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7.5. Maximum Principle for the One-Step-Theta Method 327

(ii) C r > 0 for at least one r ∈{1,..., M 1 }:
We set
M 1

A j := (A h ) jk ,
k=1
so that condition (3) (i) for A h means that A j ≥ 0for j =1,... ,M 1 .
Therefore, we have
C r =1 + τΘA j > 0 (7.101)

for the indices r of the first time level, where the “global” index r
corresponds to the “local” spatial index j. For the following time
levels, the relation

C r =1 − 1+ τ(Θ + Θ)A j = τA j ≥ 0 (7.102)
holds, i.e., (3) (i) and (ii).
(4) For every r 1 ∈{1,... , M 1 } satisfying
∗
M 1

(C h ) rs = 0 (7.103)
r=1
there exist indices r 2 ,...,r l+1 such that
=0 for i =1,... ,l
(C h ) r i r i+1
and
M 1

(C h ) r l+1 s > 0 . (7.104)
s=1
To avoid too many technicalities, we adopt the background of a finite
difference method. Actually, only matrix properties enter the reason-
ing. We call (space-time) grid points satisfying (7.103) far from the
boundary, and those satisfying (7.104) close to the boundary.Due to
(7.101), all points of the first time level are close to the boundary
(consistent with the fact that the grid points for t 0 = 0 belong to the
parabolic boundary). For the subsequent time level n, due to (7.102),
apoint (x i ,t n ) is close to the boundary if x i is close to the bound-
˜
ary with respect to A h . Therefore, the requirement of (4) ,thata
∗
point far from the boundary can be connected via a chain of neigh-
bours to a point close to the boundary, can be realized in two ways:
Firstly, within the time level n, i.e., the diagonal block of C h if A h
satisfies condition (4) . Secondly, without this assumption a chain of
∗
neighbours exist by (x, t n ), (x, t n−1 )up to(x, t 1 ), i.e., a point close
to the boundary, since the diagonal element of −I + τΘA h does not
vanish due to (7.100). This reasoning additionally has established the
following: