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Optimal Higher Order Time Discretizations 73
Proof. Using Von Neumann analysis [RM67] and spectral theory of selfadjoint
operators (namely the spectral theorem [RS78]), it is sufficient to look
at the (λ-parameterized) family of difference equations (u n is now a sequence
of complex numbers):
u n+1 − 2u n + u n−1
∆t 2 + λP m (λ∆t 2 )u n =0, λ ∈ σ(A h ), (12)
where σ(A h ) is the spectrum of A h . The characteristic equation of this recurrence
is
r 2 − [ 2 − Q m (λ∆t 2 ) ] r +1=0.
This is a second degree equation with real coefficients. The product of the roots
being 1, the two solutions have modulus less than 1 – which is equivalent to
the boundedness of u n – if and only if the discriminant of this equation is
non-positive, in which case the roots belong to the unit circle. This leads to
Q m (λ∆t 2 )[4 − Q m (λ∆t 2 )] ≥ 0or
0 ≤ Q m (λ∆t 2 ) ≤ 4.
If (9) holds, since σ(A h ) ⊂ [0, ‖A h ‖], λ∆t 2 ∈ [0, 4] which proves that (9) is a
sufficient stability condition. The second part of the proof is left to the reader.
⊓⊔
Remark 2. The equality σ(A h )=[0, ‖A h ‖] holds, for instance, when one uses
a finite difference scheme of the wave equation with constant coefficients in
the whole space. The Fourier analysis proves that the spectrum of A h is, in
this case, purely continuous.
⊓⊔
The finiteness of α m for each m is quite obvious. However, its value is
difficult to compute explicitly, except for the first values of m. One has, in
particular,
α 1 =4, α 2 =12, α 3 =2(5+5 1 3 − 5
2
3 ) ≃ 7.572, α4 ≃ 21.4812,... (13)
For the exact – but very complicated – expression of α 4 , we refer to [CJRT01]
or [Jol03]; other values of α m are given in the column “k = 0” of Table 1 on
page 88. It is particularly interesting to note that for the fourth order scheme,
one is allowed to take a time step which is √ α 2 /α 1 (≃1.732) times larger
than for the second order scheme, which almost balances the fact that the
cost of one time step is twice larger. In the same way, with the scheme of
order 8, one can take a time step √ α 4 /α 1 (≃ 2.317) times larger (while each
time step costs four times more). Surprisingly, the scheme of order 6 seems
less interesting: the stability condition is more constraining that for the fourth
order scheme.
From the theoretical point of view, it would be interesting to know the
behaviour of α m for large m. For this we first identify the limit behaviour of
the polynomials Q m (x). One easily checks that