Highly Accurate Pseudospectral Approximations of the Prolate Spheroidal Wave Equation for Any

18 Pages·2016·0.54 MB·English

by  H. Alıcı

JSciComput(2017)71:804–821
DOI10.1007/s10915-016-0321-7
Highly Accurate Pseudospectral Approximations of the
Prolate Spheroidal Wave Equation for Any Bandwidth
Parameter and Zonal Wavenumber
H.Alıcı1,2 · J.Shen2
Received:4September2016/Revised:22October2016/Accepted:5November2016/
Publishedonline:24November2016
©SpringerScience+BusinessMediaNewYork2016
Abstract Theprolatespheroidalwaveequation(PSWE)istransformed,usingsuitablemap-
pings, into three different canonical forms which resemble the Jacobi, Laguerre and the
Hermitedifferentialequations.TheeigenpairsofthePSWEareapproximatedwiththecor-
respondingclassicalorthogonalpolynomialasabasisset.Itisobservedthatforanyzonal
wavenumbermtheJacobitypepseudospectralmethodsarewellsuitedforsmallbandwidth
parameterscwhereastheHermiteandLaguerrepseudospectralmethodsareappropriatefor
verylargecvalues.Moreover,Jacobipseudospectralmethodsworkwellforanyparameter
valuessuchthatm ≥ c.Ournumericalresultsconfirmthatforanyvaluesofm,theJacobi
[(α,β)=(±1/2,m)]andtheLaguerre(γ=±1/2)pseudospectralmethodsformulatedin
this article for the numerical solution of the PSWE with small and very large bandwidth
parameters, respectively, are highly efficient both from the accuracy and fastness point of
view.
Keywords Prolate spheroidal wave equation · Classical orthogonal polynomials ·
Pseudospectralmethods
MathematicsSubjectClassification 33E10·33C45·65L60·65L15
B
H.Alıcı
[email protected];[email protected]
J.Shen
[email protected]
1 DepartmentofMathematics,HarranUniversity,63290S¸anlıurfa,Turkey
2 DepartmentofMathematics,PurdueUniversity,47907WestLafayette,IN,USA
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1 Introduction
ThePSWE
(cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7)
1 d d m2 π π
− cosθ + +c2sin2θ y =λ(c,m)y, θ ∈ − , (1)
cosθ dθ dθ cos2θ 2 2
arisesinmanyareasofphysicssuchasthenuclearshellmodel,atomicandmolecularphysics,
thestudyoflightscatteringinopticsandtheoreticalcosmologicalmodels[3].In(1),λ(c,m)
standsfortheeigenvalueparameter,mthezonalwavenumberandcthebandwidthparameter.
MorebackgroundonthePSWEcanbefound,forexample,in[1,14,19,20].Here,weare
lookingforthesquareintegrablesolutionsofthePSWE.Singularitiesof(1)aswellasthe
unboundednessoftheterm,containingthezonalwavenumber,atη =±π/2implythatthe
eigenfunction y(η)mustvanishattheboundaries.Clearly,suchaneigenfunctionwillbein
thespaceL (−1,1)ofthesquareintegrablefunctionswhichsuggeststheuseoftheDirichlet
2
conditions (cid:6) π(cid:7)
y ± =0 (2)
2
attheboundaries.Then,thetransformation
η=sinθ, η∈(−1,1) (3)
takestheequationintothefollowingequivalentform
(cid:3) (cid:4)
m2
(1−η2)y(cid:4)(cid:4)−2ηy(cid:4)− +c2η2 y =−λ(c,m)y, y(±1)=0, (4)
1−η2
whosesolutionsaretheprolatespheroidalwavefunctions(PSWFs),whichareveryuseful
forapproximatingsolutionsofsomeODEsandPDEs.Inparticular,it isobservedin[25]
that for problems involving bandlimited functions, spectral methods based on the PSWFs
requirefewerpointsperwavelengthtoachievethesameaccuracywhencomparedtomethods
basedonorthogonalpolynomialssuchasChebyshevorLegendrepolynomials.Basedonthis
observation,therearenumerousstudiesontheuseofPSWFsasabasissetinspectralschemes
asanalternativetotheclassicalbasessuchasChebyshevandLegendrepolynomials.In[16],
theauthorsintroducedanewclassofnumericaldifferentiationschemesconstructedviathe
PSWFs which requires fewer points per wavelength (compared to existing differentiation
schemes based on orthogonal polynomials) to achieve the same accuracy when it is used
toapproximatederivativesofbandlimitedfunctions.In[9],theauthorshaveexaminedthe
meritsofusingPSWFsasbasisfunctionswhensolvinghyperbolicPDEsusingpseudospectral
methods and concluded that one might gain from using the PSWFs over the traditional
ChebyshevorLegendremethodsintermsofaccuracyandefficiencyformarginallyresolved
broadband solutions. the authors of [23] have proposed a new well-conditioned prolate-
collocation scheme which significantly outperforms Jacobi polynomial-based methods in
approximating highly oscillatory bandlimited functions. In [5], author used PSWFs as an
alternativetoChebyshevandLegendrepolynomialsforspectralelementandpseudospectral
algorithms.HeshowedthatwhencomparedtoChebyshevorLegendrepolynomials,prolate
functionsgivemoreuniformspatialresolutionandwhenusedasthespatialdiscretization
fortime-dependentPDEsincombinationwithexplicittime-steping,PSWFsallowalonger
stabletimestepthanLegendrepolynomials.
Ontheotherhand,insolvingtheHelmholtzequationsinspheroidaldomains,oneneeds
tosolvetheeigenvalueproblem(4)withawiderangeofthezonalwavenumberm andthe
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bandwidthparameterc=ka.Indeed,applyingtheseparationofvariablesprocessinprolate
spheroidalcoordinates,
(cid:8) (cid:8)
x =a (ξ2−1)(1−η2)cos(φ), y =a (ξ2−1)(1−η2)sin(φ), z =aξη, (5)
oneobtainsthePSWEin(4)astheangularpartoftheHelmholtzequation.Therefore,itis
veryimportanttobeabletoobtainaccuratenumericalsolutionsof(4)forawiderangeof
wavenumberk,particularlyverylargek.Inparticular,onecanadopttheFourier-likespectral
methoddevelopedin[22]forwhichfastandaccuratesolutionoftheeigenvalueproblem(4)
isessential.
AlgorithmstocomputetheeigenpairsofthePSWEarestill anactiveresearcharea[3,
7,8,15,18,21]. Nevertheless, the computation of the eigensolutions is a challenging task
especiallywhenthebandwidthparametercisverylarge.Morerecently,Ogburnetal.[18]
and Huang et al. [15] have approximated the eigenvalues of the PSWE with very large
bandwidthparameterbyafinitedifferenceandpseudospectraltypeschemes,respectively.In
[18]agridof20000pointshasbeenappliedwithhighorderfinitedifferencestoapproximate
theeigenvalueswhenc=107.Howevertheirmethodisveryexpensivesinceithasoccupied
agigabyteofmemoryandspentfivehoursonalaptop.Ontheotherhand,in[15]theauthors
usedradialbasisfunctionsonauniformgridtakingintoaccountthereflectionsymmetryof
theproblem.Thatis,theyusedtwodifferentbasisfunctionstotreattheevenandoddstates
ofthePSWE.Thenumericalresultsarequitesatisfactorywhencomparedtothoseof[18],
however,themethodincludesascalingparameterwhoseoptimumvaluemustbedetermined
e√mpirically.InordertoovercomethisissuetheyusedtheHermitefunctionbasisscaledby
cinapseudospectralpicturewhichalsotakescareofthesymmetry.Furthermore,authors
appliedperiodizedHermitebasisforsmallvaluesofthebandwidthparameterc ∈ [0,100]
sincetheHermitefu√nctionsarenotsuitableforthisparameterrange.But,inthiscasethe
fixedscalingfactor cistoolargeandshouldbechosenbytrial-and-error.Thenumerical
resultsobtainedbyapplyingthelasttwoapproachesarealsoquitesatisfactory.
NumericalsolutionofthePSWEforsmallbandwidthparametersisalsodiscussedin[2].
Therein,wehavetabulatedseveralei√genvaluesofthePSWEwhenthezonalwavenumber
m =0andbandwidthparameterc= 10.Besides,wehavereportedthesmallesttruncation
sizesnecessarytoobtainthegroundstateeigenvalueλ (c,m)accuratewithinthemachine
0
epsilonasfunctionsofc andm inaseparatetablewhichconfirmsthatthetruncationsize
N considerablyincreasesasthebandwidthparametercgetslarger.Although,themethodof
[2]yieldshighlyaccurateresultsforsmallbandwidthparameters,itbecomesuselesswhen
c>1000.
Therefore, in this article, we construct several pseudospectral formulations of PSWE
based on the classical orthogonal polynomials so that the lowest twenty eigenvalues with
anybandwidthparametercandzonalwavenumberm canbecomputedwithinthemachine
accuracyofquadrupleprecisionarithmeticwithatruncationsizenotexceeding15inasplit
second. Moreover, not only the lower eigenvalues but also the higher modes, as high as a
thousand,canbecomputedwithinthemachineaccuracyjustinaminute.
InSect.2,wegiveageneralpseudospectralformulationofahypergeometriclikeequation
sinceintheforthcomingSects.3and4wetransformthePSWEintotheequationsofthiskind.
Then,inSect.3,wetransformthePSWEintotwoseparateequationsresemblingtheJacobi
differentialequationandhence,weconstructtwodifferentJacobipseudospectralmethods
forsmallvaluesofthebandwidthparameter.Thefirstoneapproximatesthefullspectrum
atoncewhilethesecondonetakescareofthesymmetryoftheproblemandthuscomputes
theevenandoddeigenvaluesetsseparately.InSect.4,wetransformthePSWEintoanother
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twoequationswhichappearmuchsimilartotheHermiteandLaguerredifferentialequations.
Therefore,weconstructtheHermiteandLaguerrepseudospectralformulationsofthePSWE
for very large bandwidth parameters. The latter of the methods can separate the even and
oddstateswhichhalvesthetruncationsizeandreducethecostbyafactoroffour.Section
5,demonstratesanddiscussesthenumericalresultsobtainedbythemethodsconstructedin
Sects.3and 4wherethecomparisonwitheachotheraswellastheliteratureresultsaredone.
Thelastsectionconcludesthepaperwithsomeremarks.
2 AGeneralPseudospectralFormulationforHypergeometricLike
Equation
Considertheequation
σ(x)y(cid:4)(cid:4)+τ(x)y(cid:4)+q(x)y =λr(x)y, x ∈(a,b)⊆R (6)
whereq(x)andr(x) > 0continuousfunctions,λisaneigenvalueparameterandσ andτ
arepolynomialsofdegreeatmosttwoandone,respectively.Thelastequationreducestothe
equationofhypergeometrictype(EHT)
σ(x)y(cid:4)(cid:4)+τ(x)y(cid:4)+λy =0 (7)
when q(x) = 0 and r(x) = −1. It is known that the n−th degree classical orthogonal
polynomials y(x) = p (x) denoted by the names Jacobi, Laguerre and Hermite are the
n
solutionsoftheEHTforthespecificvaluesoftheparameter
(cid:9) (cid:10)
λ:=λ =−n τ(cid:4)+ 1(n−1)σ(cid:4)(cid:4) . (8)
n 2
Theyareorthogonalinthesensethat
(cid:11)
b
p (x)p (x)ρ(x)dx =δ h2 (9)
m n mn n
a
whereδ istheKronecker’sdelta,h isthenormalizationconstantandtheweightfunction
mn n
ρ(x)>0isthesolutionofthePearsonequation[σ(x)ρ(x)](cid:4) =τ(x)ρ(x).Thesepolynomial
solutionshaveexactlynrealanddistinctrootsin(a,b)whichareinterlaced,i.e.sortingall
therootsinascendingorder,therootsof pn+1(x)alternatewiththoseof pn(x).Now,for
equation(6)weproposeanapproximatesolutionoftheform
(cid:12)N
y(x)= (cid:14) (x)y (10)
n n
n=0
where
(cid:14) (x)= pN+1(x) , n =0,1,...,N (11)
n (x−x )p(cid:4) (x )
n N+1 n
are the set of Nth degree Lagrange interpolating polynomials in which pN+1(x) is the
(N + 1)−st degree polynomial solution of (7) and y = y(x ) are the exact values of
n n
the function y(x) at the roots {xn}nN=0 of pN+1(x). The zeros of the classical orthogonal
polynomialsmaybedeterminedastheeigenvaluesofthetridiagonalmatrix
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Table1 Thecoefficientsoftherecursionformulaforthenormalizedclassicalorthogonalpolynomials
Polyn. An Bn
(cid:13)
2 n(n+α)(n+β)(n+α+β) β2−α2
Jacobi
2n+α+β (2n+α+β−1)(2n+α+β+1) (2n+α+β)(2n+α+β+2)
√
Laguerre − n(n+γ) 2n+γ+1
(cid:14)
Hermite n/2 0
⎡ ⎤
B A 0
0 1
⎢⎢A1 B1 A2 ⎥⎥
R =⎢⎢⎢ A2 B2 ... ⎥⎥⎥ (12)
⎢⎣ ... ... A ⎥⎦
N
0 A B
N N
wherethe A and B arethecoefficientsinthethreetermrecurrancerelation
n n
An+1φn+1(ξ)+(Bn−ξ)φn(ξ)+ Anφn−1(ξ)=0, n =0,1,... (13)
of the normalized classical orthogonal polynomials φ (x) = p (x)/h . The procedure is
n n n
knownastheGolub-Welschalgorithm[13].InTable1,wepresentthecoefficients A and
n
B forthenormalizedJacobi,LaguerreandtheHermitepolynomials.
n
Now, inserting the approximate solution in (10) into the equation (6) and requiring its
satisfactionatthegridpointsx andusingthefactthat(cid:14) (x ) = δ weobtainthesetof
m n m mn
N +1equations
(cid:12)N (cid:9) (cid:10) (cid:12)N
σ(x )(cid:14)(cid:4)(cid:4)(x )+τ(x )(cid:14)(cid:4)(x )+q(x )δ y =λ(c,m)r(x ) δ y (14)
m n m m n m m mn n m mn n
n=0 n=0
leadingtothediscreterepresentation
B(cid:21)y=λ(c,m)y (15)
(cid:9) (cid:10)
of(6)wherethegeneralentryB(cid:21) oftheresultingmatrixB(cid:21)= B(cid:21) maybewrittenas
mn mn
(cid:9) (cid:10)
B(cid:21) = 1 σ(x )(cid:14)(cid:4)(cid:4)(x )+τ(x )(cid:14)(cid:4)(x )+q(x )δ , m,n =0,1,...,N. (16)
mn r(x ) m n m m n m m mn
m
Actuallythematriceswithentries
⎧
d(1) :=(cid:14)(cid:4)(x )= 1⎪⎪⎪⎪⎨xm −2 xn pp(cid:4)N(cid:4)N++11,,mn if m (cid:7)=n (17)
mn n m 2⎪⎪⎪⎪⎩−τn if m =n
σ
n
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and
⎧ (cid:2) (cid:5)
⎪⎪⎪⎪⎨−x −3 x στm + x −2 x pp(cid:4)N(cid:4) +1,m if m (cid:7)=n
d(2) :=(cid:14)(cid:4)(cid:4)(x )= 1 m n m m n N+1,n (18)
mn n m 3⎪⎪⎪⎪⎩ 1 (cid:26)τn (cid:9)σ(cid:4) +τ (cid:10)+N(cid:9)τ(cid:4)+ 1(N +1)σ(cid:4)(cid:4)(cid:10)(cid:27)if m =n
σ σ n n 2
n n
areknownasthepseudospectraldifferentiationmatricesoforderoneandtwo,respectively
whichcanbederivedbykeepinginmindthat pN+1(x)satisfiestheEHTin(7)whenλ =
λN+1andpN+1(xm)=0form =0,1,...,N[2].In(17)and(18)weadopttheabbreviations
p(cid:4) = p(cid:4) (x ),σ =σ(x )andτ =τ(x ).
N+1,k N+1 k k k k k
Thus,inserting(17)and(18)into(16)weobtainanexplicitrepresentationfortheentries
⎧
B(cid:21) =− 1 ⎪⎪⎪⎪⎨(xm12−σmxn)2 pp(cid:4)N(cid:4)N++11,,mn if m (cid:7)=n (19)
mn 6rm ⎪⎪⎪⎪⎩ τn(cid:9)τ −2σ(cid:4)(cid:10)−2N(cid:9)τ(cid:4)+ 1(N +1)σ(cid:4)(cid:4)(cid:10)−6q if m =n
σ n n 2 n
n
ofthematrixB(cid:21)wherer =r(x )andq =q(x ).Bymeansofthesimilaritytransformation
k k k k
B= S−1B(cid:21)SwhereSisadiagonalmatrixwithentries
(cid:28)
σ
Smn = rm p(cid:4)N+1,mδmn, m =0,1...,N (20)
m
itispossibletoavoidthecomputationoftheterms p(cid:4) (x ).Moreover,thenewmatrixB
N+1 m
issymmetricwhoseentriesaregivenby
⎧ (cid:28)
⎪⎪⎪⎪⎨(x −12x )2 σrmrσn if m (cid:7)=n
1 m n m n
Bmn =−6⎪⎪⎪⎪⎩ 1 (cid:26)τn(cid:9)τ −2σ(cid:4)(cid:10)−2N(cid:9)τ(cid:4)+ 1(N +1)σ(cid:4)(cid:4)(cid:10)−6q (cid:27)if m =n. (21)
r σ n n 2 n
n n
Therefore,theeigenvaluesof(15)andhence,theapproximateeigenvaluesof(6),maybe
determinedbythesymmetricmatrix-eigenvalueproblem
Bv =λ(c,m)v (22)
sincethesimilarmatricessharethesamespectrum.Clearlyaneigenvector yof(15)isgiven
by
y= Sv (23)
intermsofaneigenvectorvofthesymmetricmatrixB.
3 TheJacobiPseudospectralMethodsforthePSWE
In(4),togetridofthesingulartermproportionalto1/(1−η2)wetransformthedependent
variablebythemapping
y(η)=(1−η2)m/2u(η) (24)
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leadingtotheequation
(1−η2)u(cid:4)(cid:4)−2(m+1)ηu(cid:4)+q(η)u =μu, μ=m(m+1)−λ(c,m) (25)
with
q(η)=−(cη)2. (26)
Notethatthesolutionu(η)doesnothavetosatisfyanyboundaryconditionssincethetrans-
formationin(24)takescareoftheboundaryconditionsin(2).
Noticethat,thesystem(1)isreflectionsymmetric,i.e.,itisinvariantunderthereplacement
oftheindependentvariablebyitsnegative.Thus,thespectrumλ (c,m)ofPSWEcanbe
n
decomposedintotwosubsetscontainingmerelythesymmetricλ (c,m)andanti-symmetric
2n
λ2n+1(c,m) statessothatthecorrespondingeigenfunctions y2n(η)and y2n+1(η)areeven
andoddfunctionsofη,respectively.Byseparatingtheevenandoddstates,onedealswith
twomatricesofsizeN×N insteadofamatrixofsize2N×2N togettheeigenvalueswhich
reducesthecostbyafactoroffour.Now,startingwiththeeventransformation
η=cos2θ (27)
thesystemin(1)maybewrittenas
(cid:2) (cid:5)
(1−η2)y(cid:4)(cid:4)+ 1(1−3η)y(cid:4)− m2 + c2(1−η) y =−1λ(c,m)y, y(−1)=0.
2 2(1+η) 8 4
(28)
Thenproposingasolutionoftype
y(η)=(1+η)m/2u(η) (29)
satisfyingtheboundaryconditionatη=−1,wegettheequation
(cid:2) (cid:3) (cid:4) (cid:5)
(1−η2)u(cid:4)(cid:4)+ m+ 1 − m+ 3 η u(cid:4)+q(η)u =μu, (30)
2 2
whereμ = 1[m(m+1)−λ (c,m)]and
n 4 2n
c2
q(η)= (η−1) (31)
8
fortheevenstatesofthePSWE.Ontheotherhand,forthetreatmentoftheoddstatesfirst
letting
y(θ)=sinθφ(θ) (32)
whereφisnecessarilyanevenfunctionofθ,wetransformtheEq.(1)into
(cid:3) (cid:4)
m2
−φ(cid:4)(cid:4)+(tanθ −2cotθ)φ(cid:4)+ +c2sin2θ φ =[λ(c,m)−2]φ (33)
cos2θ
with the transformed boundary conditions φ(±1) = 0. The evenness of the function φ
suggeststheapplicationofthesamemapsη=cos2θ andφ(η)=(1+η)m/2u(η)toreach
attheequation
(cid:2) (cid:3) (cid:4) (cid:5)
(1−η2)u(cid:4)(cid:4)+ m− 1 − m+ 5 η u(cid:4)+q(η)u =μu (34)
2 2
whereμn = 41[(m+1)(m+2)−λ2n+1(c,m)]andq(η)hasalreadygivenin(31).Notice
thatwhenc=0,Eq.(25)resemblestheJacobidifferentialequation
(1−η2)y(cid:4)(cid:4)+[β−α−(α+β+2)η]y(cid:4)+n(n+α+β+1)y =0 (35)
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with(α,β)=(m,m)whereas(30)and(34)aremuchakintotheJacobidifferentialequation
with(α,β) = (−1,m)and(α,β) = (1,m),respectively.Actually,theequationsin(30)
2 2
and(34)canbeputtogethertogivetheequation
(1−η2)u(cid:4)(cid:4)+[β−α−(α+β+2)η]u(cid:4)+q(η)u =μu, (36)
where
(cid:9)(cid:29) (cid:30)(cid:29) (cid:30) (cid:10)
1
μ= α+β+ 1 α+β+ 3 −λ(c,m) . (37)
4 2 2
Here (α,β) = (−1,m) and (α,β) = (1,m) yield the even and odd states, respectively,
2 2
whichcanbeseenonreturningbacktotheoriginalvariableθ by(27),(29)and(32).Infact,
theinterrelations
P(m,m)(η)=(−1)n n!(2n+m)! P(−21,m)(1−2η2) (38)
2n (2n)!(n+m)! n
and
P(m,m)(η)=(−1)n n!(2n+m+1)! P(21,m)(1−2η2) (39)
2n+1 (2n+1)!(n+m)! n
betweentheJacobipolynomialswithintegerparametersmandn,supportsthatseparationof
theevenandoddstateshalvesthetruncationsizesinceoneonlyneedstouseP(±21,m)(1−2η2)
n
insteadof P(m,m)(η)or P(m,m)(η)intheconstructionoftheLagrangepolynomialsin(11).
2n 2n+1
Notethat,equation(36)isoftheform(6)withr(η) = 1,σ(η) = 1−η2 andτ(η) =
β−α−(α+β+2)η.Hence,inthiscase,thematrixin(21)readsas
⎧ (cid:14)
⎪⎪⎪⎨−2 (1−ηm2)(1−ηn2) if m (cid:7)=n
(η −η )2
B = m n (40)
mn ⎪⎪⎪⎩[β−α−(α+β+2−)6η(n1]−[βη−2α)−(α+β−2)ηn]−13N(N+α+β+3)+q(ηn)if m =n.
n
Theeigenvaluesμ ofthematrixin(40)leadstotheapproximateeigenvalues
n
λ (c,m)=m(m+1)−μ (41)
n n
ofPSWEwhenq(η )=−(cη )2whereη aretherootsoftheJacobipolynomialP(m,m)(η)
n n n N+1
whileityieldstheeven
λ (c,m)=m(m+1)−4μ (42)
2n n
andodd
λ2n+1(c,m)=(m+1)(m+2)−4μn (43)
stateswhenq(η )=c2(η −1)/8whereη aretherootsoftheJacobipolynomialP(−21,m)(η)
n n n N+1
and P(21,m)(η),respectively.
N+1
4 TheHermiteandLaguerrePseudospectralMethodsforthePSWE
EigenfunctionsofthePSWEareconfinedtoasmallintervalaroundtheoriginforverylarge
valuesofthebandwidthparameterc.Ontheotherhand,thezerosoftheJacobipolynomials
are denser at the boundaries and coarser in the middle of the interval (−1,1). Thus, if
the Jacobi pseudospectral methods described in the last section are used, most of the grid
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pointsarewastedintheintervalwheretheeigenfunctionsaretooclosetozero.Althoughthe
Jacobipseudospectralmethodsareveryefficientforsmallbandwidthparametersc<1000,
theyarenotfeasibleforverylargecvalues.However,theprolatefunctionscanaccurately
be approximated by the Hermite or the Laguerre functions with γ = ±1/2 when c tends
to infinity [4,10,12,15,19]. Therefore, in this section, we construct the Hermite and the
LaguerrepseudospectralformulationsofthePSWEforverylargevaluesofc>1000where
theformercomputesthefullspectrumatonceandthelatterseparatesitastheevenandodd
inordertohalvethetruncationsize N.
SincetheHermitepolynomialsaredefinedovertherealline,weapplythetransformation
x =αarctanhη, x ∈(−∞,∞) (44)
totheequationin(4),whereαisanoptimizationparameter.Itisimportanttonotethatthe
optimumvalueα ofthescalingparameterαwillnotbechosenempiricallywhichwillbe
opt
explainedlater.Thisleadstotheequation
−y(cid:4)(cid:4)+ 1 V (x/α)y = 1 λ(c,m)W(x/α)y, y(±∞)=0 (45)
α2 α2
overthereallinewith
V(t)=c2sech2ttanh2t+m2 and W(t)=sech2t. (46)
Finally,letting
y(x)=e−x2/2u(x) (47)
satisfyingtheboundaryconditionsatinfinity,wegetanequationoftype(6)
u(cid:4)(cid:4)−2xu(cid:4)+q(x)u =μr(x)u, μ= 1 λ(c,m) (48)
α2
withσ(x)=1,τ(x)=−2x,λ=μ,
1
q(x)=x2−1− V (x/α) (49)
α2
and
r(x)=W(x/α). (50)
Noticethat(48)resemblestheHermitedifferentialequationu(cid:4)(cid:4)−2xu(cid:4)+2nu = 0andin
thiscase(21)readsas
⎧
⎪⎪⎪⎨−2cosh((xxm/−α)xco)2sh(xn/α) if m (cid:7)=n
m n
Bmn =⎪⎪⎪⎩cosh2(x /α)(cid:2)−2(cid:29)x2+N(cid:30)+q(x )(cid:5)if m =n (51)
n 3 n n
whereq(xn)arethevaluesof(49)attherootsofHN+1(x).Thentheeigenvaluesμn of(51)
leadstotheapproximateeigenvalues
λ (c,m)=α2μ (52)
n n
ofthePSWEforverylargevaluesofthebandwidthparameterc.Ontheotherhand,reflection
symmetriccharacterof(4)suggeststheuseofeventransformation
x =(αarctanhη)2, x ∈(0,∞) (53)
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JSciComput(2017)71:804–821 813
inordertoseparatethesymmetricandanti-symmetricstates.Thisleadstotheequation
xy(cid:4)(cid:4)+ 1y(cid:4)− 1 V (cid:29)√x/α(cid:30)y =− 1 λ(c,m)W(cid:29)√x/α(cid:30)y, y(∞)=0 (54)
2 4α2 4α2
whereV andW aredefinedin(46).Lastly,proposingasolutionoftype
y(x)=xae−x/2Φ(x) (55)
satisfyingtheboundaryconditionatinfinitywherethepolynomialtermhasbeenintroduced
tocopewiththeartificialsingularityatx =0,weobtaintheequation
(cid:3) (cid:4) (cid:2) (cid:5)
xΦ(cid:4)(cid:4)+ 2a+ 1 −x Φ(cid:4) + 1 2a(2a−1) −(4a+1)+x− 1V (cid:29)√x/α(cid:30) Φ
2 4 x c
1 (cid:29)√ (cid:30)
=− λ(c,m)W x/α Φ. (56)
4α2
Thenforsimplicityletting2a+ 1 = γ+1andchoosingγ = ±1 (ora = 0,a = 1)we
2 2 2
eliminatethetermproportionalto1/x inordertoendupwiththeequation
xΦ(cid:4)(cid:4)+(γ+1−x)Φ(cid:4)+q(x)Φ =μr(x)Φ, μ=− 1 λ(c,m) (57)
4α2
whichisalsoanequationoftype(6)with
(cid:2) (cid:5)
1 1 (cid:29)√ (cid:30) (cid:29)√ (cid:30)
q(x)= x−2(γ+1)− V x/α and r(x)=W x/α . (58)
4 c
Clearly the solutions in (55) with γ = −1 and γ = 1 yield even and odd states of the
2 2
PSWE,respectively,whichcanbeseenonreturningbacktotheoriginalvariableηvia(53).
Toapproximatetheeven
λ (c,m)=−4α2μ (59)
2n n
andodd
λ2n+1(c,m)=−4α2μn (60)
stateeigenvaluesofthePSWEweonlyneedtodiagonalizethematrix
⎧ √ (cid:29)√ (cid:30) (cid:29)√ (cid:30)
⎪⎪⎪⎪⎨−2 xmxncosh(x −xmx/α)2cosh xn/α if m (cid:7)=n
m n
Bmn =⎪⎪⎪⎪⎩cosh2(cid:29)√x /α(cid:30)(cid:26) 1 (cid:9)1−(γ−x )2(cid:10)− 1N +q(x )(cid:27)if m =n. (61)
n n n
6x 3
n
whenγ = −1 andγ = 1,respectivelywhereq(x )arethevaluesofthefunctionq(x)in
2 2 n
(58)atthezerosofL(γ) (x).
N+1
In[15]theauthorsusedtheevenandoddindexedHermitefunctionstoapproximatethe
symmetric and anti-symmetric state eigenvalues of the PSWE, respectively. On the other
hand,ourHermitepseudospectralformulationyieldsthefullspectrumatonce.Nevertheless,
theLaguerrepseudospectralformulationofthepresentstudycanseparatetheeven(γ=−1)
2
andodd(γ= 1)states.Actually,theapproachof[15],i.e.theuseofevenandoddindexed
2
HermitefunctionsinapseudospectralpictureandthepresentLaguerrepseudospectralmeth-
odswithγ=±1 canberegardedasequivalentmethodsifweremembertheinterrelations
2
H (x)=(−1)n22nn!L(−1/2)(x2) (62)
2n n
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