formula by the factor that gets rid of the operators in the bottoms of the compact finite difference formula, and write out in terms of mesh point values.

6.

You will need to replace the specification of by one of the form . Figure out the value of q for which the MF is scheme is the most accurate for a given number of points. Then test various values of D to figure out for which value of D the scheme is most accurate for the same number of computed points.

7.

Hold a competition between the FTCS scheme at N=1/6, the CN scheme, and the MF scheme to see what method can achieve a maximum error at t=0.5 of 0.01 with the least number of total points computed. Same for a maximum error of 0.001, and of 0.0001, and of 0.00001, and of 0.000001.

Hand in the MF code only, and ensure it is extremely neat and very solidly commented. Add evidence supporting the values of p, q, C, and D. Also hand in tables of spatial and temporal number of points, total number of points, and actual maximum error showing the results of the requested competition. Comment on all results.