0

We have already met the process of smoothing in connection with the heat
equation: starting with bounded initial data, *f*(*x*), the solution of the
heat equation (para.1)

represents the effect of smoothing *f*(*x*), so that (in fact analytic) and .

A general process of smoothing can be accomplished by convolution with
appropriate smoothing kernel *Q*(*x*)

such that
is sufficiently smoother than *f*(*x*) is, and

With the heat kernel, the role of is played by time *t* > 0.
A standard way to construct such filters is the following. We start with a
-function supported on, say, (-1,1), such that it has a unit mass
and zero first *r* moments,

Then we set and consider

Now, assume *f* is (*r* + 1) - differentiable in the
neighborhood of *x*; then, since is supported on and satisfies (smoo.4) as well,
we have by Taylor
expansion

The first *r* moments of vanish and we are left with

i.e., converges to *f*(*x*) with order *r*+1 as
. Moreover, is as smooth as
is, since

has many bounded derivatives as *Q* has, i.e., starting with differentiable
function *f* of order *r* + 1 in the neighborhood of *x*, we end up with
regularized function in .

__: For regularization - choose a unit mass
kernel, see Figure 3.10,
__

__
Then is a
regularization
of f(x)
with first order convergence rate
__

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To increase the order of convergence, one requires more vanishing moments,
(smoo.4),(which
yield more oscillatory kernels).
We note that this smoothing process is purely local -- it involves
-neighboring values of function f, in order to yield
a -regularized function with
.
The convergence rate here is r+1.
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Figure 3.10: Unit mass mollifiers
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We can also achieve local regularization with spectral convergence. To this
end we set
where is a -function supported on .
Figure 3.11 demonstrates such a mollifier.
In this case the support of the mollifier is kept fixed; instead, by
increasing __

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__

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Figure 3.11: A spectral unit mass mollifier
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Consider now
Then is because is; and the convergence
rate is spectral, since by (2.1.34)
and since was chosen as we are left with a residual
term which does not exceed
Thus, the convergence rate is as fast as the __

__
We can as easily implement such smoothing in the Fourier space: For example,
with the heat kernel we have
so that for any __

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__

__
Note: Since we can
deal with any
unbounded f by splitting . To
obtain spectral accuracy we may use
Clearly is and the familiar Fourier estimates
give us
We emphasize that this kind of smoothing in the Fourier space need not be
local; rather or are negligibly small
away from a small interval centered around the origin depending on
or . (This is due to the uncertainty principle.)
__

__
The smoothed version of the pseudospectral approximation of (meth_ps.16)
reads
i.e., in each step we smooth the solution either in the real space
(convolution) or in the Fourier space (cutting high modes). We claim that this smoothed
version is stable hence convergent under very mild assumptions on the smoothing
kernel . Specifically, (smoo.18) amounts in the Fourier space,
compare (meth_ps.3)
The real part of the matrix in question is given by
where
is interpreted as the smoothed differentiation operator. Now, looking at
(smoo.20) we note:
__

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__

- For
*p*= 0 we are back at the spectral analysis, compare (meth_spec.52), (meth_spec.53) and the real part of the matrix in (smoo.20) - the aliasing free one - is bounded. - We are left with |
*p*| = 1: in the unsmoothed version, these terms were unbounded since as or . With the smoothed version, these terms are bounded (and stability follows), provided we have

For example, consider the smoothing kernel where

This yields the smoothed differentiation symbols

which corresponds to the second order center differencing in (app_ps.36);
stability is immediate by (smoo.21) for

Yet, this kind of smoothing reduces the overall spectral accuracy to a second
one; a fourth order smoothing will be

In general, the accuracy is determined by the low modes while stability has
to do with high ones. To entertain spectral accuracy we may consider
smoothing kernels other than trigonometric polynomials ( finite
difference), but rather, compare (smoo.17)

An increasing portion of the spectrum is differentiated *exactly*
which yields spectral accuracy; the *highest* modes are not
amplified because of the smoothing effect in this part of the spectrum.

We close this section noting that if the
*differential* model contains some dissipation - e.g.,
the parabolic equation

then stability follows with no extra smoothing. The parabolic dissipation
compensates for the loss of ``one derivative'' due to aliasing in
first order terms.
To see this we proceed as follows: multiply

by and sum to obtain

Suppressing excessive indices, ,
we have for the RHS of (smoo.27)

Now, the first sum on the right gives us the usual loss of one derivative and
the second are compensates with gain of such quantity. Petrovski type
stability (gain of derivatives) follows. We
shall only sketch the details here. Starting with the first term on the right
of (smoo.28) we have

while for the second term

and this last term dominates the RHS of (smoo.29).

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Thu Jan 22 19:07:34 PST 1998