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EGN 5456 Homework 6 Fall 1996
Numerical Solution of Heat Conduction in a Bar

Problem Statement

Find the temperature distribution T in a bar of length with a heat conduction coefficient . The ends of the bar are kept at zero degrees for all time. However, initially the center of the bar is at a larger temperature . You can assume that this initial temperature decreases linearly to zero at each end. Plot the temperature distribution T for times 0, 0.125, 0.25, 0.375, and 0.5 in a single graph against x.

Analytical Solution

First solve the problem analytically.

• Find the separation of variables solution as a Fourier sine series. Rewrite this solution as a complex Fourier series, just to see that it can be done.
• Use the Fourier sine series solution to create a function subprogram T_ana that takes x, t, , and as arguments and returns the exact temperature at that point and time. Some points to consider:
  • The first time this subprogram is used, it should read the value of from an input file. But do not keep reading it in.
  • Pay particular attention to the best place to terminate the Fourier series. Sum exactly as many terms as needed to get the most accurate solution.
  • Avoid underflow, which may impact performance.
  • Since the series converges slowly for time t=0, you may want to do this time separately.
• Use your function in a computer program to plot the exact solution T(x) in a single graph for times 0, 0.125, 0.25, 0.375, and 0.5. Use a plotting software package of your choice.

Solution

Numerical Solution

Next solve the problem numerically using a finite difference method.

• Discretize the bar into J+1 x-locations spaced apart by defining

  

• Similarly, restrict the times to N+1 discrete times spaced apart by defining

  

  Here is the last time you want to compute.

• You need to determine all (J+1)(N+1) temperature values and then plot the ones corresponding to the requested times. Do that by first initializing the initial values as given. (It is best to do this using your exact solution function; then you can compute a different initial condition using exactly the same program).
• Next compute the values for later times using the formulae

  

  (the left hand side approximates , the right hand side ) This formulae can be used first for n=0 to compute the values of for all . The end values and can be found from the given Dirichlet boundary condition. (Again it is neatest to do this using your exact solution function). Next, now that all values are known for n=1, the values of can be computed for all values of , and so on.

• You will need to check at each time level whether it needs to be output.
• Try a value for J such as 16 or 32.
• Try 3 different values for N. The first value should give (as accurately as possible) , the second , and the third .
• For each of the three cases, make a separate plot of T versus x for times 0, 0.125, 0.25, 0.375, and 0.5. In these graphs, present the exact solution by a continuous line and the computed values by symbols. (You may want to use more than J points to plot the exact solution accurately. About 50 points is a good choice.)
• Comment on the differences in the results.

Follow the recommended programming techniques.
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