EGN 5456 Project Fall 1999

This project is 25% of your grade.

Solve the following convection equation:

u_t + a u_x + b u_y = u

with a=2 and b=0.5, in a rectangular domain of length and height h=1, for . Take as initial condition:

and as boundary conditions:

Derive the exact solution for the P.D.E. in general and in particular for this special case.

Then write a Peaceman-Rachford scheme to compute a finite difference solution.

Optimize the ratio of mesh size to time step for a given computational effort (= points computed) to give best results for a given effort.

Next create a Mitchell-Fairweather scheme from the Peaceman-Rachford one.

Optimize the ratio of square mesh size to time step for the same computational effort as for the Peaceman-Rachford scheme to give the best results.

See whether the Mitchwell-Fairweather scheme gives better results than the Peaceman-Rachford

Increase the effort by a factor 10 and compare again.

Program requirements:

• One input file. No exceptions.
• One output file (additional output files for plotting as needed permitted.) No exceptions.
• Read from the input file and print to the output file: a, b, , h, , , maximum Courant number (taken as 1), , and the (taken as 5) desired output times (0, 0.25, 0.5, 0.75, 1). No exceptions.
• All input variables checked for proper sign, value, whether program limits are exceeded, etcetera.
• Program must work correctly when the parameters in the input file are changed. No exceptions.
• Exactly the following subroutines/functions (no more, no less, no exceptions):
  • main does the actual initialization and time advance.
  • output does the output of the computed solution at the requested times and writing to the plot files.
  • exact is the exact solution, and should be used to set the initial and physical boundary conditions.
  No exceptions.
• Each subroutine/function in a separate file. A makefile must also be provided.
• The only two-dimensional, (j,l), array storage that may be used are , , and . All other storage must be scalar or one-dimensional. No exceptions.
• The coefficients of the tridiagonal system may not be stored in an array, only the processed bidiagonal coefficients C^* and D^* may be stored as arrays. No exceptions.
• No common blocks. No exceptions.
• No reading from the input file except in main. No exceptions.
• All array storage declared in main using symbolic constants (C) or parameters (Fortran) for the bounds. No exceptions.
• Print out rms and maximum errors whenever output is done.
• No large tables of numbers.
• Heavily commented so that anybody can easily read, understand, and modify it.
• One time march from 0 to 1 does all five output times. No exceptions.
• Extremely neat.
• Satisfies good programming requirements.

If any requirement marked as ``No exceptions'' is not met, the maximum assigned for the program is 10%. (Total for the project 50% or less.)

Contour lines of constant temperature are a good way to plot the temperature evolution. Otherwise you might want to plot the temperature evolutions at and the ones at as six individual graphs.