Optimum Choice of NGP, CIC and QS Algorithms in One Dimensional Electrostatic Particle Simulations

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1 Journal of American Science 2010;6(10) Oimum Coice of NGP, CIC and QS Algorims in One imensional Elecrosaic Paricle Simulaions Ferdowsi Universiy of Masad, earmen of Elecrical Engineering, Scool of Engineering, Masad IRAN Teleone number: (0098) Absrac: Several algorims of differen order and comlexiy are available aricle simulaions. In is sudy, e acual resuls of measuremens of self eaing imes as a measure of accuracy of a single secies one dimensional elecrosaic simulaion of ermal lasmas are resened using zero order weigings (NGP), linear weiging (CIC), and quadraic sline (QS) weiging. Oimal range of arameers is found o be along v / x 1.5 zero order weigings and v / x 0.5 linear and quadraic sline weiging. Considerable increase in self eaing ime is acieved roug runcaion in k-sace. Te resuls are useful in e oimum coice of simulaion algorim and indicae a using a iger order simulaion algorim is igly recommended, esecially wen k-sace runcaion is used. [Journal of American Science 2010;6(10): ]. (ISSN: ). Key words: Plasma simulaion, self eaing imes, coice of simulaion algorim, NGP, CIC, QS Nomenclaure T e =elecron emeraure T i =ion emeraure =Bolzman's consan m e =elecron mass mi =ion mass =lasma frequency =ebye leng x =simulaion grid size =simulaion ime se n 0 =aricle densiy N =number of aricles in a ebye leng N C =number of aricles in a grid sacing =self-eaing ime v =elecron ermal emeraure k max =maximum allowable mode k =las mode ke las 1. Inroducion Te coice of wic algorim o use in aricle simulaions involving many aricles is igly imoran since i deermines bo e comuaional exense and e accuracy involved since eac algorim naurally inroduces some errors due o discreizaion of bo ime and sace. Paricle simulaions ave been carried ou o sudy e beavior of elecron beams in vacuum ubes and ave laer develoed ino e simulaions of bo elecrosaic and elecromagneic lasmas. Linear inerolaion or oer iger order inerolaion scemes are almos always used in aricle-in-cell simulaion because of eir lower noise caracerisics and accuracy relaive o e neares-grid-oin meod. Te iger order inerolaion scemes are cosen because of eir oimal ermance, balancing a smaller number of aricles agains more comuer oeraions er aricle er ime se. However, is is no always e case. Parker (2002) resened large-scale gyrokineic aricle simulaions, were someimes neares-grid-oin inerolaion is used wi resuls virually idenical o ose of linear inerolaion using e same number of aricles. He resened a comarison and analysis of neares-grid-oin and linear inerolaion scemes sowing wy neares-grid-oin inerolaion can be oimal. Li e al. (2009) roosed imroved aricle-in-cell (PIC) algorims including volume weiging cloud-in-cell model, geomery rofile considered exlosive elecron emission model and divergence error diffused erfecly maced layer boundary in order o increase e recision of e algorim and decreased e numerical noise. Teir model resuled in e develomen of a user-friendly, 2.5-dimensional PIC code called UNIPIC o simulae ig ower microwave :// 623 americansciencej@gmail.com

2 Oimum Coice of NGP, CIC and QS Algorims source devices and used eir geomery oimizaion in x-y, z-r and r-φ coordinae sysems. Resuls of simulaion of a relaivisic magneron, magneically insulaed line oscillaor, virual caode oscillaor ave all sown good ysics image and beam-wave ineracion caracerisics. In e elecrosaic simulaions, e model consiss of carged moving aricles exeriencing bo e ce due o emselves and due o e exernally alied fields. Maxwell's equaions and Newon's Lorenz equaions are bo alied o follow e moion of ese aricles. Simulaions are done bo in discree sace and ime. Tus, e models used affec e accuracy and sabiliy of e simulaions. Te simles model used is e zero order aricle and field weiging called neares-gid-oin (NGP). A beer weiging would be firs order weiging a smooes e densiy and field flucuaions a e exense of accessing wo grid oins eac aricle wice eac se. Tis is called cloud-in-cell (CIC). A iger order weiging is quadraic sline (QS) a furer rounds off e rougness of e aricle sae. Te non-ysical self eaing imes of a one dimensional elecrosaic simulaion of ermal lasma are resened using zero order weiging or e neares-grid-oin (NGP), linear weiging cloud-in-cell (CIC-PIC), and quadraic sline weiging (QS) in a momenum conserving code called ES1 a was wrien around 1972 (Birdsall and Langdon, 2005). I is noewory a even energy conserving codes sow self-eaing 0. In suc codes, energy is conserved only in e limi of zero. ES1 is a "aricle-in-cell" simulaion of e Vlasov Eq. "1-imensional" (in sace, lus e corresonding velociy dimension) eriodic roblems. One may use ES1 o develo insig ino lasma beavior and e roeries of e Vlasov equaion, including surrising nonlinear resuls. ES1 racks several ousand individual aricles in ase sace. Tus, i effecively solves a Klimonovic-uree Equaion wic in e limi of a very large number of aricles sould converge o e equivalen roblem of solving e Vlasov Eq. a smoo f(x,v,). I as been found a e emeraure of ermal lasmas increases linearly wi ime. Self eaing ime is defined as e ime aken e ermal energy 2 ( Torv ) of e sysem o double in value. A one dimensional elecrosaic lasma model consising of a mobile elecron secies and immobile neuralizing ion background was used in is sudy. For is sysem, is is e ime in wic e average kineic energy of an elecron increases by 0.5 T. Tis increase in energy is numerical in origin and of a socasic naure. I arises due o e flucuaions in e ce due o e resence of finie grids in sace and ime. Teree, e self eaing ime srongly deends on and x. A Maxwellian velociy loader wi firs and second momen correcion was used (Giomer 1971). Hockney (1971) emirically obained e self-eaing imes a wo dimensional lasma wi ions and elecrons using Te Ti, m / m i e 64 wi furer refinemens by Hockney e al. (1974). Quadraic sline weigings were added o e scoe in is sudy and our resuls indicae a self eaing imes are longes v / x 3/2 NGP and v / x 1/ 2 CIC and QS. 2. Coice of in e deerminaion of self eaing ime Resuls of simulaions carried ou wi ES1 e yical grow in ime of e ermal energy indicae a e energy increases linearly wi ime 0.6. Tis imlies a random rocess. Teree, is sudy will be resriced o is range of values of.te grow observed in ermal energy larger n values of is like wi n 1. Tis imlies some oer m of grow as ye unexlained. Te same cange in grow aern was observed e ermal energy 0.6 using bo CIC-PIC and QS weigings. Tus, was deermined from similar isories wi uniy sloe. In obaining, e zero in ime and e iniial ermal energies were assigned o e beginning of grow linear in ime. 3. eendence of on n, n ( x ) 0 0 Te resuls of simulaions self-eaing imes are sown in Fig. 1 in unis of elecron lasma frequency N n and 0 N ( C N n x ) were n is varying. Here 0 0 N C denoes e number of aricles in a grid, sacing, N denoes e number of aricles in a ebye leng, n0 is e aricle densiy, x is e simulaion grid size and is e ebye leng. 624

3 Journal of American Science 2010;6(10) frequency divided by N C N / x are sown in Fig. 3, 4 and 5 differen values of. Te dased line drawn roug e differen gras indicaes v / x 1.5 NGP and v / x 0.5 CIC and QS. Te longes eaing imes occur a abou ese values of v / x. Fig. 1 Self eaing imes vs n, n ( x ) NGP, CIC sowing linear deendence Fig. 2 sows e resuls of simulaions self-eaing imes loed as elecron lasma frequency N C N wo differen raios of / x 0.5 and / x 2.0 NGP. Similar resuls were observed CIC. Fig 3. Self eaing imes divided by / x several values of N C NGP N Fig. 2 Self eaing imes vs N C N differen raios of / x indicaing a linear deendence of on N C N fixed / x NGP 4. eendence of / x and oimum coice of v / x Te self eaing imes loed as elecron lasma Fig 4 Self eaing imes divided by N C N / x several values of CIC :// 625 edior@americanscience.org

4 Oimum Coice of NGP, CIC and QS Algorims Fig 5 Self eaing imes divided by N C N / x several values of QS 5. Comarison of resuls and e gain of going o a iger order lasma simulaion algorim As a comarison of e differen algorims sudied, raios of self-eaing imes of quadraic-sline (QS) o neares-grid-oin (NGP) and cloud-in-cell (CIC) o neares-grid-oin (NGP) are loed in Figs. 6, 7 and, resecively ,0.2,0.3 Fig. 6 Raios of self-eaing imes / x 0.1 Fig. 7 Raios of self-eaing imes vs / x 0.2 Fig. 8 Raios of self-eaing imes vs / x 0.3 Tese figures indicae a CIC self eaing imes are as muc as 70 imes longer an NGP, and a e self eaing imes of QS is as muc as 650 imes longer an NGP. Suc increases in eaing imes come a e exense of muc longer comuaional imes. Acual measuremen of e cos of running e lasma simulaions er ime se on a comuer a e Lawrence Livermore Laboraory sow a T=5, 11.6 and 24 microseconds/aricle/ime se NGP, CIC and QS, resecively. Hence, we need a measure of accouning is cos. Teree, e gain of using a iger order weiging sceme e simulaions can be defined as follows: 626

5 Journal of American Science 2010;6(10) Gain increase in self eaing ime increase in comuer simulaion ime Going roug e oimal a, w e will ge gains as sow n in Table 1. Noe a e gai ns of using iger order weigings, i.e. going from NGP o CIC or going from CIC o QS are rougly one order of magniude, muc less an e gains in e self eaing imes. Table 1 A comarison of gains in going o a iger order simulaion algorim indicaing raios of increase in self-eaing imes or reduced error in energy over increase in cos deermined on a comuer a e Lawrence Livermore Laboraory CIC/NGP Gain QS/NGP Gain QS/CIC Gain Increase in gain due o k-sace smooing Te self eaing imes of one dimensional ermal lasma simulaions can be considerably increased by smooing e carge densiy in k-sace. Te smooing facor used was simle Fourier sace runcaion, were all e modes beyond k las are droed as sown in Fig. 9. Fig 9. Smooing facor used in k-sace runcaion Fig. 10 sows e self-eaing imes loed as elecron lasma frequency k / k e max las differen scemes used by keeing everying fixed and varying k las eac sceme. Te gain in self-eaing ime due o k-sace runcaion is almos roorional o k / k NGP and is close o bu no quie max las 2 roorional o ( k / k ) CIC and max las 3 ( k / k ) QS. Tus, k-s ace runcaion furer max las increases e gain of CIC/NGP, QS/ NGP and QS/CIS. Fig. 10 Self-eaing imes loed as elecron lasma frequency k max / k NGP, CIC and QS. las 7. Conclusion Te self eaing imes of one dimensional elecrosaic lasma simulaion were found o be e longes v / x 1.5 NGP and v / x 0.5 CIC and QS. Rougly seaking, e quadraic sline self eaing imes are one order of magniude longer an cloud-in-cell, wile cloud-in-cell self eaing imes are one order of magniude longer an e neares-grid-oin algorim, considering bo e gain in eaing imes and e increased comuaional cos. Smooing by Fourier sace runcaion considerably increases e self-eaing imes and is increase is rougly roorional o 1 ( / ) n k k max las were n is e order of e weiging sceme, i.e. n=0 NGP, n=1 CIC and n=2 QS. Teree, using a iger order algorim is igly recommended, esecially wen k-sace runcaion is used. :// 627 edior@americanscience.org

6 Oimum Coice of NGP, CIC and QS Algorims Acknowledgemen I am igly indebed o Professor C. K. Birdsall e suor and advice is sudy. I am also graeful of e Vice Cancellor of Researc and Tecnology of e Ferdowsi Universiy of Masad e gran rojec. References [1] Parker, S. E. (2002), "Neares-grid-oin inerolaion in gyrokineic aricle-in-cell simulaion," Journal of Comuaional Pysics, Vol. 178, No. 2, [2] Li, Y., Wang, H., Liu, C., Zang,., Wang, J., Wang, Y., (2009), "2.5-dimensional elecromagneic aricle-in-cell code-unipic ig ower microwave simulaions," Qiangjiguang Yu Lizisu Hig Power Laser and Paricle Beams 21 (12), [3] Birdsall, C. K., Langdon, A. B., (2005), Plasma Pysics via Comuer Simulaion, Insiue of Pysics, Series in Plasma Pysics, Taylor & Francis Grou, [4] Giomer, S. (1971), "Commens on numerical simulaion of e Weibel insabiliy in one and wo dimensions," Pys. Fluids, Vol. 14, July 1971, [5] Hockney, R. W., (1971), "Measuremen of collisions and eaing imes in a wo dimensional ermal lasma," J. Com. Pys., Vol. 8, Aug. 1971, [6] Hockney, R. W., Goel, S. P., Easwood, J. W. (1974), "Quie ig resoluion comuer models of a lasma," Journal of Com. Pys., Vol. 14, (Feb. 1974), /1/