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7.5. Maximum Principle for the One-Step-Theta Method 329

˜
∗
(2) If A h additionally satisfies (1.32) (6) , then the following weak
maximum principle holds:

n n
max (˜ u ) r ≤ max max (u 0 ) r , max (ˆ u ) r ,
h
h
r∈{1,...,M} r∈{1,...,M 1 } r∈{M 1 +1,...,M}
n=0,...,N n=0,...,N
where
n
u
n
˜ u := h .
h
ˆ u h
˜
(3) If A h satisfies (1.32) (1), (2), (3) (i), (4), (5), (6), (7),thena strong
maximum principle in the following sense holds:
n
If the components of ˜ u , n =0,... ,N, attain a nonnegative maxi-
h
mum for some spatial index r ∈{1,...,M 1 } and at some time level
k ∈{1,... ,N}, then all components for the time levels n =0,... ,k
are equal.
Proof: Only part (3) needs further consideration. Theorem 1.9 cannot
be applied directly to (7.97), since C h is reducible. Therefore, the proof of
Theorem 1.9 has to be repeated: We conclude that the solution is constant
at all points that are connected via a chain of neighbours to the point where
the maximum is attained. According to (4) # these include all grid points
(x, t l )with x ∈ Ω h and l ∈{0,... ,k}. From (7.100) and the discussion of
(7) we see that the connection can also be continued to boundary values
up to level k.

The additional condition (7.100), which may be weakened to nonstrict
inequality, as seen above, actually is a time step restriction: Consider again
the example of the five-point stencil discretization of the heat equation on
2
a rectangle, for which we have (A h ) jj =4/h . Then the condition takes
the form
τ 1
< (7.105)
h 2 4(1 − Θ)
for Θ < 1. This is very similar to the condition (7.87), (7.88) for the explicit
Euler method, but the background is different.
As already noted, the results above also apply to the more general form
(7.67) under the assumption (7.69). The condition (7.100) then takes the
form
for j =1,... ,M 1 .
τΘ(A h ) jj ≤ b j

Exercises

7.14 Formulate the results of this section, in particular condition (7.100),
for the problem in the form (7.67) with B h according to (7.69) (i.e.