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7.4. Stability 321

Till now we have considered only homogeneous boundary data and right-
hand sides. At least for the one-step-theta method this is not a restriction:
Theorem 7.26 Consider the one-step-theta method under the assumption
of Theorem 7.24,with λ i ≥ 0,i =1,... ,M, and with τ such that the
method is stable. Then the solution is stable in initial condition u 0 and
right-hand side q in the following sense:
h
n

n
u h ≤ u 0 h + τ q t k − Θτ . (7.93)
h h h
k=1
Proof: From the solution representation (7.70) we conclude that
n
n−k −1
n
n
u h ≤ E h,τ u 0 h +τ E h,τ (I +τΘA h )
h
h h h h q (t k −Θτ) h
k=1
(7.94)
using the submultiplicativity of the matrix norm.
We have the estimate

(I +ΘτA h ) −1 w i h =

1

w i h ≤ w i h ,

1+ Θ τλ i
and thus as in the proof of Theorem 7.24, (1) ⇒ (3),
(I +Θ τA h ) −1 h ≤ 1
concludes the proof.

The stability condition requires step size restrictions for Θ < 1 , which
2
have been discussed above for Θ = 0.
The requirement of stability can be weakened to
E h,τ h ≤ 1+ Kτ (7.95)

for some constant K> 0, which in the situation of Theorem 7.24 is equi-
valent to
|R(−λτ)|≤ 1+ Kτ,

for all eigenvalues λ of A h . Because of
n
(1 + Kτ) ≤ exp(Knτ),
in (7.93) the additional factor exp(KT ) appears and correspondingly
exp(K(n − k)τ) in the sum. If the process is to be considered only in a
small time interval, this becomes part of the constant, but for large time
horizons the estimate becomes inconclusive.
On the other hand, for the one-step-theta method for 1 < Θ ≤ 1the
2
estimate E h,τ h ≤ 1 and thus the constants in (7.93) can be sharpened
to E h,τ h ≤ R(−λ min τ), where λ min is the smallest eigenvalue of A h ,