It is shown that the eigenvalues Z sub i of the pseudospectral Fourier approximation to the operator sin(2x) curly d/curly dx satisfy (R sub e) (Z sub i) = + or - 1 or (R sub e)(Z sub I) = 0. Whereas this does not prove stability for the Fourier method, applied to the hyperbolic equation U sub t = sin (2x)(U sub x) - pi  x  pi; it indicates that the growth in time of the numerical solution is essentially the same as that of the solution to the differential equation

Tal-Ezer, H.

English

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N,_JAC,,s-/Jz, _s,_2
NASA Contractor Report 172538
I_%SE REPORT NO. 85-8 NASA-CR-172538
19850012419
ICASE
THE EIGENVALUES OF THE PSEUDOSPECTRAL FOURIER U2.'.........
APPROXIMATION TO THE OPERATOR sln(2x) _ i _ --_
Hillel Tal-Ezer
Contract No. NASI-17070
February 1985
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING
NASA Langley Research Center, Hampton, Virginia 23665
Operated by the Universities Space Research Association
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https://ntrs.nasa.gov/search.jsp?R=19850012419 2020-03-20T18:47:00+00:00Z

THE EIGENVALHES OF _ PSEUDOSPECTRAL FOURIER
APPROXIMATION TO THE OPERATOR sin(2x) _--
_x
Hillel Tal-Ezer
School of Mathematical Sciences, Tel-Aviv University
Abstract
In this note we show that the eigenvalues Zi of the pseudospeetral
Fourier approximation to the operator sin(2x) -_ satisfy
Re Zi = ± 1 or Re Zi = O.
Whereas this does not prove stability for the Fourier method, applied to the
hyperbolic equation
Ut = sin(2x)U x - _ < x < _;
it indicates that the growth in time of the numerical solution is essentially
the same as that of the solution to the differential equation.
To be submittedfor publicationin Mathematicsof Computation.
Research was supported in part by the National Aeronautics and Space
Administration under NASA Contract No. NASI-17070 while the author was in
residence at the Institute for Computer Applications in Science and
Engineering, NASA Langley Research Center, Hampton, VA 23665.
- 2-
I. Introduction
Let us consider the problem
Ut - GU = 0 0 .<x .<27
U(x,O) = U°(x) (1.1)
where
G = a(x)_-_ . (1.2)
In the Fourier pseudospectral (collocation) method,we seek a trigonometric
polynomial of degree N , UN_that satisfies
(UN)t - GNUN = 0
o
UN(X,O) = UN(X) (1.3)
where
GN = PNG;
PN is the pseudospectral projection operator [5]. It is known [2] that
when a(x) does not change sign in the interval, the semidiscrete solution
of (1.3) is stable. When a(x) changes sign in the interval, the
situation is much more complicated. Gottlieb, Orszag and Turkel [I] have
proved stability for the case where a(x) is of the form
a(x) = _ sin(x) + B COS(X) + y. (1.4)
In [4], Tadmor argues that this stability proof results from the special
form of a(x) in (1.4) and cannot be extended. In the next section we
prove a theorem related to the problem of stability of (I.I) where a(x)
is a second degree trigonometric polynomial.
- 3-
2. The Theorem and Its Proof
Theorem: Considering (i.i), (1.2), where a(x) = sin(2x), then the eigen-
values _Nx of GN satisfy
Proof:
The projected subspace VN that results from using the operator PN
is spanned by the following 2N basis functions
VN = Sp{l,cos(x).....cos(Nx), sin(x),...,sin(N-l)x)}. (N even) (2.2)
Define the following four subspaces of VN
W1 = Sp{COS(X), cos(3x),...,cos((N-I)x)}
W2 = Sp{sin(x), sin(3x),...,sin((N-l)x)} (2.3)
W3 = S {sin(2x, sin(4x),...,sin((N-2)x)}P
W4 = S {l,cos(2x),...,cos(Nx)}.P
It is easily verified that
VN = W1 @ W2 @ W3 @ W4 (2.4)
and each W. is invariant of GN; therefore we can discuss separately1
the four matrices which represent GN in each one of the subspaces Wi.
-4 -
Define now
BM = [GN]w. I $ i $ 4 (M = _); (2.5)I
1
then by using elementary trigonometric relations we get that BM are
tridiagonal matrices whose elements are:
-I -3 \
1 0 -S
3 " "
M 1 • • -N+3 "
B1 = _
• 0 -N+I
N-3 ×N_
2 2
/ 1-3 ii
1 0 -5
3 • "
M 1 • • -N+3 "
B2 = _
o 0 -N+
N-3-N+I/N__× N_-
2 2
/ 0 -4 21
i 2 0 -
M 1
B3 = [ • • -N+4
0 -N+
N-4 0 /(N-i) × (N-I)
0 -2 1
0 0 -4
2 " "
M 1
B4 = _ • • -N+2 '
0 0
-5-
let A by any tridiagonal matrix:
aI cI
b2 a2 c2
A = (2.6)
bn_ I an_ I Cn_ I
b a
n n
and let _ by the submatrix
aI c1
b2 a2 c2
- (2.7)
bk_ I ak_ I Ok_ I
bk ak
Upon defining
qk(A) = det _ (2.8)
it is easily verified that
qk+l (A) = ak+l qk (A) - bk+l Ck qk-I (A) (2.9)
-6-
and
qn(A) = det A.
M i = 1,2,3,4In the following we treat each one of the matrices Bi,
separately.
H
LemmaI: The matrix B1 has one zeroeigenvalue,and all itsother
• I.=1.
eigenvalues 11 satisfy Re i
Proof: For any M define
polynomial of 2BM is given byThe characteristic
QM(t) : det CM _2.10)
and using (2.8)
%_(I): %(%_).
We define now the following family of polynomials (in the variable I)
P0= l Pl= - (I+ i)
(2.11)
Pk+l = - lPk . (4k2 - l) Pk-i 1 .<k <
Note that from (2.9) and the structure of CM
Pk = qk(% ) 2 .< k < M; (2.12)
- 7-
however (2.12) is not true for k = M; rather we have
QM(X) = (2hi- 1 - _) PM-1 + (4(M-1)2 - 1) PM-2 2 < M . (2.15)
From (2.11) we get
O.M(X) = (2M - 1) PM-1 + PM 2 < M. (2.14)
Using (2.14) and (2.13) results in
QM+I(x) = - x PM+ (2,'4+1)O_ 2 < M. (2.is)
Finally we solve (2.15) for PM in terms of %1 (_)' QM+I (X) and
substitute the result in (2.14). We thus get the polynomials QM(1), M >.2
that satisfy the following recursion formula
Q2(t) = 1(X-2) ; Q3(1) = - t(t2-4t + 13)
(2.16)
O,M+I(1) = (2-t) QM(),) + (2M-l) 2 ON_l(1 ) 3 < M .
k_
It is easy to verify now that I = 0 is an eigenvalue of 2KI. In fact
I = 0 is a root of Q2(1) and Q3(X) and therefore of any QM(1). We
define now
x = 1(2 - 1) (a) (2.17)
and
1 _(x) • (i) '`4-1P_(x) = 7 (b)
to get
R2 = - x ; R3 = x 2 _ 9
and
2
_+1 = :__ - (z'4-1) %t-1 M>.a. (2.18)
- 8 -
The relation (2.18) defines RM(X) as a family of orthogonal polynomials
on the real axis. Therefore, for every M the roots of RM(X) are real,
which implies by (2.17)(a) that 2 X are imaginary. Therefore, the
of the matrices 2BM for any M have real part equal to 2.eigenvalues
This completes the proof of Lemma i.
Lemma 2: For any M the matrix BM has one zero eigenvalue and the real
part of the others is -I.
Proof: The proof is an immediate result of the fact that in view of (2.9)
qk(-BM - Xl)
satisfy the same recurrence formula as qk(BM - %1).
Lemma 3: The eigenvalues of BM are purely imaginary.
Proof: Define the matrix
l/d
Then it is clear that
D-l B_'_D
is a skew symmetric matrix, and therefore its eigenvalues are purely
M
imaginary. The same is of course true for B3.
Lemma 4: The eigenvalues of BM are purely imaginary.
- 9 -
M M it followsthat if Pk isProof: From the definitionof B3 and B4
M 12Pkcharacteristicpolynomialof (B3)k×k then is the characteristic
o_M. M
polynomialof [B4)(k+2)×(k+2)"Thus the eigenvaluesof B4 are purely
imaginary.
The proof of Lemma 4 concludesthe proof of the theorem.
- i0 -
References
[I] D. Gottlieb, S. Orszag, E. Turkel, Stability of pseudospectral and
finite difference methods for variable coefficient problems,
Math. Comp., Vol. 37, No. 156, (1981), pp. 293-305.
[2] D. Gottlieb, S. Orszag, Numerical Analysis of Spectral Methods;
Theory and Applications, CBMS-NSF Regional Conference Series in
Applied Mathematics, SIAM Publisher, Philadelphia, PA 1977.
[3] E. Issacson, H.B. Keller, Analysis of Numerical Methods, John Wiley
and Sons, Inc., New York. (1966).
[4] E. Tadmor, Finite-difference, spectral and Galerkin methods for
time-dependent problems, ICASE Report No. 83-22, NASA CR-172149, 1983.
[5] R.G. Voigt, D. Gottlieb, M.Y. Hussaini, Spectral Methods for
Partial Differential Equations, Proceedings of a Symposium.
August 16-18, 1983, SIAM, Philadelphia, PA, 1984.

!. Re_x_t No NASA CR-!72538 _. Government Ace,s,on No. _. n,_ip;,nl"! _Ulog No.
ICASE Report No. 85-8
4. Title Ind Subtitle 5. Rlporl Dill
February 1985
The Eigenvalues of the Pseudospectral Fourier 6. PIdorm;n o Organization Code
Approximation to the Operator sin(2x) _
7. Author(s} 6. Perform;rigOrganlzit;onRepot1No.
Hillel Tal-Ezer 85-8
10.Work UnitNo.
9, PerformingOrganizationNJmeandAdd,eu
Institute for Computer Applications in Science
and Engineering 11.Contr=cto,GrantNo.
Mall Stop 132C, NASA Langley Research Center NAS[-17070
Hampton, VA 23665 13. Type of R,port lad P_iodCov,,,d
12. SponsoringAgencyNameandAddress Contractor Report
National Aeronautics and Space Administration 14. Sponln,lngAgencyCod_
Washington, D.C. 20546 505-31-83-01
15. _pplement*ry Notes
Langley Technical Monitor: J. C. South, Jr.
Final Report
16. Abstract
In this note we show that the elgenvalues Zi of the pseudospectral Fourier
approximation to the operator sln(2x) _x satisfy
Re Zi = _ ! or Re Zi = O.
Whereas this does not prove stability for the Fourier method, applied to the
hyperbolic equation
U = sin(2x)U - _ < x < _;t x
it indicates that the growth in time of the numerical solution is essentially the
same as thgt of the solution to the differential equation.
17, Key Words {Suggested by AuthorJs)) 18. Distribution Stitement
pseudospectral 64 - Numerical Analysis
stability
Fourier
variable coefficients Unclassified - Unlimited
Fo; sate by the NationalTechnical Inf0=malionService.Sprin£1ield.V,£inia Z2161
NASALangley.1985




The eigenvalues of the pseudospectral Fourier approximation to the operator sin (2x) d/dx

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