Homework 4: Approximation of PDEs

The good news is that you aren't doing any programming this time.  The bad news is:


Part A

You are attempting to solve a conduction problem involving a rod with internal heat source and cooling at the surface by a fluid bath. Within the rod the conductivity (k) can be assumed constant and the conduction equation is:

Heat flux at the surface from the metal rod to the fluid is given by the expression:

where for example we might have:

Tfluid = 400K

and

The rod is divided into the following set of finite volumes:

Construct the finite volume approximation to the conduction equation for:

In the class examples and the second homework, the flux term used at the rod surface was only first order accurate.  Use what you've learned about generating finite difference approximations to produce a second order approximation to the flux terms at the surface of this rod.  Apply the method of undetermined coefficients to obtain a second order approximation to the temperature derivative at the surface, using volume cetered temperatures.  Next apply flux matching to evaluate the surface temperature as a function of volume centered temperature, h, and the fluid temperature.  Substitute your expression for Tsurf into h*(Tsurf-Tfluid) to get a final expression for the surface flux, needed by the outer ring finite volume conduction equations.  Show all of these steps in your homework report.

Clearly define any notation that you introduce. Clearly show all steps and define additional equations (e.g. flux matching, special boundary conditions) needed to obtain the same number of equations as unknown temperatures. How many unknown temperatures have you introduced for the specific cylindrical grid above?  

Part B

For the same conduction problem use Taylor series expansions to construct a difference approximation at the interior nodal point circled below, that is at least third order accurate. Remember that for this approach we are only evaluating temperatures at the point where the grid lines cross.


Show the steps in your derivation. Clearly define any notation that you introduce and mark on a figure the location of any temperatures that you use. Give the expression for the error term, showing only the terms involving the lowest powers of mesh spacing.
 

Note: You should use Mathmatica, Maple, or MathCad to work through this problem. If you do, provide sufficient commented output from the program to show and explain all of the steps that you went through in generating the solution.  PDF output from these programs is best for me.  However, I can read any native Mathematica file and MathCad .mcd files, saved in MathCad 2001 or older format.