Computational Physics — 3rd/4th Year Option AngusMacKinnon September26,2002 Computational Physics Course Description Theuseofcomputersinphysics,aswellasmostotherbranchesofscienceandengineering,hasincreased manytimesalongwiththerapiddevelopmentoffasterandcheaperhardware.Thiscourseaimstogivethe studentathoroughgroundinginthemaincomputationaltechniquesusedinmodernphysics. Itisparticu- larlyimportantinthiscoursethatthestudentsshouldlearnbydoing.Thecourseisthereforedesignedsuch that a significant fraction ofthe students’time isspent actually programmingspecific physicalproblems ratherthanlearningabstracttechniques. T(cid:0) hecoursewillcoverproblemsin4(5)broadsections: (cid:0) Ordinarydifferentialequations,suchasthoseofclassicalmechanics. Partialdifferentialequations,suchasMaxwell’sequationsandtheDiffusionandSchro¨dingerequa- (cid:0) tions. Matrixmethods,suchassystemsofequationsandeigenvalueproblemsappliedtoPoisson’sequation (cid:0) andelectronicstructurecalculations. MonteCarloandothersimulationmethods,suchastheMetropolisalgorithmandmoleculardynam- (cid:0) ics. (Iftimepermits:)ComputerAlgebra;anintroductionusingMapletotheusesandabusesofalgebraic computinginphysics. Thisisnotashortcourseincomputingscience,norinprogramming.Itfocusesspecificallyonmethods forsolvingphysicsproblems.Studentswillbeexpectedtobefamiliarwithbasicprogramming:successful completionofthe1styearcomputingLab.issufficient. Thereisnorequirementthatthepracticalworkbe doneusingMicrosoftC++onthedepartmentalcomputers,butanyonewishingtousesomeotherprogram- minglanguageorcomputershouldconsultthelecturerbeforehand. Thisistomakesurethereisbothhelp availablefromdemonstratorsandthatitwillbepossibletoassesstheworksatisfactorily. Adetailedtimetableforthecourseisgivenbelow. Eachofthe4(5)sectionsofthecoursewillbeaccompaniedbyashortproblemsheetincludingpractical exercisesandlongerprojects. Studentswillbeexpectedtodothefirstprojectand2others.Therewillbeachoiceofprojectsavailable inthelatersections. Thisyearyouwillberequiredtosubmityourprogramelectronicallyinadditiontoincludingitasan appendix in your project reports. You should also include an example input file and the corresponding output file. This will enable the assessors to check and run your program if necessary. Your program shouldbesentasaplaintexte-mailattachment(s)1toComputational-Physics, A. MacKinnon orph.compphys@ic.ac.ukbythestateddeadline.Donotsendittomypersonale-mailaddress.The first project is to be handed in to Martin Morris in the Computing Lab by 5.00pm on the 28th October. (cid:1)(cid:3)(cid:2)(cid:5)(cid:4)(cid:7)(cid:6) Thesewillbeassessedandreturnedbythedemonstratorswhoshouldprovidethestudentswithfeedback whichwillbeofuseinthelongerprojectslateron. Thisprojectwillcountfor ofthefinalresult. 1UsingOutlookclick Insert then File 1 During the Lab. sessions staff and demonstrators will be available, but the students are expected to do the practical work and the problems outside these times as well. I am aware that there is an General Relativitylecturescheduledfor3.00pmonTuesdays.NeverthelesstheLab.willbereservedforthiscourse forthewholeafternoonanddemonstratorswillbeavailableduringthisperiod. Thereportonthe2ndproject,whichshouldbechosenfromthe2offeredonPartialDifferentialEqua- tions,shouldbesubmittedby5.00pmonNovember18th. The3rdprojectmaybechosenfromthoseofferedonMatrixandMonteCarlomethods(orpossiblyon ComputerAlgebra)andthereportshouldbesubmittedby5.00pmonFridayDecember13th,thelastday ofterm. This timetablehas beendesignedtomakesurethatstudentsareworkingatasteadyrateduring theterm. (cid:1)(cid:9)(cid:8)(cid:10)(cid:4)(cid:7)(cid:6) From10.00am–12.00noononMonday6thJanuary2003,therewillbeashortwrittentestunderexam (cid:1)(cid:11)(cid:8)(cid:10)(cid:4)(cid:12)(cid:6) conditions. This will count for of the final result. The 2nd and 3rd projects will also count for . Thusthedistributionofmarksforthevariouspartsofthecoursewillbeapproximately Project1 10 Project2 30 Project3 30 Test 30 Week Lectures Practicals Deadlines Sep. 30 Thu. 9.00am Fri. 2.00pm Fri. 4.00pm Oct. 7 Fri. 4.00pm Tue. 2.00–5.00pm Oct. 14 Fri. 4.00pm Tue. 2.00–5.00pm Oct. 21 Fri. 4.00pm Tue. 2.00–5.00pm Oct. 28 Fri. 4.00pm Tue. 2.00–5.00pm Mon. 5.00pm1stProject Nov.4 Tue. 2.00–5.00pm Nov.11 Thu. 9.00am Tue. 2.00–5.00pm Nov.18 Thu. 9.00am Tue. 2.00–5.00pm Mon. 5.00pm2ndProject Nov.25 Thu. 9.00am Tue. 2.00–5.00pm Dec. 2 Thu. 9.00am Tue. 2.00–5.00pm Dec. 9 Thu. 9.00am Tue. 2.00–5.00pm Fri. 5.00pm3rdProject Jan. 6 Test: Mon. 10.00am–12.00noon For those with surnames beginning with A–K the test will be in lecture theatre 2 and for those with surnamesbeginningwithL–Zinlecturetheatre3. Therewillbenoexaminationinthesummer. AcopyofthetimetableaswellascontactdetailsforthedemonstratorsispostedontheExchangeserver. ThiscanbeaccessedfromOutlookunder//PublicFolders/AllPublicFolders/Physics/ComputationalPhysics. FurtherinformationisavailableontheWorldWideWebunder http://www.cmth.ph.imperial.ac.uk/angus/Lectures/compphys/. Chapter 1 Ordinary Differential Equations 1.1 Types of Differential Equation In this chapter we will consider the methods of solution of the sorts of ordinary differential equations (ODEs)whichoccurverycommonlyinphysics. ByODEswemeanequationsinvolvingderivativeswith respect to a single variable, usually time. Although we will formulate the discussion in terms of linear ODEsforwhichweknowtheanalyticalsolution,thisissimplytoenableustomakecomparisonsbetween thenumericalandanalyticalsolutionsanddoesnotimplyanyrestrictiononthesortsofproblemstowhich themethodscanbeapplied.Inthepracticalworkyouwillencounterexampleswhichdonotfitneatlyinto thesecategories. Theworkin thissectionisalso consideredin chapter16ofPressetal. (1992)orchapterII ofPotter (1973) Weconsider3b(cid:13)(cid:15)a(cid:14) sicdifferentialequations: (cid:14)(cid:23)(cid:22) (cid:4) (cid:24) (cid:14)(cid:25)(cid:22)(cid:26)(cid:14)(cid:28)(cid:27)(cid:30)(cid:29) (cid:31)(cid:7)!#"%$ (cid:16)%& (cid:13)(cid:17)(cid:16)(cid:19)(cid:18)(cid:21)(cid:20) (cid:20) (cid:13)(cid:15)(cid:14) TheDecayequation (1.1) $ (cid:14)(cid:23)(cid:22) (cid:4) (cid:24) (cid:14)(cid:25)(cid:22)(cid:26)(cid:14)(cid:28)(cid:27)(cid:30)(cid:29) (cid:31)(cid:7)!#" (cid:16)%& (cid:13)(cid:17)(cid:16) (cid:20) (cid:18)’(cid:20) (cid:13)(cid:15)(cid:14) TheGrowthequation (1.2) (cid:14),(cid:22) (cid:4) (cid:24) (cid:14)(cid:25)(cid:22)(cid:26)(cid:14)(cid:28)(cid:27)(cid:30)(cid:29) (cid:31)(cid:7)!#".- (cid:16)%& (cid:13)(cid:17)(cid:16)(cid:19)((cid:21))+* )+* TheOscillationequation (1.3) whicharerepresentativeofmostmorecomplexcases. Higherorderdifferentialequationscanbereducedto1storderbyappropriatechoiceofadditionalvari- ables. Thesimplestsuchchoiceistodefinenewvariablestorepresentallbutthehighestorderderivative. Forexample,thedampedharmonicoscillato(cid:13)(cid:17)r0 (cid:14)equatio(cid:13)(cid:15)n(cid:14) ,usuallywrittenas / (cid:14)(cid:23)(cid:22) (cid:4) (cid:13)1(cid:16) 0 (cid:18)32 (cid:13)(cid:17)(cid:16)(cid:19)(cid:18)54 (1.4) (cid:14) (cid:22)(cid:26)(cid:13)(cid:15)(cid:14)17(cid:28)(cid:13)(cid:17)(cid:16) 6 canberewrittenintermsof andvelocity(cid:13) intheformofapairof1storderODEs 6 2 4 (cid:14)1(cid:22) (cid:4) (cid:13)(cid:17)(cid:16) (cid:18) / 6 (cid:18) / (cid:13)(cid:15)(cid:14) (1.5) $ (cid:22) (cid:4) (cid:13)1(cid:16) 6 (1.6) 8 8 Similarlyany thorderdifferentialequationcanbereducedto 1storderequations. (cid:14) Such systems of ODEs can be written6in a very concise notati8on by defining a vector, y say, whose elementsaretheunknowns,suchas and i(cid:13)n(1.1). AnyODEin unknownscanthenbewritteninthe generalform " (cid:16)%&;(cid:22) (cid:4) (cid:13)(cid:17)(cid:16)9(cid:18) : y f y (1.7) 3 OrdinaryDifferentialEquations 4 8 (cid:16) whereyandfare –componentvectors. Rememberthatthereisnosignificanceintheuseoftheletter intheaboveequations. Thevariableis notnecessarilytimebutcouldjustaseasilybespace,asin(1.9),orsomeotherphysicalquantity. Formallywecanwritethesolutionof(1.7)as "<(cid:16)%&;(cid:22) "<(cid:16) (cid:27) &=$3>3? " "<(cid:16)CBD& (cid:16)CBE&(cid:15)(cid:13)(cid:17)(cid:16)CB : y y ?A@ f y (1.8) (cid:16)%(cid:27)GFH(cid:16) "<(cid:16)%& by integrating both sides over the interval . Although (1.8) is formally correct, in practice it is usuallyimpossibletoevaluatetheintegralontheright–hand–sideasitpresupposesthesolutiony . We willhavetoemployanapproximation. (cid:16) (cid:16)I(cid:22) (cid:4) All differential equations require boundary conditions. Here we will consider cases in which all the (cid:16) boundary conditions are defined at a particular value of (e.g. ). For higher order equations th*e boundaryconditionsmaybedefinedatdifferentvaluesof . Themodesofaviolinstringatfrequency (cid:13)(cid:17)0J(cid:14) 0 obeytheequation (cid:22)L$ * (cid:14) (cid:13)1K 0 M 0 (cid:14)N(cid:22) (cid:4) (1.9) withboundaryconditionssuchthat atbothendsofthestring. We shallconsidersuchproblemsin chapter3. 1.2 Euler Method (cid:16)P(cid:22)(cid:26)(cid:16)RQ(cid:28)S=TU$V(cid:16)CQ O Consideranapproximatesolutionof(1.8)overasmallinterval bywritingtheintegralas > ?AWYX(cid:12)Z " "<(cid:16) B& (cid:16) B&(cid:17)(cid:13)(cid:17)(cid:16) B (cid:1) (cid:16) " "<(cid:16) Q & (cid:16) Q &\[ : O : ? W f y f y (1.10) "](cid:16)CQ(cid:28)S#TJ&U(cid:22) "](cid:16)CQ1&=$ (cid:16) " "<(cid:16)CQ(cid:17)& (cid:16)CQ1&^[ toobtain O : y y f y or,inamoreconcisenotation, Q(cid:10)S#T (cid:22) Q $ (cid:16) " Q (cid:16) Q &;(cid:22) Q $ (cid:16) Q [ O : O y y f y y f (cid:16) (1.11) O Wecanintegrateoveranylargerintervalbysubdividingtherangeintosectionsofwidth andrepeating (1.11)foreachpart. Equivalentlywecanconsiderthatweha(cid:13)veapproxQ(cid:28)iS=mTat$edQthederivativewithaforwarddifference (cid:1) [ (cid:13)(cid:17)(cid:16)=_Q (cid:16) y_ y O y _ (1.12) _ WewillalsocomeacrosscentredandbackwardQ(cid:28)S=diTff$ereQ(cid:7)nce(cid:30)eTs, (cid:13) c f (cid:16) (cid:13)(cid:17)(cid:16) _Q (cid:1)a‘b yQ $ OQ(cid:7)egyT centred y_ _ (cid:16) (1.13) _ bd y O y backward respectively. Q (cid:22) "<(cid:16)CQ(cid:17)& (cid:16) (cid:16)%Q(cid:23)(cid:22) (cid:16) Q (cid:22) " Q (cid:16)CQ(cid:17)& Herewehaveusedanotationwhichisveryc8goOmmonincomput:ationalphysics,inwhichwecalculate y y atdiscretevaluesof givenby ,andf f y . Inwhatfollowswewilldropthevectornotationyexceptwhenitisimportantforthediscussion. OrdinaryDifferentialEquations 5 1.2.1 OrderofAccuracy (cid:14)(cid:30)"<(cid:16)%& (cid:16) Q HowaccurateistheEulermethod? Toquantifyth(cid:13)(cid:17)i(cid:14)swecon(cid:16)s0id(cid:13)er0 (cid:14)aTaylorexpansionof around (cid:14) Q(cid:28)S=T (cid:22)h(cid:14) Q (cid:16) O [ [ [ (cid:18) O (cid:13)(cid:17)(cid:16) _Q (cid:18) f (cid:13)(cid:17)(cid:16) 0 _Q (cid:18) _ _ _ _ (1.14) _ _ andsubstitutethisinto(1.11) (cid:13)(cid:15)(cid:14) (cid:16)C0 (cid:13)(cid:17)0(cid:5)(cid:14) (cid:14) Q (cid:16) O [ [ [ (cid:1) (cid:14) Q $ (cid:16)%jk"A(cid:14) Q (cid:16) Q & (cid:18) O (cid:13)(cid:17)(cid:16)i_Q (cid:18) f (cid:13)(cid:17)(cid:16) 0 _Q (cid:18) O : _ _ _ _ (cid:13)(cid:15)(cid:14) (1.15) _ _ (cid:22)l(cid:14)(cid:10)Q (cid:16) (cid:18) O (cid:13)(cid:17)(cid:16) _Q : _ _ (1.16) _ (cid:16) O wherewehaveused(1.7)toobtainthefinalform. Hence,weseethatthetermin intheexpansionhas beencorrectlyreproducedbytheapproximation,butthatthehigherorderQ termsarewrong. Wetherefore (cid:16) describetheEulermethodas1storder8accurate. O Anapproximationtoaquantityis thorderaccurateifthetermin intheTaylorexpansionofthe quantityis correctlyreproduced. The orderofaccuracyofa method is theorder ofaccuracywith which theunknownisapproximated. Notethatthetermaccuracyhasaslightlydifferentmeaninginthiscontextfromthatwhichyoumight use to describethe resultsofan experiment. Sometimesthe term orderof accuracyisused to avoidany ambiguity. (cid:16)%0 TheOleading order deviation is called the truncation error. Thus in (1.2.1) the truncation error is the termin . 1.2.2 Stability TheEulermethodis1storderaccurate. Howeverthereisanotherimportantconsiderationinanalysingthe method: stability. Let us suppose that at some time the actual numerical solution deviates from the true (cid:14) solutionOofthedifferenceequation(1.11)(N.B.nottheoriginaldifferentialequation(1.7))bysomesmall amount ,due,forexample,tothefiniteaccuracyofthecomputer.Thejnaddingthisinto(1.11)gives (cid:14)(cid:10)Q(cid:28)S#T (cid:14)(cid:10)Q(cid:28)S#Tm(cid:22)(cid:26)(cid:14)(cid:10)Q (cid:14)nQo$ (cid:16)qp

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