In this section we turn our attention to the behavior of the Fourier method (weighted.6) in terms of the -norm. Table 3.1 suggests that when measured with respect to the standard (weight-free) -norm, the Fourier approximation may grow linearly with the number of gridpoints N.
Table 3.1:
Amplification of at t=10,
The main result of this section asserts that this is indeed the case.
The right hand side of (weak-in.1) tells us that the Fourier method may amplify the -size of its initial data by an amplification factor -- that is , the Fourier method is algebraically stable. The left hand side of (weak-in.1) asserts that this estimate is sharp in the sense that there exist initial data for which this amplification is attained -- that is, the Fourier method is weakly - unstable.
We turn to the .
Let denote the solution of the Fourier method (weighted.6)
subject to arbitrary initial data, . We claim that we
can bound the ratio in terms of
the condition number,
, of the weighting matrix H,
. Indeed
Here, the first and last equalities are Parseval's identities;
the second and forth inequalities are straightforward by
the definition of a weighted norm; and the third
is a manifestation of the weighted -stability stated in
Theorem 3.1.
The estimate (weak-in.2) requires to upper-bound the condition
number of the weighting matrix H. We recall that the weighting
matrix H is the direct sum of
the matrices
given in (weighted.21a)-(weighted.21b), whose -norms equal the
squared -norms of the corresponding Jordan blocks,
Inserting this into (weak-in.2) we arrive at
Thus it remains to upper bound the condition number of the
Jordan blocks, . For the sake of completeness
we include a brief calculation of the latter.
The inverse of are upper-triangular
Toeplitz matrices,
for which we have,
This means that , and together with the
straightforward upper-bound, ,
the right hand side of the inequality (weak-in.1)
now follows with .
The above -algebraic stability is essentially due to
the upper-bound on the size of the inverses
of Jordan blocks stated in (weak-in.4b). Can this upper-bound be improved?
an affirmative answer to this question depends on the regularity of the data,
as shown by the estimate
which yields an bound for -data,
Noting that the rest of the arguments in the proof of algebraic
stability are invariant with respect to the -norm
(-- in particular, the weighted -stability stated above),
we conclude the
following extension of the right inequality in (weak-in.1).
Corollary 3.1 tells us how the smoothness of the initial data is related to the possible algebraic growth; actually, for -initial data with , there is no -growth. However, for arbitrary data () we remain with the upper bound (weak-in.4b), and this bound is indeed sharp for, say, . (In fact, the latter is reminiscent of the unstable oscillatory boundary wave we shall meet later in (weak-in.20)).
These considerations lead us to the question whether the linear -growth upper-bound offered by the right hand side of (weak-in.1) is sharp. To answer this question we return to take a closer look at the real and imaginary parts of our system (weighted.7a).
We recall that according to (weighted.9a) the real part,
,
satisfies,
Summing by parts against we find
The boundary conditions (weighted.9b),
, imply that
the second term on the right is positive;
using Cauchy-Schwartz to upper bound the first term yields
,
which in turn implies that the real part of the system (weighted.7a)
is -stable
Figure 3.1: Fourier Solution of .
In contrast to the -bounded real part, it will be shown below that the imaginary part of our system experiences an linear growth, which is responsible for the algebraically weak -instability of the Fourier method.
The imaginary part of our system, , satisfies
the same recurrence relations as before
the only difference lies in the augmenting boundary conditions which now read
Trying to repeat our argument in the real case, we sum by parts
against ,
but unlike the previous case, the judicious minus sign in the
augmenting boundary conditions (weak-in.6b) leads to
the lower bound
This lower bound indicates (but does not prove!)
the possible -growth
of the imaginary part.
Figure 3.1 confirms that unlike the -bounded real
part, the behavior of the
imaginary part is indeed markedly different --
it consists of binary oscillations which form a growing modulated
wave as .
These binary oscillations suggest to
consider , in order to gain a better
insight into the growth of the underlying modulated wave. Observe that
(weak-in.6a)-(weak-in.6b) then recasts into the centered difference scheme
which is augmented with first order homogeneous extrapolation at the
'right' boundary
We note in passing that
{i} The 's, and hence the 's, are symmetric
-- in this case they
have an odd extension for
;
{ii} No additional boundary condition is required
at the left characteristic boundary ; and finally,
{iii} Though (weak-in.9a)-(weak-in.9b) are independent of the frequency spacing
-- in fact any will do, yet the choice of
will greatly simplify the
formulae obtained below.
These simplifications will be advantageous throughout the rest of this
section.
Clearly, the centered difference scheme (weak-in.9a) could be viewed
as a consistent approximation
to the linear wave equation
The essential point is that is an inflow
boundary in this case, and that the boundary
condition (weak-in.9b) is inflow-dependent in the sense that it is
consistent with the
interior inflow problem. Such inflow-dependent boundary condition renders
the related constant coefficient approximation unstable.
To show that there is an -growth in this case requires a
more precise study,
which brings us to the
.
We decompose the imaginary components, ,
as the sum of two contributions -- a stable part, ,
associated with the evolution of the initial data; and an unstable
part, , which describes the
unstable binary oscillations propagating from the boundaries
into the interior domain,
Here, is
governed by an outflow centered difference scheme
which is complemented by stable boundary extrapolation,
As before, we exploit symmetry to confine our attention to the 'right half' of
the problem, .
A straightforward -energy estimate confirms
that this part of the imaginary components is -stable,
. In fact, the scheme (weak-in.10) retains high-order
stability in the sense that
We close our discussion on the so called "s"-part by noting that (weak-in.10)
is a second-order accurate approximation to the
initial-value problem
Observe that the initial condition is nothing but
a trigonometric interpolant in the frequency -space',
which coincides with the initial value of the imaginary components,
.
Using the explicit solution of this initial value problem, we end up with
a second order convergence statement which reads
We now turn our attention to the unstable oscillatory part,
. It is
governed by an inflow centered difference scheme,
which is coupled to the previous stable "s"-part (weak-in.10), through the
boundary condition
The boundary condition (weak-in.14b) is the
first-order accurate extrapolation we met
earlier in (weak-in.9b) -- but this time, with the
additional inhomogeneous boundary data.
And as before, a key ingredient in the -instability
is the fact that such
boundary treatment is
inflow-dependent.
Specifically, we claim:
the inflow-dependent extrapolation on the left of (weak-in.14b)
reflects
the boundary values on the right of (weak-in.14b),
which 'inflow' into the interior domain with
an amplitude amplified by a factor of order .
To prove this claim we proceed as follows. Forward differencing
of (weak-in.14a) implies that
satisfy the stable difference scheme
Clearly, this difference scheme is consistent with, and hence convergent
to the solution of the initial-boundary value problem
Observe that describes a boundary wave which is prescribed on the
boundary of the computed spectrum,
, and propagates into the interior domain
of lower frequencies ,
We conclude that the forward differences, ,
form a second-order accurate approximation of this boundary wave,
Returning to the original variables,
,
the latter equality reads
which confirms our above claim regarding the amplification of a boundary wave
by a factor of .
The a priori estimates (weak-in.11) and (weak-in.18) provide us with
precise information on the behavior of
the imaginary components, :
their initial value at t=0 propagate by the stable "s"-part
and reaches the boundary of the computed spectrum at with
the approximate boundary values of (weak-in.13),
; the latter propagate
into the interior spectrum as a boundary wave of the form (weak-in.17),
, whose
primitive in (weak-in.18) describes the unstable
oscillatory ""-part of the solution. Added all together we end up with
Thus, the unstable ""-part
contributes a
wave which is modulated by binary oscillations; the amplitude of these
oscillations start with amplification
near the boundary of the computed spectrum, , and
decreases as they propagate into the interior domain of lower
frequencies. Moreover, for any fixed t >0, only those modes
with wavenumber k such that , are affected by
the unstable "" part.
Put differently, we state this as
There are two different cases to be considered, depending on the smoothness of the initial data.
Figure 3.9: Fourier solution of .
Figure 3.2 demonstrates this result for a prototype case of smooth initial data in Besov -- in this case, initial data with cubically decaying imaginary components, . As told by (weak-in.19), the temporal evolution of these components should include an amplified oscillatory boundary wave, , consult Remark 3 below. This amplification is confirmed by the quadratic decay of the boundary amplitudes, . Note that despite this amplification, the boundary wave and hence the whole Fourier solution remain bounded in this smooth case.
Figure 3.3 demonstrates this result for a prototype case of
nonsmooth initial data
with imaginary components given by,
, that is, initial data
represented by a strongly peaked dipole at
.
According to (weak-in.19), the evolution of these
components in time yields
In this case the oscillatory boundary wave,
,
is added to the -initial conditions, ,
which is responsible for the -growth of order
.
This linear -growth is even more apparent with the
'rough' initial data we met earlier in Figure 3.1.
1. .
The last Theorem confirms the -instability indicated previously by
the lower bound (weak-in.8),
By the same token, summation by parts of the imaginary part (weak-in.7),
leads to the upper bound
which shows that had the
boundary values of the computed spectrum -- which in this case consist of the
last single mode , were to remain relatively small,
then the imaginary part - and consequently the whole Fourier approximation
would have been -stable.
For example, the rather weak a priori bound will suffice
What we have shown (in the second part of Theorem
3.2)
is that such an a priori bound does not hold for
general nonsmooth -initial data, where according to (weak-in.19),
.
We recall that there are various procedures which enforce stability of the Fourier method, without sacrificing its high order accuracy. One possibility is to use the skew-symmetric formulation - consult §3.4 below. Another possibility is based on the observation that the current instability is due to the inflow-dependent boundary conditions (weak-in.9b) -- or equivalently (weak-in.6b), and the origin of the latter could be traced back to the aliasing relations (app_ps.7). We can therefore de-alias and hence by (weak-in.21) stabilize the Fourier method by setting , or more generally, . De-aliasing could be viewed as a robust form of high-frequency smoothing. This issue is dealt in §3.5 below. Figure 3.4a shows how the de-aliasing procedure (-- setting ), stabilizes the Fourier method which otherwise experiences the unstable linear growth in Figure 3.4b. With (weak-in.21) in mind, we may interpret these procedures as a mean to provide the missing a priori decaying bounds on the highest mode(s) of the computed spectrum, which in turn guarantee the stability of the whole Fourier approximation.
2. .
The situation described in the previous remark is a special case of
the following assertion:
Assume that a(x)
consists of a finite number, say m modes.
Then the corresponding Fourier
approximation (meth_ps.17) is -stable, provided the
last m modes were filtered so that the following a priori bound holds
It should be noted that our present discussion of a(x) with m=1 modes
is a prototype case for the behavior of the Fourier method,
as long as the corresponding
Fourier approximation is based on an odd number of 2N+1 gridpoints;
otherwise the case of an
even number of gridpoints is -stable.
The unique feature of this -stability is due to the
fact that Fourier differentiation matrix in this case,
-- being even order
antisymmetric matrix, must have zero as a double eigenvalue, which
in turn inflicts a 'built-in' smoothing of the last
mode in this case, namely,
Table 3.2 confirms the usual linear weak -instability already
for a 2-wave coefficient.
Table 3.2:
Amplification of at t=5
with even number of gridpoints.
3. .
Consider the case of sufficiently smooth initial data so that the
imaginary components decay of order ,
In this case, we may approximate the corresponding initial interpolant
, and (weak-in.19) tells
us the Fourier approximation takes the approximate form
Observe that ,
(with ), where
as .
This lower bound is found to be
in complete agreement with the
-stability statement of Corollary 3.1
(apart from the
factor for ) -- an enjoyable sharpness.