2. (Finite Difference Schemes for the One-Dimensional Heat Equation)

An interesting and useful application of the Excel driver is in displaying the values of the approximate solutions (using FDM in this case) to the heat equation as discussed in Chapter 13. In particular, I personally have found it useful for the following cases:

 C1: Having written FD code, we wish to see if it is producing correct results. To this end, displaying the results in an easy format is a boon.

 C2:Wewould like to compare the accuracy of the ADE family of schemes with a baseline case such as the Crank–Nicolson method or an analytical solution of a PDE.

 C3: To display the accuracy of an approximate method as a function of some independent variable. An example is plotting option price in the binomial method as a function of the number of steps. The same exercise is applicable to the finite difference method.

Answer the following questions:

a) Use the two ADE schemes that we introduced in Chapter 13 to approximate the solution of the heat equation (see equation (13.14) in Chapter 13).

b) Compare the solutions in part

a) with the exact solution (13.15) and the solution produced by the Crank–Nicolson method.

c) Display the four results from parts

a) and

b) as four curves in a single Excel sheet.
Compare the results.

d) Experiment with various initial conditions in equation (13.14) and repeat steps a), b)
and c).