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7.5. Maximum Principle for the One-Step-Theta Method 325

where
 
I + τΘA h
.
 . 0 
 −I + τΘA h . 
 
. .
 . . 
C h =  . .  ,
 
0 . .
 . . 
 . . 
−I + τΘA h I + τΘA h
again with Θ:=1 − Θ,
 
τq (Θτ)+(I − τΘA h )u 0
h
 τq ((1 + Θ)τ) 
 h 
 
 . 
p =  . .  .
h
 
.
 
.
 . 
τq (N − 1+Θ)τ)
h
Since the spatial discretization is performed as in the stationary case, and
in the nth step the discretization relates to t = t n−1 +Θτ and also the
approximation
A h u(t n−1 +Θτ) ∼ ΘA h u(t n−1 )+ΘA h u(t n )
enters the formulation (7.66), we can assume to have the following structure
of the right-hand side of (7.66):
ˆ
n
q ((n−1+Θ)τ)= −A h (Θˆ u n−1 +Θˆ u )+f((n−1+Θ)τ)for n =1,...,N.
h
h
h
(7.98)
n
Here the ˆ u ∈ R M 2 are the known spatial boundary values on time level
h
t n , which have been eliminated from the equation as explained, e.g., in
Chapter 1 for the finite difference method. But as noted, we allow also for
the case where such values do not appear (i.e., M 2 = 0) then (7.98) reduces
to
q ((n − 1+Θ)τ)= f((n − 1+Θ)τ) for n =1,... ,N .
h
For the continuous problem, data are prescribed at the parabolic boundary
i
Ω ×{0}∪ ∂Ω × [0,T ]; correspondingly, the known values ˆ u are collected
h
with the initial data u 0 ∈ R M 1 to a large vector
 
u 0
 ˆ u 0 
h
 1 
ˆ u h  ,
 
ˆ u h = 
.
 . 
 . 
ˆ u N
h