NumericalAlgorithmsforPhysics:ProjectList

AndreaMignonePhysicsDepartment,UniversityofTorino

AA2018-2019

WritingtheProject•  InthefollowingIwillshowalistofprojectsforthiscourse.

•  Eachprojectshouldbenomorethan≈10pageslong(excludingthecodelistedintheappendix).

•  A.pdffileisstronglyrecommended.

•  Thedocumentstructureshouldconsistsof

–  Anabstract/Introductorypartwherethephysicalproblemisexplainedandwhyweneedtoresorttonumericalintegration.

–  Atestsectionwherethenumericalmethodisvalidatedagainstknownanalytical/referencesolution.

–  Amodelstudyoftheproblemincludingplots.–  Afinalsummary/discussion.–  Anappendixincludingthecodeusedfortheproject,usingfixed-sizefonts(afontwhose

lettersandcharacterseachoccupythesameamountofhorizontalspace,e.g.,Courier,CourierNew,LucidaConsole,Monaco,andConsolas).

Project#1:FinitePotentialWell•  Considerthetime-independentSchrödingerequationinonedimension,

whereψ(x)isthewavefunction,mistheparticlemass,EitsenergyandV(x)isthe

potentialenergy.Theprobabilitytofindtheparticlebetweenxandx+dxis•  Considerthepotentialwellgivenby

•  InthiscasetheSchrödingerequationbecomes

•  Sincethepotentialissymmetric,weassumethatthewavefunctionshavedefined

parity(oddoreven).

Project#1:FinitePotentialWell:Wavefunctions•  ForanevenwavefunctionwemusthaveInx=aweimposecontinuityconditions:BysolvingthisequationweobtaintheeigenvaluesE.Rememberthatαandβare

functionsofE.

•  Foranoddfunction,andimposingthesamecontinuityconditionsweget

Project#1:FinitePotentialWell:Purpose•  Computetheeigenvaluesofthefinitepotentialwell.

•  Usedimensionlessunitsbyintroducingsothattheequationstobesolved(inη)are:•  Lookforboundstateswhichsatisfy–K<η<0and,toavoiddealingwithsingularities

inthetangentitisbettertorewritetheequationsas

Project#1:FinitePotentialWell:GraphicalSolution•  Agraphicalrepresentationofthesolutionisgivenbelow:

Project#2:RealisticProjectileMotion•  Weconsiderthemotionofaparticlesubjecttodragforce

whereisthedragforceduetoairresistance(forthepresentcalculationyoucanuseB = 4.e-5 m-1).

•  Thisforceisalwaysoppositetovelocityandthereforeremembertowriteitinvectorcomponents.

•  Foragiveninitialvelocityv0anddistanceLtoatarget,determinetheangles(ifany)youmustorientyourcannonatinordertohitthetarget.

y

xθ

L

Project#3:RealisticPendulum•  Hereweconsidertheequationofapendulumforarbitraryamplitudeandsubjectto

dampingaswellasdrivingforce:

•  HereqisameasureofdampingwhileFDandΩDaretheamplitudeandfrequencyofthedrivingterm.Transformtheproblemintoasystemofcoupled1storderODE:

•  Sothatourvectorofunknownsis,Y = (θ, ω).

•  Withzerodrivingforcethemotionisdamped.–  WithFD=0.5therearetworegimes:–  AninitialtransientdecaywherethemotionwithangularfrequencyOmegaisdamped–  Afollowingphasewherethependulumsettlesinintoasteadyoscillationinresponsetothe

drivingforce.

Project#3:RealisticPendulum•  ThebehaviorchangesdramaticallywhenFD=1.2sincethemotionisnolongersimple

evenatlongtimes.•  Thesystemdoesnotsettleintoarepeatingsteadystatebehaviorandthisisan

indicationofchaoticbehavior.

Project#4:theDoublePendulum•  Adoublependulumconsistsofonependulum

attachedtoanother.Itisanexampleofasimplephysicalsystemwhichcanexhibitchaoticbehavior.

•  Consideradoublebobpendulumwithmassesm1 andm2 attachedbyrigidmasslesswiresoflengthsL1andL2.Further,lettheanglesthetwowiresmakewiththeverticalbedenotedθ1andθ2.

•  Thepositionofthetwomassesaregivenby

Project#4:theDoublePendulum•  Aftersometediousalgebra,theequationsofmotioncanbewrittenas:

•  Usingm1 = m2 andL1 = L2,studythedoublependulummotionbydirectintegrationoftheequationsofmotion.

•  Trytoaddressthefollowingissues:

–  Isenergyconserved?–  Canyoudeterminearangeofinitialconditionthatleadsthesystemtochaos?–  Doesthependulumflip?

Project#5:Three-bodyproblem•  Considerthesimplestthree-bodyproblemgivenbytheearth,theSunandJupiter.

•  WeknowthatwithoutJupiter,theEarth’sorbitisstableanddoesnotchangeintime.

•  OurobjectiveistoquantifyhowmucheffectthegravitationalfieldofJupiterhasonEarth’smotion.

•  ChangetheJupitermassbyafactorof10,100and1000:doyoustartseeinganeffect?

Project#6:Lane-EmdenEquation•  Inastrophysics,theLane–EmdenequationisadimensionlessformofPoisson's

equationforthegravitationalpotentialofaNewtonianself-gravitating,sphericallysymmetric,polytropicfluid.ItisnamedafterastrophysicistsJonathanHomerLaneandRobertEmden.

•  Asphericallysymmetricstarinhydrostaticequilibriummustobeythehydrostaticbalanceequation

wheremassisrelatedtodensityby

•  Assuminganadiabatic“quasi-static”changeofstateofthegasfollowing:

onecanrewritethepreviousequationasThisistheLane-Emdenequation.•  Theequationiswrittenintermsofdimensionlessvariabledefinedby

Project#6:Lane-EmdenEquation•  ExactsolutionstotheLane-Emdenexistforn=0,1,5(seee.g.,wikipedia)but

otherwisenumericalintegrationmustbeused.•  Boundaryconditions:togetauniquesolutiontotheLane-EmdenEquation,weneed

tospecifytwoboundaryconditionsforthis2ndorderODE.Arealisticmodelcannothavea‘cusp’attheoriginwhichmeansthat

•  Becauseρ≈θn,onlyθ≥0canberealistic:thisdefinesthesurfaceofthepolytropeisdefinedastheradiusatwhichθfirstbecomeszero(orquitesmall).Thisisdesignatedasξs,so

•  Ifthesurfaceseemstobeapproachinginfinityinsize,(e.g.forn=5)thecodeshouldstoptheintegration.

•  Notethat,atthebeginning,anindefinitevalueof1/ξ*dθ/dξexists.Thisissolvedbyexpandingθas

Project#6:Lane-EmdenEquation•  SolvetheLane-Emdenequationsfordifferentvalues

ofn=0,1,3/2,3.•  Verify(whenpossible)againstanalyticalsolution

(e.g.n=1àθ=sinξ/ξ).•  Findtheradiusofthedifferentpolytropesandmake

plotofthedensityinunitsofthecentraldensity.

•  Forwhitedwarfswithhighdensities,theequationofstateiswellapproximatedbyapolytropicequationofstatewithindexn=3.TheconstantKinthepolytropicequationofstatethenis

•  Whatisthemassofahighdensitywhitedwarf?Ifyouhavedoneeverything

•  right,youwillhaverediscoveredtheChandrasekharMass!

Project#7:Particle(s)inEMFields•  Studythemotionofoneormore(non-relativistic)particlesinafixedelectromagnetic

field:

•  Thepreviousequationiswritteninthec.g.ssytem(widelyusedinastrophysics)butdimensionlessunitsarerecommended.

•  Theequationofmotion(possibly)involvespropagationinall3direction(x,y,z).

•  Theelectricandmagneticfieldvectoraregivenexternallyandparticlesdonotinteractwitheachother.

•  TheprojectinvolvesdirectintegrationoftheequationofmotionusingRK-typeintegratorsand/orthegeneralizationofsymplecticintegrationschemetothecaseofvelocity-dependentforce(theBorisalgorithmistheprogenitorofsuchschemes).

Project#7:Particle(s)inEMFields•  Provideaprojectwithatleastthreedifferentcases.Asuggestionisgivenhere:

Case Npart E B Notes

Simplegyration 1 (0,0,0) (0,0,1) Considerbothperpendicularandparallelpropagation.Checkyourresultswithanalyticalformula.

ExBdrift 1 (0,E,0)withE<1

(0,0,1) Considerpropagationalongy-direction:whatmotiondoyousee?Doestheparticleaccelerate?Explain.

Xpoint 103 (0,0,1/2) (y/L,x/L,0)L=103.

Placeparticlesuniformlyinthesquaredomain[-L,L]2andinitializeparticlevelocityto0.1usingrandomlynumbersdistributedangles.Describeyourresults:-  Whatkindofmotionisobserved?-  Doparticlesaccelerate?Where?

Project#8:PhysicsofPartiallyIonizedHydrogen•  Inavarietyofastrpphysicalscenarios(protoplanetarydisks,interstellarmedium,

stellarinteriors,supernovae,etc…)theinternalenergyoftheplasmaissubjecttoradiativecoolingduetoavarietyofprocesses,includingbremmstrahlung,collisionalionizationandexcitation,etc…

•  Forauniformgasdistribution(nospatialvariation)andassumingapartiallyionizedhydrogengas,thisequationmaybesimplifiedandwrittenas

whereΛ(n,T)isthecoolingfunction,neisthenumberofelectrons,nisthehydrogennumberdensity.•  Thegasinternalenergyincludesastandardkinetictermplustheionizationenergy

(neutralatomshaveapotentialenergythatislowerthanthatofionsbyanamountχ0=13.6eV).

•  Herex(T)istheionizationfraction,definedby

Project#8:PhysicsofPartiallyIonizedHydrogen•  Indilutegases(asitisthecaseforseveralastrophysicalenvironmentssuchasthe

interstellarmediumoraproto-planetarydisk),thedegreeofionizationxcanbecomputedusingCollisionalexcitationequilibrium(CIE),accordingtowhich

whereα=157890.0,whileχ0=13.6.•  Thecoolingfunctionisusuallytabulatedbut,asaverycrudeapproximation,wecan

usethefollowingempiricalrelation:

whereC=10-22erg/(cm3s).

Project#8:PhysicsofPartiallyIonizedHydrogen•  AssuminganinitialtemperatureofT=106Kandacloudofconstantdensityn=1cm-3,

solvetheinternalenergyequation

•  Notethatateachstep,thetemperaturemustbefoundfromtheinternalenergyby

invertingtheexpressionfortheinternalenergy:

wheren(thegasnumberdensity)isfixedthroughouttheevolution.Arootfindingalgorithmmustbeused.•  StopwhenyoureachT≈103K.

•  Produceaplotofthetemperatureasafunctionoftime.

Project#9:Potentialflowaroundcylinder•  Thepotentialflowaroundacircularcylinderisaclassicalsolutionfortheequationsof

aninviscid,incompressiblefluidflow.•  Farfromthecylinder,theflowisunidirectionalanduniform.•  Theflowisincompressibleandhasnovorticity:sothatitsvelocitycanthusbewritten

1.  asthegradientofapotential:2.  usingthestreamfunction:•  BoththevelocitypotentialandthestreamfunctionssatisfytheLaplaceequation:

Project#9:Potentialflowaroundcylinder•  Solvetheprobleminpolarcoordinates(r,θ)

•  Discretizedthepreviousequationonadomaindefinedbya≤r≤5a,0≤θ≤2π[orsimilar,seta=1]

•  Usetheexactsolution(simple)toprescribethe

boundaryconditions.•  OracombinationorDirichlet+Neumannb.c.

•  Computeerror.

a

Project#10:PoissonEquationinAxialsymmetry•  Threedimensionalproblemswithaxialsymmetrycanbetreatedusingcylindrical

coordinates(r,z).

•  GeneralizetheiterativealgorithmspresentedinCh.10totosolvethePoissonequationin2Dcylindricalcoordinates

•  Applytheresultingdiscretizationtoanumberofproblemssuchas:–  Uniformlychargeddisk–  Uniformlychargedring–  Dipolefield(assumechargesaredistributedonafinitesizespheres).

•  Compareresultswithknownanalyticalsolutionsontheaxis.