Partial Differential Equations - Modelling and ... - ResearchGate

72 J.Ch. Gilbert and P. Joly
u h (t n+1 ) − 2u h (t n )+u h (t n−1 )
∆t 2 +
]
m−1
∑
+ A h
[u h (t n )+2 (−1) k ∆t 2k
(2k +2)! Ak hu h (t n ) = O(∆t 2m ).
k=1
This identity leads to the scheme (7)–(8).
Using again Horner’s rule for the representation of the polynomial P m ,
reduces the calculation of u n+1
h
to m successive applications of the operator
A h (∆t), according to the following algorithm:
Step 1. Set u n,0
h
= u n h .
Step 2. Compute
Step 3. Set u n+1
h
u n,k
h
= u n,k−1
h
− 2 ∆t2 A h u n,k−1
h
, k =1, ··· ,m.
(2k + 1)(2k +2)
= u n,m
h
.
In other words, since the most expensive step of the algorithm is the application
of the operator A h (a matrix-vector multiplication in practice), the
computational cost for one time step of the scheme of order 2m is only m
times larger than the computational cost for one time step of the scheme of
order 2.
2.2 Stability Analysis
The stability analysis of the higher order scheme (7) is similar to the one of
the second order scheme but it is complicated by the fact that one must verify
that the operator A h (∆t) is positive, which already imposes an upper bound
on ∆t.
Theorem 2. A sufficient stability condition for scheme (7) is given by
where we have defined
with
∆t 2 ‖A h ‖≤α m , (9)
α m =sup{α |∀x ∈ [0,α], 0 ≤ Q m (x) ≤ 4}, (10)
m−1
∑
Q m (x) =xP m (x) =x +2 (−1) l x l+1
(2l +2)! . (11)
This condition is necessary as soon as the spectrum of A h is the whole interval
[0, ‖A h ‖].
l=1