CHAPTER TREFETHEN
Chapter
Fourier sp ectral metho ds
An example
Unb ounded grids
Perio dic grids
Stability
Notes and references
Think globally Act lo cally
BUMPER STICKER
CHAPTER TREFETHEN
Finite di erence metho ds like nite element metho ds are based on lo cal
representations of functions usually by low order p olynomials In contrast
sp ectral metho ds make use of global representations usually by high order
p olynomials or Fourier series Under fortunate circumstances the result is a
degree of accuracy that lo cal metho ds cannot match For large scale compu
tations esp ecially in several space dimensions this higher accuracy may be
most imp ortant for p ermitting a coarser mesh hence a smaller numb er of data
values to store and op erate up on It also leads to discrete mo dels with little
or no arti cial dissipation a particularly v aluable feature in high Reynolds
number uid ow calculations where the small amountofphysical dissipation
maybeeasilyoverwhelmed byany dissipation of the numerical kind Sp ectral
metho ds have achieved dramatic successes in this area
Some of the ideas b ehind sp ectral metho ds have been intro duced several
times into numerical analysis One early prop onentwas Cornelius Lanczos in
the s who showed the p ower of Fourier series and Chebyshev p olynomials
in a variety of problems where they had not b een used b efore The emphasis
was on ordinary di erential equations Lanczos s work has been carried on
esp ecially in Great Britain byanumb er of colleagues suchasC W Clenshaw
More recently sp ectral metho ds were intro duced again by Kreiss and
Oliger Orszag and others in the s for the purp ose of solving the par
tial di erential equations of uid mechanics Increasingly they are becoming
viewed within some elds as an equal comp etitor to the better established
nite di erence and nite element approaches At present however they are
less well understo o d
Sp ectral metho ds fall into various categories and one distinction often
made is between Galerkin tau and collo cation or pseudosp ectral
sp ectral metho ds In a word the rst twowork with the co e cients of a global
expansion and the latter with its values at p oints The discussion in this b o ok
is entirely con ned to collo cation metho ds which are probably used the most
often chie y b ecause they o er the simplest treatment of nonlinear terms
Sp ectral metho ds are a ected far more than nite di erence metho ds by
the presence of b oundaries which tend to intro duce stability problems that are
ill understo o d and sometimes highly restrictive as regards time step Indeed
di culties with b oundaries direct and indirect are probably the primary rea
son why sp ectral metho ds have not replaced their lower accuracy comp etition
CHAPTER TREFETHEN
in most applications Chapter considers sp ectral metho ds with b oundaries
but the presentchapter assumes that there are none This means that the spa
tial domain is either in nite a theoretical device not applicable in practice
or p erio dic In those cases where the physical problem naturally inhabits a
p erio dic domain sp ectral metho ds maybe strikingly successful Conspicuous
examples are the global circulation mo dels used by meteorologists Limited
area meteorological co des since they require b oundaries are often based on
nite di erence formulas but as of this writing almost all of the global circula
tion co des in use which mo del ow in the atmosphere of the entire spherical
earth are sp ectral
AN EXAMPLE TREFETHEN
An example
Sp ectral metho ds have b een most dramatically successful in problems with
p erio dic geometries In this section we presenttwo examples of this kind that
involve elastic wave propagation Both are taken from B Fornb erg The
pseudosp ectral metho d comparisons with nite di erences for the elastic wave
equation Geophysics Details and additional examples
can be found in that pap er
Elastic waves are waves in an elastic medium such as an iron bar a build
ing or the earth and they come in two varieties P waves pressure or
primary characterized by longitudinal vibrations and S waves shear or
secondary characterized by transverse vibrations The partial di erential
equations of elasticity can be written in various forms such as a system of
two second order equations involving displacements For his numerical simu
lations Fornb erg choseaformulation as a system of ve rst order equations
Figures and show the results of calculations for two physical
problems In the rst a P wave propagates uninterruptedly through a p erio dic
uniform medium In the second an oblique P wave oriented at hits a
horizontal interface at whichthewave sp eeds abruptly cut in half The result
is re ected and transmitted P and S waves For this latter example the actual
computation was p erformed on a domain of twice the size shown which is
a hint of the trouble one may be willing to go to with sp ectral metho ds to
avoid coping explicitly with b oundaries
The gures show that sp ectral metho ds may sometimes decisively outp er
form second order and fourth order nite di erence metho ds In particular
sp ectral metho ds are nondispersive and in a wave calculation that prop erty
can be of great imp ortance In these examples the accuracy achieved by the
sp ectral calculation on a grid is not matched by fourth order nite dif
ferences on a grid or by second order nite di erences on a
grid The corresp onding di erences in work and storage are enormous
Fornberg picked his examples carefully sp ectral metho ds do not always
p erform so convincingly Nevertheless sometimes they are extremely impres
sive Although the reasons are not fully understo o d their advantages often
hold not just for problems involving smo oth functions but even in the presence
of discontinuities
The gures in this section will app ear in the published version of this b o ok only with p ermission
UNBOUNDED GRIDS TREFETHEN
a Schematic initial and end states
b Computational results
Figure Sp ectral and nite di erence simulations of a P wave
propagating through a uniform medium from Fornb erg
UNBOUNDED GRIDS TREFETHEN
a Schematic initial and end states
b Computational results
Figure Sp ectral and nite di erence simulations of a P wave
incident obliquely up on an interface from Fornb erg
UNBOUNDED GRIDS TREFETHEN
Unb ounded grids
We shall b egin our study of sp ectral metho ds by lo oking at an in nite un
b ounded domain Of course real computations are not carried out on in nite
domains but this simpli ed problem contains many of the essential features
of more practical sp ectral metho ds
Consider again the set of square integrable functions v fv g on the
j
h
unb ounded regular grid hZ As mentioned already in x the foundation
of sp ectral metho ds is the sp ectral di erentiation op erator D
h h
ays One is by means of the which can be describ ed in several equivalent w
Fourier transform
SPECTRAL DIFFERENTIATION BY THE SEMIDISCRETE FOURIER TRANS
Compute v
Multiply by i
Inverse transform
Dv F i F v
h
h
Another is in terms of band limited interp olation As describ ed in x
one can think of the interp olantasaFourier integral of band limited complex
exp onentials or equivalently asan in nite series of sinc functions
SPECTRAL DIFFERENTIATION BY SINC FUNCTION INTERPOLATION
P
Interp olate v by a sum of sinc functions q x v S x x
k
k h k
Di erentiate the interp olant at the grid points x
j
x Dv q
j j
Recall that the sinc function S x de ned by
h
sin x h
S x
h x h
UNBOUNDED GRIDS TREFETHEN
is the unique function in L that interp olates the discrete delta function e
j
j
e
j
j
and that moreover is band limited in the sense that its Fourier transform has
compact supp ort contained in h h
For higher order sp ectral di erentiation we multiply F v by higher
h
powers of i or equivalently di erentiate q x more than once
tal reason is Why are these two descriptions equivalent The fundamen
that S x is not just any interp olant to the delta function e but the band
h
limited interp olant For a precise argument note that both pro cesses are
obviously linear and it is not hard to see that b oth are shift invariantin the
m m
sense that D K v K Dv for any m The shift op erator K was de ned
can be written as a convolution in Since an arbitrary function v
h
P
sum v v e it follows that it is enough to prove that the two
k
j
k j k
pro cesses give the same result when applied to the particular function e That
equivalence results from the fact that the Fourier transform of e is the constant
function h whose inverse Fourier transform is in turn precisely S x
h
Since sp ectral di erentiation constitutes a linear op eration on it can
h
also b e viewed as multiplication by a biin nite To eplitz matrix
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
D
B C
h
B C
B C
B C
B C
B C
B C
B C
B C
A
As discussed in x this matrix is the limit of banded To eplitz matrices
corresp onding to nite di erence di erentiation op erators of increasing orders
of accuracy see Table on p In this chapter we drop the subscript
on the symbol D used in x We shall be careless in this text ab out
the distinction between the op erator D and the matrix D that represents it
Another way to express the same thing is to write
Dv a v a
h
UNBOUNDED GRIDS TREFETHEN
S x
*o h *o
S x
x h
*o o * *o ****oooo ****oooo *o o * *o *o
*o
a b
Figure The sinc function S x and its derivative S x
x
h h
as in
The co e cients of can be derived from either the Fourier
transform or the sinc function interpretation Let us b egin with the latter
The sinc function has derivative
cos x h sin x h
S x
h x
x x h
with values
if j
S x
h x j
j
if j
jh
at the grid points See Figure This is precisely the zeroth column
of D since that column must by de nition contain the values on the grid of
the sp ectral derivative of the delta function The other columns corresp ond
ing to delta functions centered at other points x contain the same entries
j
appropriately shifted
Now let us rederive by means of the Fourier transform If
d
Dv a v then Dv a v and for sp ectral di erentiation we want
a i Therefore by the inverse semidiscrete Fourier transform
Z
h
i j h
i e d a
j
h
For j the integral is giving a while for j integration by parts
Think ab out this Make sure you understand why represents a column rather than a rowof
D
UNBOUNDED GRIDS TREFETHEN
yields
h
Z
i j h i j h
h
e ie
a i d
j
ij h ij h
h
h
The integral here is again zero since the integrand is a periodic exp onential
and what remains is
i j i j
a e e
j
j h h h
j
i j i j
e e
jh jh
as in
The entries of D are suggestiveoftheTaylor expansion of log x and
this is not a coincidence In the notation of the spatial shift op erator K of
D can be written
D K K K K K K
h h
which corresp onds formally to
log K log K D
h h
K
log log K
h h
K
hD
Therefore formally e K and this makes sense byintegrating the deriva
tive over a distance h one gets a shift See the pro of of Theorem
i j h
If v e for some h h then Dv i v Therefore i is an
j
eigenvalue of the op erator D On the other hand if v has the same form with
h h then will be indistinguishable on the grid from some alias
wave number h h with h for some integer and the
result will b e Dv i v In other words in Fourier space the spatial di eren
tiation op erator b ecomes multiplication by a p erio dic function thanks to the
discrete grid and in this sense is only an approximation to the exact di er
entiation op erator for continuous functions Figure shows the situation
graphically For band limited data however the sp ectral di erentiation op
erator is exact in contrast to nite di erence di erentiation op erators which
are exact only in the limit dashed line in the Figure
i j h
Actually this is not quite true by de nition an eigenvector must b elong to and e do es not
Strictly sp eaking i is in the sp ectrum of D but is not an eigenvalue However this technicalityis
unimp ortant for our purp oses and will b e ignored in the present draft of this b o ok
UNBOUNDED GRIDS TREFETHEN
eigenvalue
i
h h h h
Figure Eigenvalue of D divided by i corresp onding to the
i x
eigenfunction e as a function of The dashed line shows corre
sp onding eigenvalues for the nite di erence op erator D
i h
C C
i h
i h
i h
a D nite di erence b D sp ectral
Figure Eigenvalues of nite di erence and sp ectral rst order
di erentiation matrices as subsets of the complex plane
Figure compares the sp ectrum of D to that of the second order nite
di erence op erator D of x
D is a b ounded linear op erator on withnorm
h
kD k max ji j
h
h h
see xx Notice that the norm increases to in nity as the mesh is re ned
UNBOUNDED GRIDS TREFETHEN
This is inevitable as the di erentiation op erator for continuous functions is
unb ounded
So far wehave describ ed the sp ectral approximation to the rst derivative
op erator x but it is an easy matter to approximate higher derivatives to o
For the second derivative the co e cients turn out to b e
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
D
B C
h
B C
B C
B C
B C
B C
B C
B C
B C
A
To derive the entries of this matrix one can simply square D this leads to
in nite series to be summed One can di erentiate a second time Or
one can compute the inverse Fourier transform ofa asfollows Two
integrations by parts are involved and terms that are zero have b een dropp ed
For j
Z
h
i j h
a e d
j
h
Z
i j h
h
e
d
ij h
h
h
j ij ij i j h
e e e
ij h j h j h
h
For j the integral is simply
a
h h
i j h
The e ect of D on a function v e is to multiply it by the square
j
of the factor asso ciated with D as illustrated in Figure Again one has
a periodic multiplier that is exactly correct for h h The dashed
line shows the analogous curve for the standard centered three p oint nite
UNBOUNDED GRIDS TREFETHEN
eigenvalue
h h h h
Figure Eigenvalue of D corresp onding to the eigenfunction
i x
e as a function of The dashed line shows corresp onding eigen
values for the nite di erence op erator
C C
h h
a nite di erence b D sp ectral
Figure Eigenvalues of nite di erence and sp ectral second
order di erentiation matrices as subsets of the complex plane
di erence op erator of x The sp ectrum of D must be real since D is
symmetric it is the interval h
The developments of this section are summarized and generalized to the
mth order case in the following theorem
UNBOUNDED GRIDS TREFETHEN
SPECTRAL DIFFERENTIATION ON AN UNBOUNDED REGULAR GRID
m
Theorem The mth order sp ectral di erentiation op erator D is a
with norm b ounded linear op erator on
h
m m
kD k
h
m m m
If m is odd D has the imaginary sp ectrum i h i h and can
be represented by an in nite skew symmetric To eplitz matrix with entries
m
a a h for j
j
m m m
If m is even D has the real sp ectrum h and can be
represented by an in nite symmetric To eplitz matrix with entries
m
a a h for j
j
The purp ose of all of these sp ectral di erentiation matrices is to solve
partial di erential equations In a sp ectral collo cation computation this is
done in the most straightforward way p ossible one discretizes the continuous
problem as usual and integrates forward in time by a discrete formula usually
by nite di erences Spatial derivatives are approximated by the matrix
D This same prescription holds regardless of whether the partial di erential
equation has variable co e cients or nonlinear terms For example to solve
u a x u by sp ectral collo cation one approximates a x u at each time step
t x x
by a x Dv For u u one uses D v where v denotes the p ointwise
t x
j
square v v Alternative discretizations may also be used for b etter
i i
stability prop erties see This is in contrast to Galerkin or tau sp ectral
metho ds in which one adjusts the co e cients in the Fourier expansion of v to
satisfy the partial di erential equation globally
Sp ectral approximations with resp ect to time can also sometimes b e used see H Tal Ezer Sp ectral
metho ds in time for hyp erb olic equations SIAM J Numer Anal pp
UNBOUNDED GRIDS TREFETHEN
EXERCISES
Co e cients of D Determine the matrix co e cients of the third order sp ectral
di erentiation matrix D Compare your result with the co e cients of Table
R
a Compute the integral S x dx of the sinc function One could do this by
h
complex contour integration but instead be cleverer than that and nd the answer
by considering the Fourier transform The argument is quite easy b e sure to state it
precisely
b By considering the trap ezoid rule for integration Exercise explain why the
answer ab ove had to come out as it did Hint what is the integral of a constant function
PERIODIC GRIDS TREFETHEN
Perio dic grids
To b e implemented in practice a sp ectral metho d requires a b ounded domain In this
section we consider the case of a p erio dic domain or equivalently anunb ounded domain
on which we p ermit only functions with a xed p erio dicity The underlying mathematics
of discrete Fourier transforms was describ ed in x The next chapter will deal with more
general b ounded problems
N N
x x x
N N N N
*********ooooooooo *********ooooooooo
x
h
Figure Space and wavenumb er domains for the discrete Fourier trans
form
To rep eat some of the material of x our fundamental spatial domain will now be
as illustrated in Figure Let N b e a p ositiveeven integer set
h N even
N
and de ne x jh for any j The grid p oints in the fundamental domain are
j
x x x h
N N
and the invaluable identity is this
N
h
were de ned in x as was the discrete Fourier transform The norm k k and space
N
N
X
i j h
v F v h e v
N j
j N
the inverse discrete Fourier transform
N
X
i j h
v F v e v
j j
N
N
and the discrete convolution
N
X
v w v w h
k j j k
j N
PERIODIC GRIDS TREFETHEN
Recall that since the spatial domain is p erio dic the set of wave numbers is discrete
namely the set of integers Z and this is whywehavechosen to use itself as a subscript
in and We take N N as our fundamental wave number
domain The prop erties of the discrete Fourier transform were summarized in Theorems
and recall in particular the convolution formula
d
v w v w
As was describ ed in x the discrete Fourier transform can b e computed with great
e ciency b y the fast Fourier transform FFT algorithm for which a program was given on
p The discovery of the FFT in was an imp etus to the development of sp ectral
metho ds for partial di erential equations in the s Curiously however practical imple
mentations of sp ectral metho ds do not always make use of the FFT but instead sometimes
p erform an explicit matrix multiplication The reason is that in large scale computations
whichtypically involvetwo or three space dimensions the grid in each dimension mayhave
as few as or p oints or even fewer in so called sp ectral element computations and
these numb ers are low enough that the costs of an FFT and of a matrix multiplication may
b e roughly comparable
Now we are ready to investigate D the sp ectral di erentiation op erator
N N N
for N p erio dic functions As usual D can b e describ ed in various ways One is by means
N
of the discrete Fourier transform
SPECTRAL DIFFERENTIATION BY THE DISCRETE FOURIER TRANSFORM
Compute v
Multiply by i except that v is multiplied by
N
Inverse transform
D v F for N i v otherwise
N
N
The sp ecial treatment of the value v is required to maintain symmetry and app ears
N
only in sp ectral di erentiation of odd order
Another is in terms of interp olation by a nite series of complex exp onentials or
equivalently p erio dic sinc functions
PERIODIC SPECTRAL DIFFERENTIATION BY SINC INTERPOLATION
Interp olate v by a sum of p erio dic sinc functions
N
X
v S x x q x
k N k
k N
Di erentiate the interp olant at the grid p oints x
j
D v q x
N j j
PERIODIC GRIDS TREFETHEN
In the second description wehave made use of the p erio dic sinc function on the N p oint
grid
x
sin
h
S x
N
x
tan
h
which is the unique p erio dic function in L that interp olates the discrete delta function
e on the grid
j
j
e
j
j N N
and which is band limited in the sense that its Fourier transform has compact supp ort
contained in h h and furthermore satis esv v Compare
N N
For higher order sp ectral di erentiation on the p erio dic grid wemultiplyv by higher
powers of i or equivalently di erentiate q x more than once If the order of di erentiation
is o dd v is multiplied by the sp ecial value tomaintain symmetry
N
As in the last section the sp ectral di erentiation pro cess can b e viewed as multiplica
tion byaskew symmetric To eplitz matrix D compare N
PERIODIC GRIDS TREFETHEN
h h h
cot cot cot
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
h h h h
cot cot cot cot D
B C
N
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
A
h h h
cot cot cot
In contrast to the matrix D of D is nite it applies to vectors v v
N
N N
and has dimension N N D is not only a To eplitz matrix but is in fact circulant
N
This means that its entries D wrap around dep ending not merely on i j but on
ij
N
i j mo d N
As in the last section the entries of can b e derived either by the inverse discrete
Fourier transform or by di erentiating a sinc function The latter approach is illustrated in
Figure which shows S and S for N Since N is even symmetry implies that
N N
S x for x as well as for x Di erentiation yields
N
x x
sin cos
h h
x S
N
x
x
tan
sin
h
and at the grid p oints the values are
if j
S x
N j
jh
j
cot if j
Notice that for jjhj these values are approximately the same as in Thus the
i j entry of D as indicated in is
N
if i j
D
N ij
x x
i j
ij
cot if i j
Any circulant matrix describ es a convolution on a p erio dic grid and is equivalent to a p ointwise
multiplication in Fourier space In the case of D that multiplication happ ens to b e by the function
N
i
PERIODIC GRIDS TREFETHEN
S x
o N
* *o
S x N
*o *o ****oooo ****oooo *o *o *o *o *o
*o
a b
Figure The p erio dic sinc function S x and its derivative S x
N N
To derive by the Fourier transform one can make use of summation by parts
the discrete analog of integration by parts See Exercise
i x
The eigenvectors of D are the vectors e with Z and the eigenvalues are the
N
quantities i with N N These are precisely the factors i in the de nition
of D by the Fourier transform formula The numb er is actually a double eigenvalue
N
of D corresp onding to two distinct eigenvectors the constant function and the sawto oth
N
What ab out the sp ectral di erentiation op erator of second order Again there are
various ways to describ e it This time b ecause is an even function no sp ecial treatment
of N is required to preserve symmetry in the Fourier transform description
Computev
Multiply by
Inverse transform
D v F v
N N
The sinc interp olation description follows the usual pattern
Interp olate v by a sum of p erio dic sinc functions
N
X
v S x x q x
k N k
k N
terp olanttwice at the grid p oints x Di erentiate the in
j
D v q x
j j
N
The matrix lo oks like this compare
Now that D is nite they are truly eigenvalues there are no technicalities to worry ab out as in
N the fo otnote on p
PERIODIC GRIDS TREFETHEN
h h h
csc csc csc
h
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
h h h h
D
B C
csc csc csc csc
N
h
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
B C
A
h h h
csc csc csc
h
Note that b ecause N has b een treated di erently in the two cases D is not the
N
square of D which is why we have put the sup erscript in parentheses In general the
N
mth order sp ectral di erentiation op erator can b e written as a p ower of D if m is even
N
and as a p ower of D or as D times a p ower of D ifm is o dd See Exercise
N N
N
Figures are the analogs of Figures for a p erio dic grid
We summarize and generalize the developments of this section in the following theorem
PERIODIC GRIDS TREFETHEN
SPECTRAL DIFFERENTIATION ON A PERIODIC GRID
Theorem Let N be even If m is o dd the mth order sp ectral di erentiation matrix
m
isaskew symmetric matrix with entries D
N
a a for j
j
m m
eigenvalues i i and norm
h h
m
m
kD k
N
h
m
If m is even D is a symmetric matrix with entries
N
a a for j
j
m m
eigenvalues and norm
h
m
m
kD k
N h
PERIODIC GRIDS TREFETHEN
eigenvalue i
*o *o *o *o *o *o *o *o *o *o *o *o *o *o *o *o *o. . . *o. . . *o. . . *o. . . . *o. . *o. . *o. . . *o. . *o. . *o. . o. o. o. o. o. o. o.
* * * * * * *
. *o. . *o. . *o. .
h h h h
...... *o . . . *o . . . *o . . . *o *o *o *o *o *o *o *o *o *o *o *o *o *o *o *o
*o *o *o *o
Figure Eigenvalue of D divided by i corresp onding to the eigenfunc
N
i x
tion e as a function of for N The smaller dots show corresp onding
eigenvalues for the nite di erence op erator
i h *o
o
C C * *o *o o *o *o i h *o *o *o *o *o *o *o *o *o o
o * i h * * *o *o *o *o
*o
i h
a nite di erence b DN sp ectral
Figure Eigenvalues of nite di erence and sp ectral rst order di erenti
ation matrices on a p erio dic grid as subsets of the complex plane for N
PERIODIC GRIDS TREFETHEN
eigenvalue
. . . *o. *o *o. *o. *o *o. *o. *o *o.
*o. *o. *o. *o. *o. *o.
h h h h o. o. o. o. o. o. o. o. * . . * * . . * * . . * * . . * *o . . *o *o . . *o *o . . *o *o . . *o ...... *o . . . *o *o . . . *o *o . . . *o *o . . . *o
*o *o *o *o *o *o *o *o
*o *o *o *o *o *o *o *o
*o *o *o *o
i x
Figure Eigenvalue of D corresp onding to the eigenfunction e asa
N
function of for N The smaller dots show corresp onding eigenvalues for
the nite di erence op erator
C C
h h
***ooo*o * o * o * o * o * o * **ooo **********oooooooooo*o *o *o *o *o *o *o
a nite di erence b D sp ectral
N
Figure Eigenvalues of second order nite di erence and sp ectral di eren
tiation matrices on a p erio dic grid as subsets of the complex plane for N
PERIODIC GRIDS TREFETHEN
EXERCISES
Fourier transform derivation of D Not yet written
N
D D For most values of N the matrix D would serve as quite a go o d discrete
N N
N
second order di erentiation op erator but as mentioned in the text it is not identical to D
N
a Determine D D andD for N and con rm that the latter two are not equal
N N
N
b Explain the result of a by considering sinc interp olation as in Figure
c Explain it again by considering Fourier transforms as in Figure and
J
for arbitrary J d Give an exact formula for the eigenvalues of D
N
tiation Sp ectral di eren
Making use of a program for computing the FFT in Matlab such a program is built in in
Fortran one can use the program of p write a program DERIV that computes the
m th order sp ectral derivativeofanN p oint data sequence v representing a function de ned
on or
N length of sequence p ower of input
m order of derivative integer input
v sequence to b e di erentiated real sequence of length N input
w m th sp ectral derivativeofv real sequence of length N output
Although a general FFT co de deals with complex sequences make v and w real variables
in your program since most applications concern real variables Allow m to b e anyinteger
m and make sure to treat the distinction prop erly b etween even and o dd values of m
Test DERIV with N and N on the functions
u x exp sin x and u x j sin xj
for the values m and hand in two tables one for each function of the resulting
errors in the discrete norm Put a star where appropriate and explain your results
Plot the computed derivatives with m
Sp ectral integration Mo dify DERIV to accept negativeaswell as p ositivevalues of
m For m DERIV should return a scalar representing the de nite integral of v over one
period together with a function w representing the jmjth inde nite integral Explore the
behavior of DERIV with various v and m
Hamming window Write a program FILTER that takes a sequence v and smo oths
v multiplying the transform by the Hamming window it by transforming to
k
w cos v
k k
N
and inverse transforming Apply FILTER to the function j sin xj and hand in a plot of the result
PERIODIC GRIDS TREFETHEN
Fourier transform derivation of D Not yet written
N
D D For most values of N the matrix D would serve as quite a go o d discrete
N N
N
second order di erentiation op erator but as mentioned in the text it is not identical to D
N
a Determine D D andD for N and con rm that the latter two are not equal
N N
N
b Explain the result of a by considering sinc interp olation as in Figure
c Explain it again by considering Fourier transforms as in Figures and
Sp ectral di erentiation Typ e the program FFT of Figure into your computer
t you understand how to compute b oth a and exp eriment with it until you are con den
discrete Fourier transform and an inverse discrete Fourier transform For example try as
input a sine wave and a sinc function and make sure the output you get is what you exp ect
Then makesureyou can get the input backagainbyinversion
Making use of FFT write a program DERIV N m v w which returns the m th order sp ec
tral derivativeofthe N p oint data sequence v representing a function de ned on or
N length of sequence p ower of input
m order of derivative integer input
v sequence to b e di erentiated real sequence of length N input
w m th sp ectral derivativeofv real sequence of length N output
Although FFT deals with complex sequences make v and w real variables in your program
since most applications concern real variables Allow m to b e anyinteger m and make
sure to treat the distinction prop erly b etween even and o dd values of m
Test DERIV with N and N on the functions
u x exp sin x and u x j sin xj
for the values m and hand in two tables one for each function of the result
ing errors in the discrete norm Explain your results If p ossible plot the computed
derivatives with m
Sp ectral integration Mo dify DERIV to accept negativeaswell as p ositivevalues of
m For m DERIV should return a scalar representing the de nite integral of v over one
period together with a function w representing the jmjth inde nite integral Explore the
behavior of DERIV with various v and m
Hamming window Write a program FILTER N v w whichtakes a sequence v and
smo oths it by transforming tov multiplying the transform by the Hamming window
k
w cos v
k k
N
and inverse transforming Apply FILTER to the function j sin xj and hand in the result A
plot would b e nice
STABILITY TREFETHEN
Stability
Just a few results so far The nished section will b e more substantial
Sp ectral metho ds are commonly applied to time dep endent problems according to the
metho d of lines prescription of x rst the problem is discretized with resp ect to space
and then the resulting system of ordinary di erential equations is solved by a nite di erence
metho d in time As usual we can investigate the eigenvalue stability of this pro cess by
examining under what conditions the eigenvalues of the sp ectral di erentiation op erator are
contained in the stability region of the time discretization formula The separate question of
stability in the sense of the Lax Equivalence Theorem is rather di erent and involves some
subtleties that were not present with nite di erence metho ds these issues are deferred to
the next section
In xx wehaveintro duced two families of sp ectral di erentiation matrices D and
m
m
not exactly its p owers D for an in nite grid and D and its higher order analogs D
N
N
powers for a p erio dic grid The sp ectra of all of these matrices lie in the closed left half of
the complex plane and that is the same region that comes up in the de nition of A stability
We conclude that if any equation
m
u
u
t
m
x
is mo deled on a regular grid by sp ectral di erentiation in space and an A stable formula in
time the result is eigenvalue stable regardless of the time step
By Theorem an A stable linear multistep formula must b e implicit For a mo del
problem as simple as the system of equations involved in the implicit formula can b e
solved quickly by the FFT but in more realistic problems this is often not true Since sp ec
tral di erentiation matrices are dense unlike nite di erence di erentiation matrices the
implementation of implicit formulas can b e a formidable problem Therefore it is desirable
to lo ok for explicit alternatives
On a regular grid satisfactory explicit alternatives exist For example supp ose we
solve u u by sp ectral di erentiation in space and the midp ointformula in time
t x
The stability region for the midp oint formula is the complex interval i k i k From
Theorem or Figure we conclude that the time stability restriction is h k
i e
h
k
This is stricter by a factor than the time stability restriction k h for the leap frog
formula which is based on second order nite di erence di erentiation The explanation
go es back to the fact that the sawto oth curve in Figure is times taller than the
dashed one
On a p erio dic grid Theorem or Figure lo osens slightly to
h
k
h N
Other explicit time discretization formulas can also b e used with u u so long as
t x
their stability regions include a neighb orho o d of the imaginary axis near the origin Figure
STABILITY TREFETHEN
reveals that this is true for example for the Adams Bashforth formulas of orders
The answer to Exercise b can readily b e converted to the exact stability b ound for
the rd order Adams Bashforth discretization
For u u the eigenvalues of D or D b ecome real and negative so we need a
t xx
N
stability region that contains a segment of the negative real axis Thus the midp oint rule
will b e unstable On the other hand the Euler formula whose stability region was drawn in
Figure and again in Figure leads to the stability restriction h k i e
h
k
On an in nite grid this is stricter than the stability restriction for the nite di erence
forward Euler formula considered in Example The cusp ed curve in Figure is
times deep er than the dashed one By Theorem exactly the same restriction is
also valid for a p erio dic grid
h
k
N
As another example the answer to Exercise a can be converted to the exact
stability b ound for the rd order Adams Bashforth discretization of u u
t xx
In general sp ectral metho ds on a p erio dic grid tend to have stability restrictions that
are stricter by a constant factor than their nite di erence counterparts This is opp osite
to what the CFL condition might suggest the numerical domain of dep endence of a sp ectral
metho d is unb ounded so there is no CFL stability limit The constant factor is usually
not much of a problem in practice for sp ectral metho ds p ermit larger values of h than nite
di erence metho ds in the rst place b ecause of their high order of spatial accuracy In other
words relatively small time steps k are needed anyway to avoid large time discretization
errors
EXERCISES
A simple sp ectral calculation
Write a program to solve u u on with periodic b oundary conditions by the
t x
pseudosp ectral metho d The program should use the midp oint time integration formula
and spatial di erentiation obtained from the program DERIV of Exercise
a Run the program up to t with k h N and N and initial data
v f x v f x k with b oth
cos x for jxj mo d
f x cos x and f x
otherwise
List the four errors you obtain at t Plot the computed solutions v x if
N
p ossible
b Rerun the program with k h and list the same four errors as b efore
c Explain the results of a and b What order of accuracy is observed How mightit
b e improved
Why is the ratio not Because h not h
STABILITY TREFETHEN
Sp ectral calculation with ltering
n
a Mo dify the program ab ove so that instead of computing v at each step with DERIV
alone it uses DERIV followed by the program FILTER of Exercise Take k h
again and print and plot the same results as in the last problem Are the errors smaller
than they were without FILTER
b Run the program again in DERIV FILTER mo de with k h Print and plot the
same results as in c
c What is the exact theoretical stability restriction on k for the DERIV FILTER metho d
You may consider the limit N for simplicity
Note Filtering is an imp ortant idea in sp ectral metho ds but this simple linear problem is
not a go o d example of a problem where ltering is needed
Inviscid Burgers equation
u u on with p erio dic b oundary conditions by the Write a program to solve
t x
pseudosp ectral metho d The program should use forward Euler time integration formula
and spatial di erentiation implemented with the program DERIV
a Run the program up to t with k h N and initial data v
sin x Plot the computed results sup erimp osed on a single plot at times t
b Explain the results of part a as well as you can
c Optional Can you nd a way to improve the calculation