Multi-Particle Simulation Techniques I

Transcription

1 Mult-Partcle Smulato Techques I J Qag Accelerator Modelg Program Accelerator Techology & Aled Physcs Dso Lawrece Berkeley Natoal Laboratory CERN Accelerator School, Thessalok, Greece No. 5, 08

2 Outle Itroducto of the artcle--cell method for mult-artcle smulato Deosto/terolato Self-cosstet feld calculatos FFT based Gree fucto method for oe boudary coo Multgrd method for rregular shae boudary coo Partcle adace

3 Itroducto Partcle--cell method: the method amouts to followg the trajectores of charged artcles self-cosstet electromagetc or electrostatc felds comuted o a fed mesh Partcle--cell codes are wdely used accelerator hyscs commuty: PARMELA, ASTRA, GPT, IMPACT-T, IMPACT-Z, GENESIS, GINGER. A eamle of D artcle--cell smulato a msmatched beam trasort through a FODO Courtesy of R. D. Rye

4 Goerg Equatos Sace-Charge Smulato / d t r f 3,, 0,,,,,, t r f r t r f r t t r f r r w

5 artcle adace Oe Ste Partcle-I-Cell Method artcle deosto feld calculato feld terolato

6 Partcle Deosto/Feld Iterolato q w q w J w w B B E E Partcle deosto: Feld terolato: q Grd reduces the comutatoal cost comared wth drect N-body ot-to-ot teracto Grd also rodes smoothess to the shot ose ad close collso

7 Weght Fucto for Deosto/Iterolato Satal localzato of errors - At artcle searatos large comared wth the mesh sacg, the feld error should be small Smoothess - The charge assged to the mesh from a artcle ad the force terolated to a artcle a artcle from the mesh should smoothly ary as the artcle moes across the mesh Mometum coserato - No self force

8 q q j m j m j j G w G w m j m j j d G d w G w! m w m cost w 0 - Satal localzato of errors Smoothess: Cotuty of weght fucto alue Cotuty of derate -h 0 h Mometum coserato ; ; ; ' ' t ' ' ' m de m self d d w w d F Weght Fucto for Deosto/Iterolato

9 Weght Fuctos for Deosto/Iterolato h w h w h h w NGP: CIC: TSC: R. W. Hockey ad J. W. Eastwood, Comuter Smulato Usg Partcles

10 Self-Cosstet Feld Calculatos / f r, d 3

11 Dfferet Boudary/Beam Coos Need Dfferet Effcet Numercal Algorthms ONlogN or ON FFT based Gree fucto method: Stadard Gree fucto: low asect rato beam Shfted Gree fucto: searated artcle ad feld doma Itegrated Gree fucto: large asect rato beam No-uform grd Gree fucto: D radal o-uform beam Fully oe boudary coos Sectral-fte dfferece method: Traserse regular e wth logtudal oe/erodc Multgrd sectral-fte dfferece method: Traserse rregular e

12 Feld Calculato wth Oe Boudary Coos Gree Fucto Soluto of Posso s Equato r G r, r' r' dr' r h N ' Gr ; r =, y,z r'r' G, y, z / y z Drect summato of the cooluto scales as N!!!! N total umber of grd ots

13 Hockey s Algorthm or Zero Paddg Gcr Gr G-r Gr G-r r L Ths s dfferet from a real erodc system The real calculato s doe dscrete coordate stead of cotuous coordate R. W. Hockey ad J. W. Eastwood, Comuter Smulato Usg Partcles

14 Hockey s Algorthm or Zero Paddg

15 J. Qag, M. Furma, ad R. Rye, Phys. Re. ST Accel. Beams, ol 5, 0440 October 00. A Schematc Plot of a e - Beam ad Its Image Charge y e + o e - z cathode Shfted Gree fucto Algorthm: Fr Gsr,r'r'dr' Gsr,r' Gr rs,r'

16 Test of Image Sace-Charge Calculato Numercal Soluto s. Aalytcal Soluto Shfted-gree fucto Aalytcal soluto

17 Itegrated Gree Fucto for Large Asect Rato Beam Gr Lack of resoluto alog loger sde f same umber of grds are used for both sdes Brute force: use more grd ots alog loger sde Better way: break the orgal cooluto tegral to a sum of small cell tegral ad use tegrated Gree s fucto wth each cell

18 J. Qag, S. Lda, R. D. Rye, ad C. Lmborg-Derey, Phys. Re. ST Accel. Beams, ol 9, Itegrated Gree Fucto cr N Gr ' r'cr' Gr,r' Gsr,r'dr'

19 A Comarso Eamle: Asect Rato = 30

20 Posso Soler wth Fte Boudary Coos j- j j+ Where j =,,N

21 Fte Dfferece Posso Soler: Iterate Methods A: s a sarse matr Drect Gaussa elmato: ON 3 Iterate method: OmN Where S s a aromato of A -, =,,m

22 Fte Dfferece Posso Soler: Iterate Methods Classcal terate methods: Jacob Damed Jacob Gauss-Sedel Successe Oer Relaato Problems of classcal terate methods: slow coergece

23 Ref: J. Demmel s class ote Weghted Jacob Chose to Dam Hgh Frequecy Error Ital error Rough Lots of hgh frequecy comoets Norm =.65 Error after Jacob ste Smoother Less hgh frequecy comoet Norm =.055 Error after Jacob stes Smooth Lttle hgh frequecy comoet Norm =.976, wo t decrease much more

24 Classcal sarse-matr-ector-multly-based algorthms: o low frequecy error decreases slowly after a few teratos o moe formato oe grd at a tme o take N /d stes to get formato across grd o matr-ector multlcatos are doe o the full fe grd Multgrd algorthm: Multgrd Motato o smooths out the umercal errors of dfferet frequeces o dfferet scales usg multle grds o moes the formato across grd by Wlog stes o most multcatos are doe o coarse grds

25 Comarso of Coerget Tme for a Eamle: SOR s. Mult-Grd

26 Mult-Grd Iterato Method Basc Algorthm: Relace correcto roblem o fe grd by a aromato o a coarser grd Sole the coarse grd roblem aromately, ad use the soluto as a correcto to the fe grd roblem ad buld a ew startg guess for the fe-grd roblem, whch s the terately udated Sole the coarse grd roblem recursely,.e. by usg a stll coarser grd aromato, etc. Success deeds o coarse grd soluto beg a good aromato to the fe grd

27 Multgrd Sketch o a Regular D Mesh Cosder a m + grd D for smlcty Let P be the roblem of solg the dscrete Posso equato o a + grd D Wrte lear system as A * = b P m, P m-,, P s a sequece of roblems from fest to coarsest

28 Multgrd Sketch o a Regular D Mesh Cosder a m + by m + grd Let P be the roblem of solg the dscrete Posso equato o a + by + grd D o Wrte lear system as A * = b P m, P m-,, P s a sequece of roblems from fest to coarsest

29 For roblem P : o o o b s the RHS ad s the curret estmated soluto A s mlct the oerators below. All the followg oerators just aerage alues o eghborg grd ots o Neghborg grd ots o coarse roblems are far away fe roblems, so formato moes quckly o coarse roblems The restrcto oerator R mas P to P - o o o Restrcts roblem o fe grd P to coarse grd P - by samlg or aeragg b-= R b Grahc reresetato: The rologato oerator P- mas a aromate soluto - to a o o o Iterolates soluto o coarse grd P - to fe grd P = P-- Grahc reresetato: The smooth oerator S takes P ad comutes a mroed soluto o same grd o o o Basc Oerators of Multgrd Iterato Uses weghted Jacob or Gauss-Sedel mroed = S b, Grah reresetato: both le o grds of sze -

30 Restrcto Oerator R - Detals The restrcto oerator, R, takes o a roblem P wth RHS b ad o mas t to a coarser roblem P - wth RHS b- Smlest way: samlg Aeragg alues of eghbors s better; D ths s o coarse = /4 * fe - + / * fe + /4 * fe + I D, aerage wth all 8 eghbors N,S,E,W,NE,NW,SE,SW

31 Prologato/Iterolato Oerator The rologato/terolato oerator P-, takes a fucto o a coarse grd P -, ad roduces a fucto o a fe grd P I D, learly terolate earest coarse eghbors o fe = coarse f the fe grd ot s also a coarse oe, else o fe = / * coarse left of + / * coarse rght of I D, terolato requres aeragg wth or 4 earest eghbors NW,SW,NE,SE

32 Pre-smoothg: comute aromated soluto by alyg stes of a relaato method o fe grd: X X S B AX ;,, Costruct resdual ectors: r B AX Restrct the resdual to coarser grd: r R r Two-Grd Iterato Sole eactly o the coarse grd for the error ector: E A R r Prologate/Iterolate the error ector to fe grd: E P E Comute the mroed aromato o fe grd: X X E Post-smoothg: comute aromated soluto by alyg stes of a relaato method o fe grd: X X S B AX ;,,

33 Structure of Multgrd Cycles V-Cycle W-Cycle g s the umber of two-grd teratos at each termedate stage

34 Multgrd V-Cycle Algorthm Fucto MGV b, Sole A* = b ge b ad a tal guess for retur a mroed f = comute eact soluto of P oly ukow retur else = S b, mroe soluto by damg hgh frequecy error r = A* - b comute resdual r- = Rr restrct from fe to coarser grd MGV r-, e- sole A*e = r recursely e = P-e- rologate from coarser grd to fe grd = - e correct fe grd soluto = S b, mroe soluto aga retur

35 Comlety of a V-Cycle o a D Grd At leel, the umber of ukow s - - O a seral mache o Work at each ot a V-cycle s Othe umber of ukows o Cost of Leel s O - = O4 o If fest grd leel s m, total tme s: o m SO4 = O 4 m = O# ukows = O a arallel mache PRAM o wth oe rocessor er grd ot ad free commucato, each ste the V-cycle takes costat tme o Total V-cycle tme s Om = Olog #ukows

36 Full/Nested Multgrd FMG Oerew: o Sole the roblem wth ukow o coarsest grd o Ge a soluto to the coarser roblem, P -, ma t to startg guess for P o Sole the fer roblem usg the Multgrd V-cycle Adatages: o o eed for tal guess of soluto o aod eese fe-grd hgh frequecy cycles o obta solutos at multle grd leel ca be used for error estmate or etraolato

37 Coergece Pcture of Multgrd D Error decreases by a factor >5 o each terato

38 Partcle Adace: Numercal Itegrato Cosstecy Accuracy Stablty Effcecy Eamles of umercal tegrators: Ruge-Kutta Lea frog Bors Itegrators beyod Bors Symlectc tegrators

39 Numercal Itegrato: Cosstecy D Eamle d d Euler algorthm: F F Cosstecy: uder the lmt of ->0, the dscrete model -> cotuous model

40 Numercal Itegrato: Accuracy Accuracy: local trucato errors the umercal dscrete algebrac equatos comared wth the orgal dfferetal equatos d d d d d F d O d O d O O The aboe Euler method s the st order accuracy. Hgher order accuracy ca be obtaed usg more sub stes.

41 Numercal Itegrato: Accuracy The order of accuracy ca also be eressed as the local trucato error of arables the umercal tegrato method. The m th order method deotes a umercal tegrato method that s locally correct through order h m ad makes local errors of order h m+. The error for oe ste s ~ m e h for ste s e h O h ~ h 0 O h ~ m ~ h ~ 0 ~ ~ t t 0 ~ F t, 0 F ~ t, t F ~ 0,0 0 h F ~ t, t t - 0 F tf t...

42 Numercal Itegrato: Stablty Stablty: roagato of errors e.g. roudoff error the dscrete algebrac equatos. A stable umercal tegrato algorthm s the oe that a small error at ay stage does ot kee o creasg as umber of stes creases e F e e e e e ' X F V V V X X Euler algorthm o comuter: Learzed equatos of errors: y e V e X

43 e e Numercal Itegrato: Stablty F' e e For a tegrato scheme to be umercally stable, the egealues of the error trasfer matr must le or o the ut crcle. The elct Euler method wll be ustable F'

44 Numercal Itegrator: Ruge-Kutta,...,, N t f d ,,,, h O h t h h t h h t h t h k k k k k f k k f k k f k f k oe of the most oular schemes for tegratg ODEs alcable to arbtrary ODEs 4 th order accuracy arable tme ste sze aulary storage 4 feld calculatos er ste

45 Numercal Itegrator: Lea-Frog d d f h h f easy to mlemet, low memory storage d order accuracy tme reersble stable for df h ; W W d ma / sgle feld calculato er ste

46 Numercal Itegrator: Bors Algorthm B E q d m d s t s t B t ta B E E ' ' t h m q h m qb m qh h m q h m q Loretz equatos of moto Bors Algorthm - wdely used lasma/accelerator commuty d order accuracy tme reersble

47 New Numercal Itegrator Needed to Iclude Sace-Charge Felds Relatstc Electro Beam 50 MeV electro beam trasorts free sace Bors ew algorthm ca use 00 tmes larger tme ste sze New 47

48 Fast Numercal Itegrator for Relatstc Charged Partcle Trackg Loretz Force Equatos: Wdely Used Bors Itegrator New Fast Relatstc Itegrator J. Bors, Proceedgs of the Fourth Coferece o the Numercal Smulato of Plasmas Naal Research Laboratory, Washgto, DC, 970, J. Qag, Nucl. Itrum. Meth. Phys. Res. A,

49 Aother Relatstc Itegrator Vay 49 J. V. Vay, Phys. Plasmas 5,

50 Numercal Test of a Electro Co-Mog wth a Postro Coastg Beam wth Uform Traserse Desty sace-charge electrc ad magetc felds from ostro beam: 50

51 Numercal Eamles Show d Order Accuracy of the New Fast Algorthms Sace-Charge Felds 50 MeV 00 MeV Bors Bors ~t ew + Vay tegrators d order coergece test ~t New + Vay tegrators d order coergece test All three algorthms show d order accuracy Bors algorthm shows much larger error tha the other two algorthms 5

52 Numercal Itegrator: Symlectc Itegrator Hamltoa equatos of moto: d d H H Symlectc Itegrator resere: the symlectc ature of Hamltoa equatos the hase sace structure Numercal tegrator: ξ f ξ Defe ts Jacoba matr M: j M j Symlectc coo: M t JM J J J 0 0 J J Ths corresods to more costrats

53 Summary Deosto/terolato artcle--cell method Self-cosstet sace-charge feld calculatos through solg the Posso equato at each tme ste usg the udated charge desty dstrbuto - FFT based Gree fucto methods for oe boudary coo - Sectral ad fte methods for regular shae boudary coo Numercal tegrators for artcle adace - Euler method - Ruge Kutta method - Lea-frog method - Bors method - Relatstc artcle tegrator - Symlectc tegrator 53