Page 296 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 12 PARTIAL DIFFERENTIAL EOUATIONS 273

Cells K3:K9 contain constants used in the calculations; these cells were
assigned the names shown in parentheses in column M. The formulas in cells
K6, K7, K8 and K9 are, respectively
=k/( hcap*rho) (coefficient k in general PDE, equation 12-22)

=D12-C12 (AX)
=B14-B13 (At)

=e*Dt/( DxA2) (f)
[In the spreadsheet, the range name f was used for the parameter Y in equation
12-26, since r can't be used as a name in Excel.]
The values in cells on the edges of the table of temperatures (column C and
column M) are the constant temperature values at the ends of the rod; the values
in row 13 are the initial temperature of the interior of the rod. The formula in the
remaining cells in the body of the temperature table (D14:L113) is based on
equation 12-22. For example, the formula in cell D14 is
=f*( C 1 3+E 1 3)+( 1 -2*f)*D 1 3

Experience has shown that the factorfmust be less than 1/2 in order to avoid
instability in the calculations. For a given problem, this requires adjustment of
both Ax and At.

50

$ 30
e
.c..
;
a,
20
:
10

0
0 20 40 60 80 100
Time, seconds

Figure 12-6. Temperature vs. time in a brass rod.
(folder 'Chapter 12 (PDE) Examples, workbook 'Parabolic PDE, sheet 'Temp distribution')