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4. Finite Difference Schemes For The Heat Equation
152
(b) For what mesh sizes is this scheme stable in the sense of von Neu-
mann?
Exercise 4.14 Consider the equation
u t = u xx − 9u,
with Dirichlet-type boundary conditions. Derive an explicit and an implicit
scheme for this equation. Use the von Neumann method to investigate the
stability of the methods.
Exercise 4.15 In Section 2.3.5 we introduced the concept of truncation
error for a finite difference approximation for a two-point boundary value
problem. Here we shall discuss a similar concept for difference approxima-
tions of the heat equation.
Observe that the scheme (4.4) can be written in the form
1 m m
(v m+1 − v )+ Av =0,
∆t
where A ∈ R n,n is given by (4.16).
n
The truncation vector τ m ∈ R is given by
1 m m
m
τ = (u m+1 − u )+ Au ,
∆t
n
where u m ∈ R is given by u m = u(x j ,t m ) for a solution of the continuous
j
problem (4.1).
(a) Show that under suitable smoothness assumptions on the solution u,
m

|τ | = O ∆t + O (∆x) . (4.48)
2
j
In rest of this exercise we study a more general difference scheme of the
form
1 m m
(v m+1 − v )+ θ(Av m+1 )+(1 − θ)Av =0, (4.49)
∆t
where θ ∈ [0, 1] is a parameter. Note that if θ = 0, this corresponds to the
explicit scheme (4.4), while if θ = 1, it corresponds to the implicit scheme
studied in Chapter 4.4 above.
(b) Sketch the computational molecule for the scheme when θ ∈ (0, 1).
(c) Show that for all θ ∈ [0, 1] the estimate (4.48) holds, and that the
choice θ =1/2 leads to an improved estimate of the form

m

|τ | = O (∆t) 2 + O (∆x) .
2
j
(Hint: Consider Taylor expansions at the point (x j , (t m+1 + t m )/2).)