We observe that is nothing but the trigonometric interpolant
of w(x) at the equidistant points :
This shows that is in fact a dospectral projection,
which in the usual sin-cos formulation reads
Thus, trigonometric interpolation provides us with an excellent vehicle to
perform approximate discretizations with high (= spectral) accuracy, of
differential and integral operations. These can be easily carried out in
Fourier space where the exponentials serve as eigenfunction. For example,
suppose we are given the equidistant gridvalues, , of an underlying
smooth (i.e., also periodic!) function .
A second-order accurate discrete derivative is provided by center differencing
Note that the error in this case is, , no
matter how smooth w(x) is. Similarly, fourth order approximation is given
(via Richardson's extrapolation procedure) by
The pseudospectral approximation gives us an alternative procedure: construct
the trigonometric interpolant
Differentiation in Fourier space amounts to simple multiplication,
since the exponentials are eigenfunctions of differentiation,
and we approximate
Indeed, by our estimates we have for
which verifies the asserted spectral accuracy. Similar estimates are valid
for higher derivatives. To carry out the above recipe, one proceeds as
follows: starting with the vector of gridvalues, , one computes the discrete Fourier coefficients
or, in matrix formulation
then we differentiate
or in matrix formulation
and finally, we return to the ``physical'' space, calculating
or in matrix formulation
The summary of these three steps is
where represents the discrete differentiation matrix, and similarly
for higher derivatives.
: Since (interpolation!) we apply . How does this compare with finite differences and finite-element type differencing?
In periodic second-order differencing we have
fourth order differencing yields
In both cases the second and fourth order differencing takes place in the
physical space. The corresponding differencing matrices have
finite bandwidth and this reflects the fact that these
differencing methods are local. Similarly, finite-element
differencing,
corresponds to a differencing matrix
We still operate in physical space with operations (tridiagonal
solver) and locality is reflected by a very rapid (exponential decay) away
from main diagonal. Nevertheless, if we increase the periodic center
differences stencil to its limit then we end up with global
pseudospectral differentiation
recall the Dirichlet kernel (2.1.33)
and its derivative,
so that
Hence (app_ps.31), (app_ps.34) give us
In this case is a full
matrix whose multiplication
requires operations; however, we can multiply
efficiently using its spectral representation from (app_ps.30),
Multiplication by F and can be carried out by FFT which requires
only operating and hence the total cost here is almost as good
as standard ``local'' methods, and in addition we maintain spectral accuracy.
We have seen how the pseudospectral differentiation works in the physical
space. Next, let's examine how the standard finite-difference/element
differencing methods operate in the Fourier space.
Again, the essential ingredient
is that exponentials play the role of eigenfunctions for
this type of differencing. To see this, consider for example the usual second
order centered differencing, , for which we have
The term is called the ``symbol'' of center
differencing. By superposition we obtain for arbitrary grid function
(represented here by its trigonometric interpolant)
that
It is second-order accurate differencing since its symbol satisfies
Note that for the low modes we have
error (the less significant
high modes are differenced with error but their amplitudes tend rapidly to zero). Thus we have
and this estimate should be compared with the usual
The main difference between these two estimates lies in the fact that
the last estimate
is local, i.e.,
we need the smoothness of w(x) only in the neighborhood of
,
and not in the whole interval,
.
The analogue localization in
the Fourier space will be dealt later.
Similarly, we have for fourth order differencing the symbol
In general, we encounter difference operators whose
matrix representation, D,
is periodic and antisymmetric (here ),
Matrices satisfying the periodicity property are called circulant, and they
all can be diagonalized by the unitary Fourier matrix
Indeed, with we have
and using the antisymmetry we end up with symbols
As an example, we obtain for the (linear) finite-element differencing system
This corresponds to differentiation of the forth-order Padé expansion.
In general, the symbols are trigonometric polynomials or rational functions in the ``dual variable,'' kh, which has ``exact'' representation on the grid in terms of translation operator (polynomials or rational functions), and accuracy is determined by the ability to approximate the exact differentiation symbol, ik, for , consult Figure 2.2.
Figure 2.2: The symbols of center differencing