Page 340 -

7.5. The Maximum Principle for the One-Step-Theta Method 323

According to the above considerations, it is nonexpansive only if λτ ≤ 1
holds. For large numbers λ, this is a very restrictive step size condition; see
also the discussion of (7.85) to (7.87).
Due to their better stability properties, implicit methods such as the
Crank–Nicolson and the implicit Euler methods do not have such step
size restrictions. Nevertheless, the application of implicit methods is not
free from surprises. For example, in the case of large numbers λ, an order
reduction can occur.

Exercises

7.10 Determine the corresponding domain of stability S R of the one-step-
1
theta method for the following values of the parameter Θ : 0, , 1.
2
7.11 Show the L-stability of the implicit Euler method.

7.12 (a) Show that the discretization
n
ξ = ξ n−2 +2τf(t n−1 ,ξ n−1 ) , n =2,...N
(midpoint rule), applied to the model equation ξ = f(t, ξ)with

f(t, ξ)= −λξ and λ> 0 leads, for a sufficiently small step size
τ> 0, to a general solution that can be additively decomposed into a
decaying and an increasing (by absolute value) oscillating component.
(b) Show that the oscillating component can be damped if additionally
the quantity ξ N is computed (modified midpoint rule):
∗
'
N
ξ N = 1 & ξ N + ξ N−1 + τf(t N ,ξ ) .
2
∗
7.13 Let m ∈ N be given. Find a polynomial R m (z)=1 + z +

m j
j=2 γ j z (γ j ∈ R) such that the corresponding domain of absolute sta-
bility for R(z):= R m (z) contains an interval of the negative real axis that
is as large as possible.

7.5 The Maximum Principle for the
One-Step-Theta Method

In Section 1.4 we have seen that for a discrete problem of the form (1.31)
there is a hierarchy of properties ranging from a comparison principle to
a strong maximum principle, which is in turn applied by a hierarchy of
conditions, partly summarized as (1.32) or (1.32) . To remind the reader,
∗
we regroup these conditions accordingly: