An Introductory Guide to Computational Methods for the Solution of Physics Problems: With Emphasis on Spectral Methods

227 Pages·2018·6.306 MB·English

by  George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto

George Rawitscher 
Victo dos Santos Filho 
Thiago Carvalho Peixoto
An Introductory Guide
to Computational
Methods for the
Solution of Physics
Problems
With Emphasis on Spectral Methods
An Introductory Guide to Computational Methods
for the Solution of Physics Problems
–
George Rawitscher (1928 2018)
Victo dos Santos Filho Thiago Carvalho Peixoto
(cid:129)
An Introductory Guide
to Computational Methods
for the Solution of Physics
Problems
With Emphasis on Spectral Methods
123
George Rawitscher (1928–2018) ThiagoCarvalho Peixoto
University of Connecticut Federal Institute of Sergipe(IFS)
Storrs,CT, USA NossaSenhora daGlória,Sergipe, Brazil
Victodos SantosFilho
H4DScientificResearch Laboratory
BelaVista, São Paulo, Brazil
ISBN978-3-319-42702-7 ISBN978-3-319-42703-4 (eBook)
https://doi.org/10.1007/978-3-319-42703-4
LibraryofCongressControlNumber:2018950796
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Fig. 1 The joy of teaching Fig. 2 The joy of learning
This book is dedicated to all the persons who
love to learn and teach. The pictures show
G. Rawitscher’s grandson Uly and his friend
Olive aspreschoolers. Here they areteaching
and learning, and enjoying being together.
Preface
The main purpose of this monograph is to provide an introduction to several
numerical computational methods for solving physics problems. At the same time,
it serves as an introduction to more advanced books, especially Trefethen [1] and
Shizgal [2], which we have frequently referenced. Both of these books use
MATLABintheirnumericalexamples,whichisalsothecaseforourpresentwork.
This monograph may then be seen as a good text for a course titled, for example,
“Introduction to Numerical Methods for Undergraduate Quantum Mechanics”.
Our text provides many examples of computational methods applied to solving
physics problems. These include expansions into a set of Sturmian functions
(Chap. 11), the iterative calculation of eigenvalues for a particular differential
equation(Chap.10),thephase–amplitudedescriptionofawavefunction(Chaps.8
and 12), the solution of a third-order differential equation (Chap. 12), the finite
element method to solve a differential equation (Chap. 7), the transformation of a
second-orderdifferential equationintoanintegralequation(Chap. 6),ortheuseof
expansionsintoChebyshevpolynomials(Chaps.5and6),whilekeepinganeyeon
accuracy and convergence properties (Chap. 4). Chapters 3–6 describe “spectral”
expansions,andareincludedbecausesuchexpansionsprovideveryaccurateresults
and are not usually taught in courses on computational methods.
The present monograph gathers together these various computational methods
that are best suited for solving physics problems. We examine the errors of such
methods with many examples and show that spectral methods can be faster and at
the same time more accurate than finite difference methods. This is also demon-
strated by means of the numerical examples in Chaps. 6 and 7. The monograph is
notintendedtobemathematicallyrigoroussincesomanyexcellentmathematically
rigorous books already exist describing spectral methods [3–16]. Such methods
havebeenusedtosolvemanydifferentequations,asisthecaseofVlasovequation
[17], Navier–Stokes equation [18], Fisher equation [19], Schrödinger and Fokker–
Planck equations [20] amongst others.
Spectral methods were first introduced in the 1970s. They are more advanta-
geous than other methods because they tend to converge quickly and generally
provide high accuracy, as described in Sect. 3.4.2. However, finite difference
vii
viii Preface
methods(basedontheTaylorseries)stillareofimportanceforspecificapplications
[21–22], some of which we describe in Chap. 2. Spectral methods lead to matrix
equationsthataremorecomplicatedthanthoseforthefinitedifferencemethods,but
the spectral convergence and accuracy gained [2] easily outweigh any drawback.
Inmoredetail,thecontentofthismonographisasfollows:Chaps.1and2givea
reviewofcomputationalerrorsandfinitedifferencemethods,respectively.Chapters
3 and 4 describe the collocation and Galerkin methods. An advantage of these
methods is demonstrated in Chap. 4 by giving theorems on the convergence of
spectralexpansions.AdvantagesarealsoshowninChaps.5and6,wherepractical
examples are given for the convergence of expansions in Chebyshev polynomials.
Chapter 6 describes the Lippmann–Schwinger integral equation whose solution
gives better accuracy than the equivalent Schrödinger equation, and is solved with
the aid of Green’s functions, all in coordinate space. In Chap. 7, we compare
various finite element spectral methods. Chapters 8–12 are dedicated to various
computational method examples.Chapter 8describes thephase–amplitudemethod
anditsapplicationtophysicalproblemsinvolvinginterestingpotentials.InChap.9,
we describe the solution for the problem of finding eigenvalues iteratively in a
simple example of a vibrating inhomogeneous string. Chapter 10 develops an
iterative method to obtain the energy eigenvalues of the second-order differential
equation for a vibrating string. A review of expansions in Sturmian functions is
presented in Chap. 11. Chapter 12 provides a novel method to solve a third-order
differential equation, based on spectral expansions and the implementation of the
asymptotic boundary conditions without the use of Green's functions. Finally, in
Chap.13,wepresentourfinalconsiderationsandgeneralconclusionsofthepresent
work.
Insummary,thepurposeofthismonographistoprovideacompactandsimple
introduction to several computational methods generally not taught in traditional
courses,andtoexaminetheerrorsandtheadvantagesofsuchmethodsasshownin
manyexamples.Wehopethatthismonographprovidesstudentsandteacherswith
a comprehensive foundation for a smooth transition to more advanced books.
Storrs, CT, USA George Rawitscher
São Paulo, Brazil Victo dos Santos Filho
São Paulo, Brazil Thiago Carvalho Peixoto
April 2018
References
1. L.N.Trefethen,SpectralMethodsinMATLAB(SIAM,Philadelphia,2000)
2. B.D.Shizgal,SpectralMethodsinChemistryandPhysics.ApplicationstoKineticTheoryand
QuantumMechanics(Springer,Dordrecht,2015)
3. D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods (SIAM, Philadelphia,
1977)
Preface ix
4. B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge Monographs on
Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK,
1998)
5. D.Gottlieb,J.S.Hesthaven,J.Comput.Appl.Math.128(1–2),83–131(2001)
6. J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems
(CambridgeUniversityPress,Cambridge,2007)
7. J.P.Boyd,ChebyshevandFourierSpectralMethods(Dover,NewYork,2001)
8. C. Canuto, M. Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods: Fundamentals in
SingleDomains(Springer,NewYork,2006)
9. M.O.Deville,P.F.Fisher,E.H.Mund,HighOrderMethodsforIncompressibleFluidFlow
(CambridgeUniversityPress,Cambridge,2002)
10. C.-I. Gheorghiu, Spectral Methods for Differential Problems, Casa Cartii de Stiinta,
Cluj-Napoca,Romania,2007.Availableathttp://www.ictp.acad.ro/gheorghiu/spectral.pdf
11. R.Peyret,SpectralMethodsforIncompressibleViscousFlow(Springer,NewYork,2002)
12. G.Ben-Yu,SpectralMethodsandTheirApplications(WorldScientific,Singapore,1998)
13. D.Funaro,PolynomialApproximationofDifferentialEquations(Springer,Berlin,1992)
14. D.A.Kopriva,ImplementingSpectralMethodsforPartialDifferentialEquationsAlgorithms
forScientistsandEngineers(Springer,Berlin,2009)
15. C.Shu,DifferentialQuadratureandItsApplicationinEngineering(Springer,Berlin,2000)
16. J. Shen, T. Tang, L.-L. Wang, Spectral Methods: Algoritms, Analysis and Applications
(Springer,Berlin,2011)
17. L.Gibelli,B.D.Shizgal,SpectralconvergenceoftheHermitebasisfunctionsolutionofthe
Vlasovequation:Thefree-streamingterm.J.Comput.Phys.219(2),477–488(2006)
18. P.R. Spalart, R.D. Moser, M.M. Rogers, Spectral methods for the Navier-Stokes equations
withoneinfiniteandtwoperiodicdirections.J.Comput.Phys.96(2),297–324(1991)
19. D.Olmos,B.D.Shizgal,AspectralmethodofsolutionofFisher’sequation.J.Comput.Appl.
Math.193(1),219–242(2006)
20. J. Lo, B.D. Shizgal, Spectral convergence of the quadrature discretization method in the
solution of the Schrödinger and Fokker-Planck equations: comparison with sinc methods.
J.Chem.Phys.125(19),8051(2006)
21. Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential
Equations,SteadyStateandTimeDependentProblems(SIAM,Philadelphia,2007)
22. L.N.Trefethen,FiniteDifferenceandSpectralMethodsforOrdinaryandPartialDifferential
Equations (Cornell University, Department of Computer Science, Center for Applied
Mathematics,Ithaca,NY,1996)
Acknowledgements
The authors thank all professionals and friends who have analysed the content
of the book and helped to elaborate and improve it. One of the authors (G. R.) is
grateful to the International Centre for Theoretical Physics-South American
Institute for Fundamental Research (ICTP-SAIFR), in particular to its director
Nathan Jacob Berkovits and the Secretariat of the institute, for the invitation to
teach a mini-course on spectral computational methods in São Paulo, Brazil. The
coursetookplacefrom16Marchto26April2015andconsistedoftwelvelectures.
These lectures provided the initial stimulus for writing this monograph, in coop-
eration with the two Brazilian co-authors (V. S. F. and T. C. P.) who took the
course and contributed significantly to writing the text. G. R. is very grateful to
Profs.LauroTomioandSadhanK.Adhikariforenthusiasticallysupportingthevisit
and for their dedicated hospitality. G. R. is also much indebted to his parents and
his wife Joyce, who inspired in him throughout his life a spirit of freedom and
accomplishment. The author V. S. F. thanks God and his parents for always sup-
portingandhelpinghiminhislife.V.S.F.alsothankseachmemberofhisfamily
and all of his friends (with special acknowledgement to Prof. Lauro Tomio) who
encouraged or helped him in the process of writing this monograph.
ICTP-SAIFR is a South American Regional Centre for Theoretical Physics
created through a collaboration of the Abdus Salam International Centre for
Theoretical Physics (ICTP) with the São Paulo State University (UNESP) and the
São Paulo Research Funding Agency (FAPESP). The author G. R. would like to
recognize FAPESP grant 2011/11973-4 for funding his visit to ICTP-SAIFR.
Finally,theauthorsthankPeterandHenryRawitscherfortheirvaluablehelpof
seeing through the publication and in proofreading the text and correcting the
english grammar, which greatly improved our monograph.
xi
Contents
1 Numerical Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Objective and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Accuracy in Numerical Calculations. . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 The Objective and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Order of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 The Physics of the Frictionless Pendulum . . . . . . . . . . . . . . . . 10
2.3.1 Assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 The Physics of the Descending Parachute. . . . . . . . . . . . . . . . . 13
2.4.1 Assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Galerkin and Collocation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 The Objective and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Introduction to Galerkin and Collocation Methods . . . . . . . . . . 17
3.3 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Some Useful Comments . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4.1 Details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.2 Advantage of a Non-equispaced Mesh. . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Convergence of Spectral Approximations . . . . . . . . . . . . . . . . . . . . 33
4.1 The Objective and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Fourier Spectral Differentiation on Bounded Periodic Grids. . . . 35
xiii