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4. Finite Difference Schemes For The Heat Equation
148
The interpretation of this result is that the difference scheme is a stable
dynamical system in the sense that an error in the initial data bounds
the error in the corresponding solutions. Energy arguments can also be
performed for the implicit scheme (4.39). This is discussed in Exercise 4.25
below.
4.6 Exercises
Exercise 4.1 Verify, by direct calculation, that the discrete functions
{w n } given by (4.13) are solutions of (4.4)–(4.5).
Exercise 4.2 Implement the scheme (4.2) for the heat equation and inves-
tigate the performance of the method by comparing the numerical results
with the analytical solution given by
(a) Example 3.1 on page 92.
(b) Example 3.2 on page 93.
(c) Example 3.4 on page 97.

Exercise 4.3 Repeat Exercise 4.2 using the implicit scheme (4.39). Com-
pare the numerical solutions provided by the explicit and the implicit
schemes.
Exercise 4.4 In this exercise we want to study the rate of convergence of
the explicit scheme by doing numerical experiments. Define the error by
m
e ∆ (t m ) = max |u(x j ,t m ) − v |.
j
j=0,... ,n+1
We want to estimate the rate of convergence for the scheme at time t =1/10
for the problem considered in Exercise 4.2 (a).
α
(a) Estimate, using numerical experiments, α such that e ∆ (1/10) = O((∆t) )
for ∆t =(∆x) /2.
2
(b) Repeat the experiments in (a) using ∆t =(∆x) /6.
2
(c) Try to explain the difference in the rate of convergence encountered
in the two cases above. Hint: Consider the truncation error discussed
in Exercise 4.15 below.