CD-EXCALIBUR-FMS0048-M2.3.2_OptionsParticleAlgorithms
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   :description: technical note

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ExCALIBUR,Options,for,Particle,Algorithms,M2.3.2,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Milestone,2.3.2.,This,binds,the,report,2047355-TN-01[1],along,with,a,description,of,the,features,implemented,in,the,proxy-,application,minepoch[2],as,of,September,15,,2021.,In,hotter,regions,of,the,plasma,,particle,species,have,sufﬁcient,kinetic,energy,that,they,exhibit,rel-,atively,long,periods,of,motion,between,collisions,with,other,particles.,This,collision-less,behaviour,is,often,referred,to,as,“kinetic”.,Due,to,the,lack,of,collisions,it,is,assumed,that,a,Maxwell-Boltzmann,distribution,is,not,a,good,description,of,the,particle,velocity,distribution.,This,is,in,contrast,to,cooler,regions,of,plasma,where,collisions,occur,frequently,enough,that,a,Maxwell-Boltzmann,description,is,a,suitable,description,of,the,velocity,distribution.,These,cooler,regimes,are,referred,to,as,“colli-,sional”,and,a,ﬂuid,description,,with,a,single,temperature,for,a,region,of,space,,may,be,a,suitable,approximation.,The,report,describes,a,Particle,In,Cell,(PIC),method,for,the,continuity,equation,of,a,single,species,distribution,function,f,.,The,function,f,=,f,(,(cid:126)R,,(cid:126)v),is,deﬁned,over,a,phase,space,constructed,by,the,direct,sum,of,positions,(cid:126)R,and,velocities,(cid:126)v.,It,is,assumed,that,f,can,be,written,as,f,=,f0,+,g,where,f0(,(cid:126)R,,(cid:126)v),is,a,shifted,Maxwellian,and,g(,(cid:126)R,,(cid:126)v),is,a,perturbation.,A,further,assumption,is,that,the,ﬁrst,,(cid:126)v,and,|(cid:126)v|2,moments,of,g,are,zero.,As,f0,is,a,Maxwellian,distribution,the,corresponding,evolution,is,described,by,ﬂuid,equations,and,the,report,discusses,the,evolution,of,g.,The,ﬁrst,three,moments,namely,density,,momentum,and,temperature,,which,describe,the,evolution,of,f0,,are,computed,and,stored,on,the,spatial,domain,which,is,assumed,to,be,discretised,into,cells.,The,report,mentions,that,the,it,is,possible,to,describe,g,using,properties,stored,on,a,grid,,however,as,this,is,a,particle,based,investigation,,the,focus,is,on,methods,to,represent,and,evolve,g,using,particle,based,properties.,Using,particle,based,properties,avoids,rescaling,and,resolution,issues,that,would,arise,from,using,a,grid,to,discretise,velocity,space.,The,time,evolution,of,g,is,a,function,of,f0,and,requires,that,f0,can,be,evaluated,at,the,particle,positions,as,they,act,as,tracers,through,the,ﬂuid,system.,Furthermore,as,the,particles,purely,act,as,tracers,there,are,no,particle-particle,interactions.,The,report,argues,that,the,extent,of,the,coupling,between,the,ﬂuid,and,particles,can,be,tuned,to,optimise,for,computational,efﬁciency,,i.e.,regions,with,minimal,kinetic,effects,may,not,require,solv-,ing,kinetic,equations,as,they,would,be,suitably,described,by,the,ﬂuid,system.,Omitting,particles,in,regions,removes,the,cost,of,interpolating,properties,to,an,from,particles,and,the,cost,of,advancing,particle,positions,with,stepping,techniques.,The,report,discusses,that,,as,the,ﬂuid,equations,can,exhibit,less,physically,relevant,high,speed,waves,,it,is,attractive,to,treat,time-stepping,processes,implicitly.,As,a,further,optimisation,,the,idea,is,presented,that,if,f0,is,known,,for,example,analyt-,ically,,it,can,be,integrated,so,that,only,g,must,be,computed,(using,particles,as,a,form,of,Monte,Carlo,integration),which,would,be,a,smaller,computational,cost,than,computing,f0,and,g.,The,remaining,sections,of,the,2047355-TN-01,report,discuss,details,relating,to,the,implementa-,tion,of,the,discussed,methods.,For,example,,some,existing,PIC,implementations,apply,grid,data,structures,that,conform,to,the,magnetic,ﬁeld,lines,and,capture,the,strong,anisotropy.,A,point,is,raised,that,traditional,codes,use,ﬁnite,difference,or,volume,approach,where,particles,exist,with,smooth,shapes,,such,that,the,sum,over,all,charges,deﬁnes,a,continuous,function,on,the,simula-,tion,domain.,In,contrast,to,a,ﬁnite,element,approach,where,particles,are,Dirac,Deltas,and,should,be,viewed,in,the,distributional,sense.,The,issue,is,correctly,raised,that,determining,which,FEM,cell,a,particle,resides,is,a,very,non-trivial,task,,especially,when,cell,boundaries,are,curved.,A,suggested,solution,involves,tracking,the,cell,local,positions,on,a,per,cell,basis,and,performing,the,necessary,communication,between,cells,however,this,approach,brings,additional,challenges,of,which,some,are,listed.,The,ﬁnal,sections,of,2047355-TN-01,discuss,how,the,convergence,in,the,time,stepping,of,particle,motion,is,dependent,on,the,continuity,of,the,ﬁnite,element,representation,and,raise,co-design,discussion,points.,The,second,half,of,this,report,is,the,documentation,that,accompanies,the,minepoch,software.,The,software,serves,as,a,proxy,application,to,demonstrate,advanced,particle,tracing,,drift-kinetic,species,and,low-noise,PIC.,The,standard,PIC,time,stepping,is,a,based,on,a,leapfrog,approach,that,alternates,between,inte-,grating,forward,the,particle,positions,and,both,the,electric,and,magnetic,ﬁelds.,In,the,advanced,particle,tracing,a,smaller,time,step,is,applied,for,the,particle,trajectories,such,that,N,integration,steps,are,performed,for,the,particle,motion,for,each,integration,step,of,the,electric,and,magnetic,ﬁelds.,The,proxy,application,implements,a,version,of,Drift-kinetics,based,on,long-wavelength,general,gyrokinetic,formalism.,In,this,approach,the,Drift-kinetic,species,follow,a,system,of,Euler-Lagrange,equations,that,describe,the,evolution,of,the,distribution,function.,Finally,the,proxy,application,implements,the,low-noise,PIC,which,is,described,in,the,2047355-TN-01,component,of,this,report.,2,Acknowledgement,The,support,of,the,UK,Meteorological,Ofﬁce,and,Strategic,Priorities,Fund,is,acknowledged.,References,[1],B.F.,McMillan.,Co-design,of,PIC,methods,and,other,elements,of,NEPTUNE.,Technical,Report,2047355-TN-01-3,,UKAEA,Project,Neptune,,2021.,[2],minepoch.,https://github.com/ExCALIBUR-NEPTUNE/minepoch,,2021.,Accessed:,Septem-,ber,2021.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0048,Issue:,Date:,1.00,September,15,,2021,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Will,Saunders,Wayne,Arter,Signature,N/A,N/A,Date,September,15,,2021,September,15,,2021,BD,Reviewed,By:,Rob,Akers,September,15,,2021,Advanced,Dept.,Manager,Computing,Approved,By:,Rob,Akers,September,15,,2021,Advanced,Dept.,Manager,Computing,3,Co-design,of,PIC,methods,and,other,elements,of,NEPTUNE.,Ben,F,McMillan,March,4,,2021,The,NEPTUNE,project,scope,includes,simulating,the,dynamics,of,the,plasma,and,neutrals,in,the,edge,region,of,the,plasma.,Depending,on,the,ﬁdelity,re-,quired,,and,the,situations,of,interest,,this,requires,solving,both,for,ﬂuid-like,motion,in,collisional,regions,,as,well,as,essentially,free-streaming,kinetic,be-,haviour,for,less,collisional,plasma,constituents,and,neutrals.,Various,computational,methods,to,solve,these,kinetic,problems,,and,we,will,here,be,discussing,particle-based,Monte,Carlo,methods,,where,the,computa-,tional,markers,are,used,to,follow,the,trajectory,of,a,small,subsample,of,the,physical,particles.,These,are,in,more,general,settings,referred,to,as,Lagrangian,methods,and,distinct,from,Eulerian,methods,,in,which,distributions,are,evolved,on,a,ﬁxed,grid.,Particle-In-Cell,methods,,and,Monte-Carlo,methods,designed,to,evolve,the,neutral,population,,are,quite,similar,in,concept,,and,we,will,refer,to,them,as,PIC,methods,for,the,purposes,of,this,document.,We,restrict,the,dis-,cussion,to,using,particle,methods,to,solve,kinetic,(rather,than,ﬂuid),problems.,Integrating,these,methods,with,other,components,of,the,NEPTUNE,project,may,require,careful,planning.,Interaction,of,the,proposed,particle,method,with,two,other,work,packages,was,highlighted,in,the,bidding,process,as,needing,explicit,consideration,,and,a,discussion,of,these,interactions,as,well,as,a,plan,for,going,forward,the,principle,purpose,of,this,document.,Firstly,,because,the,gyrokinetics,work,package,will,determine,the,appropri-,ate,Fokker-Planck-Maxwell,equations,that,well-magnetised,particles,follow,,it,is,critical,to,determine,whether,the,particle,methods,can,be,made,to,support,these,equations.,The,gyrokinetics,work,package,is,also,investigating,a,moment-based,numerical,scheme,to,discretise,these,equations,in,a,way,suitable,for,implicit,solution;,we,will,explain,the,relationship,between,this,scheme,and,the,moment-,based,particle,discretisation,that,is,designed,to,reduce,noise,,but,also,can,be,leveraged,as,part,of,an,implicit,method.,Secondly,,the,NEPTUNE,project,has,selected,a,spatial,representation,(for,various,plasma,quantities,and,electromagnetic,ﬁelds),based,on,ﬁnite,elements,,on,an,unstructured,grid,,with,a,view,to,using,curved,element,boundaries,in,order,to,conform,to,the,magnetic,ﬁeld,topology,and,the,physical,ﬁrst,wall,of,the,tokamak.,This,is,intended,to,be,provided,through,the,Nektar++,library.,The,advanced,particles,methods,implemented,in,this,exploratory,stage,are,however,1,based,around,simple,uniformly,spaced,regular,Cartesian,meshes.,We,outline,design,considerations,of,a,particle,solver,interfaced,with,Spectral-h/p,curvilinear,ﬁnite,element,methods.,We,also,explain,what,features,of,the,representation,of,the,electromagnetic,ﬁelds,are,desirable,for,accurate,particle,tracing.,1,What,are,PIC,methods,useful,for?,The,relative,computational,advantages,of,various,numerical,schemes,for,solving,kinetic,problems,will,determine,where,each,one,is,used,,and,for,which,class,of,particles,,or,where,kinetic,solution,is,feasible,at,all.,Particle,based,methods,are,intuitively,attractive,for,solving,kinetic,problems,,but,Eulerian,methods,converge,faster,as,a,result,of,the,Monte-Carlo,sampling,used,in,PIC,methods.,PIC,methods,are,often,favored,over,Eulerian,methods,when,the,following,conditions,apply,•,Three,velocity,space,directions,need,to,be,resolved.,•,Collision,operators,are,easily,written,in,the,form,of,a,stochastic,diﬀerential,equation,,but,not,as,a,PDE.,•,It,is,diﬃcult,to,ﬁnd,an,appropriate,velocity-space,grid,to,represent,the,particle,distribution,function,on,(eg.,multiple-beam,distributions).,•,It,is,diﬃcult,to,ﬁnd,an,optimal,spatial,grid,(as,PIC,codes,are,less,severely,impacted,by,running,on,a,non-optimal,spatial,grid).,In,general,PIC,methods,tend,to,be,relatively,easy,to,implement,and,paral-,lelise,compared,to,Eulerian,methods.,This,,and,a,willingness,to,tolerate,some,statistical,noise,,mean,that,early,examples,of,complex,plasma,kinetic,codes,tend,to,be,PIC-based,(e.g.,witness,the,evolution,of,core,and,edge,gyrokinetic,codes).,PIC,codes,are,dominant,for,neutral,particle,pushing,and,for,6D,Vlasov-Maxwell,(i.e.,non,gyrokinetic,PIC),because,except,in,special,cases,6D,Eulerian,codes,are,computationally,uncompetitive.,For,example,,if,full,fast,particle,orbits,need,to,be,tracked,to,determine,when,and,where,their,large,orbits,hit,the,wall,,PIC,is,a,natural,choice,for,this,problem.,In,edge,plasmas,,ﬂuid,modelling,of,at,least,some,of,the,species,and,some,of,the,regions,is,essential.,As,a,result,,Eulerian,and,Lagrangian,kinetic,models,need,to,be,coupled,to,ﬂuid,models.,To,provide,maximum,predictive,and,numerical,performance,,it,is,highly,desirable,for,the,Neptune,project,to,be,able,to,choose,where,and,when,to,use,each,kind,of,model,in,a,ﬂexible,and,correctly,coupled,way.,The,remainder,of,the,document,described,interfaces,between,the,numerical,discretisations,of,these,models.,2,Edge-relevant,Gyrokinetic,formalism,We,brieﬂy,introduce,the,baseline,mathematical,formulation,(independent,of,discretisation),of,the,Vlasov,equations,for,the,initial,stage,of,the,NEPTUNE,2,project.,3,Reformulation,of,kinetic,equations,in,terms,of,closure,to,ﬂuid,equations,The,Focker-Planck,equation,for,the,evolution,of,a,species,distribution,function,f,(Z),,with,Z,the,phase,space,coordinate,(representing,position,and,velocity,coordinates),may,be,written,in,the,form,+,∇Z.,(cid:105),(cid:104),˙Zf,=,S(f,),∂f,∂t,(1),with,S(f,),a,generalised,source,term,(including,self,and,non-self,collisions).,For,phase-space,conserving,equations,of,motion,(∇.,˙Z,=,0),,this,may,be,written,in,the,conservative,form,∂f,∂t,+,˙Z.∇Zf,=,df,dt,=,S(f,).,(2),As,in,the,Eulerian,representation,,it,is,often,useful,to,separately,represent,the,evolution,of,a,background,distribution,parameterised,using,ﬂuid,moments,,and,the,remaining,kinetic,response.,That,is,,we,introduce,a,splitting,f,(Z,,t),=,f0(Z,,t),+,g(Z,,t).,Choosing,the,parameterisation,f0(Z,,t),=,n(x,,t),[2πkT,(x,,t)]3/2],exp,(cid:20),m{v,−,v0(x,,t)}2/2,kT,(x,,t),(cid:21),(3),which,is,,again,,a,shifted,Maxwellian.,One,requires,that,the,ﬁrst,3,moments,of,g,vanish;,this,is,both,a,condition,on,the,initial,condition,,as,well,as,the,evolution,of,g.,By,taking,moments,of,the,Fokker-Planck,equation,,one,then,obtains,three,ﬂuid,equations,for,the,time,evolution,of,f0.,For,the,moment,,let,us,consider,these,equations,in,the,canonical,phase,space,Z,=,(R,,v),,with,a,Lorentz-like,force,F,(with,the,property,that,energy,transfer,to,the,ﬂuid,is,only,through,acceleration).,In,this,case,the,moment,equations,may,be,written,in,a,conservative,form,as,+,∇.[nv],=,(cid:90),dvS,,∂n,∂t,+,∇.[nv2,+,3nkT,],−,F,=,(cid:90),dvvS,,∂nv,∂t,(4),(5),∂[3nkT,+,nv2,0],∂t,(cid:20),v0{3nkT,+,nv2,0},+,(cid:90),+∇.,dV,(v,−,v0)(v,−,v0)2g,(cid:21),−F.v0,=,(cid:90),dvv2S.,(6),With,some,additional,transformation,velocity,and,temperature,evolution,equa-,tions,may,also,be,written,if,desired.,3,Note,that,the,kinetic,correction,(closure),term,in,the,energy,equation,would,normally,be,written,in,terms,of,a,collisional,closure,in,a,ﬂuid,scheme,,as,diﬀusive,heat,ﬂuxes,,so,switching,between,collisional,and,fully,kinetic,solutions,to,these,moment,equations,can,be,acheived,in,a,natural,way.,The,Maxwell,equation,source,term,is,a,function,of,current,,which,may,be,written,in,terms,of,the,ﬂuid,quantities,as,qnv0,,thus,the,ﬂuctuation,g,only,enters,as,a,closure,term,in,the,energy,equation.,In,the,highly,collisional,limit,,where,the,heat,ﬂuxes,are,negligible,,we,recover,a,conventional,(multi-species),set,of,plasma,ﬂuid,equations,simply,by,setting,g,→,0.,For,time,evolution,of,g,,we,simply,have,dg,dt,=,−,df0,dt,+,S(g,+,f0).,(7),We,have,considered,the,Vlasov-Maxwell,system,here,,but,conceptually,gyro-,Vlasov-Maxwell,follows,the,same,derivation,path,,and,although,the,ﬂuid,equa-,tions,diﬀer,somewhat,,the,equation,for,g,in,the,symbolic,form,above,is,equivalent,for,Vlasov,and,gyroVlasov,equations.,In,1D,,with,the,simulation,aligned,along,a,constant,background,ﬁeld,,which,is,the,initial,conﬁguration,to,be,considered,,these,systems,are,identical.,Note,that,the,above,equation,is,also,valid,for,neutral,particles.,It,is,beyond,this,point,that,the,proposed,methods,in,NEPTUNE,diﬀer.,Actually,,the,ﬂuid,equations,,and,the,coupling,to,the,gyro-Maxwell,or,gyro-,Poisson,equations,is,identical,,so,the,same,ﬂuid,and,ﬁeld,solvers,may,be,used.,However,,the,representation,of,g,diﬀers.,Firstly,,and,obviously,,in,the,Eulerian,approach,,g,is,represented,using,coeﬃcients,on,a,ﬁxed,spatial,grid,,whereas,in,the,particle-based,approach,,g,is,represented,using,markers.,Secondly,,the,Eulerian,method,described,in,Ref.,[1],performs,a,transfor-,mation,on,velocity,space,,shifting,and,rescaling,the,local,coordinate,V,=,M,(v,,v0,,T,),such,than,F0(V,),=,f0(M,−1V,)/n,is,an,unshifted,Maxwellian,of,temperature,and,density,unity.,The,grid-based,or,spectral,Eulerian,shceme,is,then,written,in,terms,of,V;,this,allows,the,grid,to,adapt,to,regions,of,low,tem-,perature,be,reﬁning,the,grid,resolution,so,the,spacing,is,constant,in,terms,of,local,thermal,velocity.,Although,it,would,also,be,possible,to,perform,this,transformation,in,the,Particle-based,scheme,,it,is,not,helpful,,because,the,eﬀective,velocity,resolution,depends,on,where,the,markers,are,in,velocity,space,,rather,than,their,coordi-,nate,labelling.,Particle,schemes,adapt,to,velocity,space,automatically,because,the,marker,phase-space,density,remains,(in,a,statistical,sense),proportional,to,the,particle,phase-space,density,,so,markers,in,cold,regions,naturally,have,low,typical,velocities.,4,4,Particle-speciﬁc,implementation,of,the,moment-,based,scheme,We,brieﬂy,summarise,the,method,to,be,implemented;,for,Vlasov-Maxwell,sys-,tems,,this,was,implemented,in,Ref.,[2].,We,consider,N,markers,(computational,macroparticles),loaded,in,phase,space,with,positions,Zi(t),with,a,phase,space,density,K(Z),=,lim,N→∞,(cid:90),(cid:88),dV,δ6(Z,−,Zi(t)),N,(8),that,represents,how,likely,we,are,to,ﬁnd,a,marker,in,a,small,region,of,phase,space,near,Z.,Due,to,Liouville’s,theorm,,we,ﬁnd,that,dK/dt,=,0.,We,store,the,initial,value,Ki,=,K[Zi(0)],at,each,marker,position,and,identify,the,eﬀective,volume,of,phase,space,associated,with,each,marker,as,1/K(Z).,To,represent,the,ﬂuctuating,distribution,function,,we,deﬁne,a,weight,gi,at,each,particle,location,with,g(Z),=,(cid:88),N,1,Ki,∂Zi(t),∂t,gi(t)δ6(Z,−,Zi(t)),=,˙Z(Zi,,t),and,if,we,evolve,gi(t),via,∂dgi,dt,=,−,df0,dt,(cid:12),(cid:12),(cid:12),(cid:12)Z=Zi,+,S(Zi),(9),(10),(11),then,by,integrating,in,an,appropriate,volume,,we,can,show,that,in,the,large-,marker,limit,,this,will,satisfy,dg,dt,=,−,df0,dt,+,S(g,+,f0).,(12),Note,that,this,the,standard,control,variates[3,,4],method.,Confusingly,,this,is,often,referred,to,as,the,‘δ,−,f,’,method,,but,‘δ,−,f,’,is,also,used,to,denote,methodologies,that,are,valid,only,when,the,ﬂuctuation,is,small.,There,is,no,such,approximation,used,here;,there,is,no,formal,restriction,on,the,size,of,g,,although,when,f,and,g,are,the,same,size,,the,numerical,advantages,may,be,lost.,The,principle,diﬀerence,between,this,control-variates,method,and,that,im-,plemented,in,codes,such,as,ORB5,and,EPOCH,is,time-evolution,of,the,back-,ground,distribution,function,f0,(however,,a,project,is,underway,to,implement,density,variation).,In,a,Galerkin,method,,moments,of,the,particle,distribution,appear,multiplied,by,shape,functions.,For,example,,consider,a,representation,of,density,n(x),=,(cid:88),nkΛk(x),(13),k,5,where,nk,are,the,nodal,density,values,and,Λ,the,In,the,weak,form,,this,may,be,equated,to,the,velocity-space,integral,of,the,particle,representation,of,the,distribution,function,f,so,we,have,(cid:90),(cid:90),dxΦ(x),(cid:88),dv,i,fi,Gi,δ6(Z,−,Zi),=,(cid:90),dxΦ(x),(cid:88),k,nkΛk(x),(14),for,all,trial,functions,Φ(x),in,the,ﬁnite,element,representation.,Because,,if,we,choose,trial,functions,in,the,same,space,as,the,density,representation,,we,have,Φ(x),=,(cid:80),h,nhΛh(x),,this,gives,us,(cid:90),∀h,,dxλh,(cid:88),i,fi,Gi,δ(x,−,xi),=,(cid:90),dxΛh(x),(cid:88),k,nkΛk(x),(15),or,(cid:88),∀h,,λh(xi),i,fi,Gi,(cid:88),=,Mhknk,k,(16),where,we,have,deﬁned,a,mass,matric,Mhk,representing,the,integral,on,the,RHS.,We,may,then,ﬁnd,a,FEM,representation,of,the,density,by,performing,the,sum,on,the,LHS,and,inverting,the,mass,matrix.,5,Computational,advantages,of,moment-based,schemes.,This,moment-based,scheme,(whether,in,Eulerian,or,Lagrangian,form),allows,a,close,connection,to,the,ﬂuid,equations,,and,allows,the,ﬂuid,and,ﬁeld,equations,to,be,partially,decoupled,from,the,kinetic,equations.,One,way,this,can,be,exploited,is,that,in,regions,where,kinetic,eﬀects,are,minimal,,the,kinetic,equations,no,longer,need,to,be,solved,at,all.,An,additional,advantage,is,that,the,coupled,Vlasov-Maxwell,system,can,be,quite,stiﬀ,in,the,sense,that,there,are,oscillations,and,relaxations,on,much,faster,timescales,than,the,dynamical,timescales,of,interest.,For,example,,Alfven,waves,propagate,at,close,to,the,speed,of,light,in,certain,edge,regions,,whereas,even,the,highest,energy,particles,typically,propagate,at,one,percent,of,this,speed.,In,the,usual,cases,,these,rapid,oscillations,are,associated,with,ﬂuid,processes,(i.e.,waves),and,the,kinetic,eﬀects,,such,as,nonlocal,thermal,conduction,,appear,on,longer,timescales.,It,is,computationally,advantageous,to,evolve,or,relax,these,waves,by,solving,low-dimensional,(3D),implicit,or,explicit,ﬂuid,equations,rather,than,(5-6D),kinetic,equations.,Even,if,the,ﬂuid,motion,needs,to,be,solved,explicitly,,this,simpliﬁcation,can,allow,a,massive,computational,speedup.,For,the,particle,method,,the,moment-based,scheme,also,allows,for,noise,reduction,by,extracting,out,the,background,distribution,and,integrating,it,an-,alytically.,That,is,,consider,the,moment,(cid:90),n,=,dV,f,=,(cid:90),dV,f0,+,(cid:90),dV,g.,(17),6,Because,f0,may,be,evaluated,analytically,,only,the,integral,over,g,needs,to,be,evaluated,using,a,Monte-carlo,method;,and,the,RMS,error,of,the,Monte-Carlo,integration,scales,like,the,amplitude,of,g,,so,if,g,is,small,,then,this,can,lead,to,a,radical,decrease,in,the,number,of,markers,needed,to,attain,a,speciﬁc,accuracy.,For,core,plasma,simulations,<,g2,>,/,<,f,2,>∼,10−4,this,allows,10,000,times,fewer,markers,to,be,used.,In,the,edge,,ﬂuctuations,are,large,,but,relatively,high,collisionality,means,that,the,departures,from,Maxwellian,are,not,necessarily,large,(e.g.,low,density,high,velocity,tails,are,present).,6,Proof,of,principle,implementation,A,1D,,two-moment,model,is,implemented,in,the,minepoch,code,that,is,on,the,github,repository.,In,the,test,case,setup,(an,input,ﬁle,’singlestream.deck’,is,given,for,this,case),is,that,a,low,temperature,single-species,collisionless,neutral,species,is,given,an,initial,density,and,velocity,perturbation,of,the,form,n,=,n0[1,+,0.3,∗,sin(4πx/Lx)],,v,=,v0[1,+,sin(6πx/Lx)].,(18),The,details,of,parameters,for,the,test,case,setup,are,described,in,detail,in,the,testing,code.,The,purpose,of,this,test,case,is,a,proof,of,principle,demonstration,that,the,control,variates,method,is,able,to,reduce,the,statistical,sampling,error,in,a,case,with,order,one,density,ﬂuctuations,(as,one,might,ﬁnd,in,the,edge):,this,is,an,advancement,over,control,variates,schemes,used,in,the,core,of,tokamaks,,which,cannot,handle,large,variations.,The,low,temperature,means,the,energy,equation,is,not,needed,,and,as,the,evolution,is,force,free,,a,simpliﬁed,momentum,evolution,equation,is,suﬃcient.,For,this,setup,,because,the,evolution,is,force-free,,the,method,of,character-,istics,may,be,used,to,ﬁnd,the,solution,analytically,(in,an,implicit,form).,We,show,the,time-evolution,of,the,density,in,the,simulation,in,ﬁgures,1,,2,and,3.,At,t,=,2.6,×,10−9,s,,with,f,=,f0,+,g,in,the,control-variates,simulation,,we,have,<,g2,>,/,<,f,2,>=,1.1,×,10−3.,That,is,,despite,the,variation,of,density,by,a,factor,of,2,in,the,x,domain,,the,kinetic,correction,to,the,particle,distribution,g,is,∼,30,times,smaller,than,the,overall,distribution,function,f,.,The,squared,noise,amplitude,scales,like,<,g2,>,/N,,,so,either,one,may,use,1000,times,fewer,markers,in,the,control-variates,simulation,for,the,same,noise,,or,1000,times,lower,squared,noise,amplitude,with,the,same,number,of,markers.,7,Finite,elements,and,particle,methods,,gyroki-,netic,examples.,Finite,element,method,meshes,are,fairly,regularly,used,as,a,basis,for,particle-,in-cell,codes.,Two,examples,of,gyrokinetic,particle-in-cell,codes,are,the,XGC[5],7,Figure,1:,Density,versus,position,at,an,intermediate,simulation,time.,code,and,ORB5[6];,the,author,of,this,document,is,a,developer,of,the,ORB5,code.,The,ORB5,code,used,a,logically,Cartesian,tensor-product,B-spline,represen-,tation,in,magnetic,coordinates.,The,elements,are,curved,in,laboratory,Cartesian,space,,and,this,allows,a,simple,structured,grid,that,nonetheless,conforms,to,the,curved,magnetic,ﬁeld,topology,of,the,tokamak,core.,This,approach,is,not,able,to,handle,the,X-point,of,scrape-oﬀ,layer,region,directly.,We,have,magnetic,coordinates,(s,,θ,,ζ),,which,are,toroidal,coordinates,with,s,∈,[0,,1],representing,the,radial,distance,from,the,magnetic,axis,,χ,the,straight-,ﬁeld,line,poloidal,angle,,and,ζ,the,(geometrical),toroidal,angle.,Taking,the,equal,spaced,case,for,simplicity,,ﬁelds,are,discretely,represented,as:,φ(s,,θ,,ζ),(cid:88),(cid:88),(cid:88),i=[0,Ni],j=[0,Nj,],k=[0,Nk],φi,j,kΛ,(cid:21),(cid:20),s,−,si,δs,Λ,(cid:21),(cid:20),θ,−,θj,δθ,Λ,(cid:21),(cid:20),ζ,−,ζk,δζ,(19),Here,,Λ(x),are,(quadratic,or,cubic),B-spline,functions,(Fig.,4),,which,serve,as,compact-support,basis,functions,for,the,ﬁnite-element,scheme.,For,quadratic,basis,functions,,the,representation,has,continuous,values,and,ﬁrst,derivatives,(i.e.,it,is,C1),and,for,cubic,,the,second,derivatives,are,also,continuous,(it,is,C2).,XGC,uses,a,block-structured,regular,mesh,,also,based,on,low-order,ﬁnite,elements.,The,blockwise,decomposition,allows,the,code,to,represent,,using,one,block,the,core,plasma,using,elements,that,conform,to,magnetic,ﬁeld,lines,to,capture,the,strong,anisotropy.,The,block,region,around,the,X-point,is,not,required,to,conform,to,the,magnetic,geometry,,but,the,low,poloidal,ﬁeld,in,this,region,simpliﬁes,the,handling,of,anisotropy.,Additional,blocks,in,the,scape,oﬀ,layer,allow,the,full,geometry,to,be,handled.,Particle-ﬁeld,operations,are,seen,in,a,radically,diﬀerent,way,in,these,ﬁnite-,8,Figure,2:,Colour,plot,of,density,versus,position,and,time,in,a,standard,(no,control,Variates),PIC,simulation.,element,based,PIC,codes,to,traditional,PIC,codes,,which,use,ﬁnite,diﬀerence,or,volume,concepts.,Traditional,PIC,(e.g.,EPOCH),considers,the,markers,to,have,a,smooth,‘cloud,of,charge’,associated,with,all,the,physical,particles,the,marker,is,representing,,leading,to,a,’shape,function’,which,may,be,used,to,deﬁne,a,continuous,particle,density,throughout,the,simulation,domain.,This,smooth,ﬁeld,may,then,be,evaluated,at,grid,points,in,a,ﬁnite,diﬀerence,scheme,,or,integrated,over,grid,volumes,in,a,ﬁnite,volume,scheme,(e.g.,EPOCH).,Finite-,element,based,PIC,schemes,,on,the,other,hand,,consider,the,markers,as,point-,particles,without,an,explicit,spatial,extent.,However,,functionally,the,smooth,ﬁeld,representation,in,a,ﬁnite,element,code,plays,a,very,similar,role,,and,the,eﬀective,charge,density,as,represented,on,the,FEM,grid,is,smooth.,All,these,codes,involve,direct,evaluation,of,the,ﬁelds,at,the,particle,posi-,tions,,which,is,logically,straightforward.,First,,one,determines,which,grid,cell,the,particle,is,in:,in,ORB5,the,particles,coordinates,are,evolved,in,magnetic,coordinates,,and,these,coordinates,can,be,individually,mapped,to,grid,indexes,in,each,direction.,The,compact,support,then,allows,an,evaluation,using,a,rela-,tively,small,number,of,basis,functions.,For,tensor-product,based,schemes,,about,two,multiplications,are,needed,per,contributing,basis,function,,so,even,though,in,the,cubic,case,64,elements,contribute,,the,per-evaluation,cost,is,not,extreme.,For,a,fully,unstructured,,curvilinear,grid,,one,can,in,principle,evaluate,the,ﬁeld,at,arbitrary,locations,,but,this,may,not,be,straightforward,or,eﬃcient.,It,may,in,general,be,complicated,or,ineﬃcient,to,determine,which,grid,cell,a,particle,is,in,based,on,its,global,coordinate,position.,The,mapping,used,to,allow,distorted,element,positions,is,the,forward,mapping,from,element-wise,local,coordinates,to,global,Cartesian,coordinates,and,the,inverse,mapping,may,be,diﬃcult,to,compute,,so,even,if,the,grid,cell,can,be,determined,,the,local,coordinate,position,in,the,cell,may,be,annoying,to,compute.,In,this,case,,it,9,Figure,3:,Colour,plot,of,density,versus,position,and,time,in,a,control-Variates,PIC,simulation.,may,be,desirable,to,keep,track,of,the,cell-local,coordinates,of,the,particle,,and,,when,it,exits,the,cell,,to,update,which,cell,it,is,in,based,on,its,crossing,of,boundaries.,Using,a,cell-local,coordinate,scheme,complicates,somewhat,the,particle,time-,stepping,in,the,sense,that:,•,The,numerical,equations,have,to,be,rewritten,in,a,general,curvilinear,form.,•,The,coordinates,are,continuous,between,cells,,but,not,necessarily,smooth,,which,will,lead,to,low,order,of,time-convergence,of,timestepping,schemes,without,any,additional,measures,being,taken.,•,One,must,keep,track,of,which,cell,the,particle,is,in,and,hand,it,between,cells,during,the,timestep,,possibly,accross,multiple,cell,boundaries.,Overall,,the,use,of,cell,coordinates,rather,than,global,coordinates,creates,a,much,tighter,coupling,and,more,complicated,interface,between,the,particle,and,FEM,code,elements.,8,Timestepping,accuracy,and,continuity,A,particle,timestepping,equation,symbolically,of,the,form,dZ,dt,=,F,(Z(t),,t),(20),usually,involves,a,dependence,on,certain,ﬁelds,(density,ﬁelds,leading,to,particle,drag,,electromagnetic,ﬁelds,for,charged,particles),that,are,represented,,in,the,10,Figure,4:,B-Splines,on,a,unit-interval,grid.,case,of,Nektar++,,using,the,FEM.,Consider,a,trajectory,Z(cid:48)(t),that,is,locally,analytic,(i.e.,smooth,to,arbitrary,order),and,the,reduced,equation,dZ,dt,=,F,(Z(cid:48)(t),,t),(21),If,F,(Z,,t),is,a,piecewise,polynomial,function,in,Z,,then,as,Z(cid:48)(t),crosses,element,boundaries,,F,will,be,non-smooth,in,time.,If,the,N-th,derivative,is,discontinuous,(a,CN,representation),then,a,simple,time-stepping,scheme,will,have,a,maximum,local,convergence,rate,of,order,N,+,2,on,timesteps,where,this,boundary,is,crossed,,and,N,+,2,globally,,since,most,intervals,will,not,cross,a,cell,boundary.,This,means,,for,example,,that,a,standard,fourth-order,Runge-Kutta,method,,with,4th,order,global,accuracy,,will,only,acheive,that,order,of,convergence,trac-,ing,particles,in,ﬁelds,with,at,least,C2,smoothness.,Although,order,of,con-,vergence,is,not,a,complete,diagnostic,of,the,usefulness,of,a,scheme,,this,indi-,cates,that,high-order,timestepping,may,not,be,appropriate.,Whether,high-order,timestepping,accuracy,is,required,is,quite,problem,dependent.,Special,purpose,integrators,may,be,built,to,break,the,time,domain,up,into,regions,so,that,in,each,time,sub-interval,the,particle,does,not,cross,any,cell,boundaries,,but,this,clearly,creates,signiﬁcant,extra,complexity.,Particle,tracing,in,magnetised,plasmas,is,a,somewhat,special-purpose,prob-,lem,,where,the,evolution,equations,may,often,be,split,into,two,portions,F0,+,F1,,with,F0,representing,(normally),the,long-wavelength,background,magnetic,ﬁeld,,and,F1,associated,with,ﬁne-scale,low-amplitude,turbulent,ﬂuctuation.,Because,of,the,extreme,anisotropy,of,transport,,it,is,often,important,to,very,accurately,solve,for,the,particle,motion,along,the,background,ﬁeld,F0,,in,order,that,small,errors,in,tracing,unperturbed,particle,orbit,do,not,overwhelm,the,small,trans-,port,eﬀect,due,to,F1.,11,Note,that,in,a,tokamak,the,particle,drifts,(in,gyrokinetic,theory),depend,on,derivatives,of,the,magnetic,ﬁeld,(∇B);,this,means,that,even,deﬁning,these,drifts,requires,a,high,order,of,continuity,of,the,magnetic,ﬁeld,representation.,This,issue,will,presumably,also,appear,in,Eulerian,gyrokinetics.,Due,to,the,long,wavelength,smoothness,of,the,background,ﬁelds,,even,if,these,derivatives,formally,lack,continuity,,the,discontinuities,may,not,be,very,large.,The,standard,solution,in,tokamak,codes,is,to,represent,the,background,ﬁelds,in,a,way,that,allows,high-order,derivatives,to,be,taken,,perhaps,on,a,simple,Cartesian,grid,,where,high-order,continuous,representations,are,straightforward,,and,then,evaluate,them,in,a,setup,phase,on,the,grid,points,on,which,the,dy-,namics,will,be,solved,,which,may,be,a,semi-structured,or,unstructured,mesh.,Note,that,,depending,on,the,timestepping,scheme,chosen,,the,particle,evo-,lution,may,need,ﬁeld,evaluation,at,intermediate,time,points.,9,Particles,in,Nektar++,,existing,implementa-,tion,,and,proof,of,principle.,Particles,have,been,implemented,as,a,ﬁlter,within,the,Nektar++,framework,,and,exploited,for,the,purposes,of,considering,erosion,wear,on,surfaces[7].,This,framework,(currently),allows,for,certain,kinds,of,particle,motion,relevant,to,solving,the,Navier-Stokes,equations,,whose,motion,is,subject,to,local,ﬂuid,forces,,but,not,for,particles,to,impact,on,the,ﬂuid,motion.,Several,kinds,of,particle,motion,are,possible,,and,are,schematically,either,ﬂuid,particles,,where,the,particles,have,the,local,ﬂuid,velocity,v,and,satisfy,the,equation,dr,dt,=,v(r,,t),(22),or,ﬁnite-sized,particles,,which,follow,equations,of,motion,dr,dt,=,V,,m,dV,dt,of,particles,subject,to,forces,due,to,the,ﬂuid.,The,ﬁlter,implements,linear,multistep,methods,to,solve,these,equations;,these,are,quite,adaptable,to,plasma,problems,,but,adaption,to,use,other,kinds,of,schemes,would,not,be,excessively,complicated.,=,F,(V,,v,,t),(23),The,existence,of,the,particles,ﬁlter,in,nektar++,,as,well,as,a,Hasegawa-,Wakatani,solver,,has,allowed,us,to,set,up,a,basic,coupled,plasma-particles,test,case,,as,a,preliminary,to,exploring,the,impact,of,e.g.,curved,simulation,elements,on,the,particle,tracing,problem.,As,a,demonstration,,we,initialise,particles,on,a,uniform,grid,and,allow,them,to,be,advected,by,the,ﬂuid,motion,(ﬁg.,5).,10,Co-design,recommendations,•,The,similarity,of,the,Particle,and,Eulerian,moment-based-schemes,mean,that,a,common,codebase,should,be,possible,and,this,will,avoid,duplication,12,Figure,5:,Particle,trajectories,on,the,(x,,y),plane.,Red,markers,show,initial,positions,,black,shows,positions,at,t,=,4.,of,eﬀort.,Until,that,point,,only,very,simple,ﬂuid,solvers,are,required.,•,Particle,methods,for,gyrokinetic,systems,are,generally,easy,to,adaptable,in,terms,of,trajectory,equations,of,the,chosen,gyrokinetic,equations,,so,it,is,not,crucial,to,ﬁx,these,at,an,early,stage,of,the,project.,•,The,representation,of,background,and,perturbed,electromagnetic,ﬁeld,needs,special,care,if,particle,orbits,are,to,be,accurately,solved.,Some,precalculation,may,be,necessary,for,derivatives,of,magnetic,ﬁelds.,Espe-,cially,if,perturbed,magnetic,ﬁelds,may,be,large,careful,thought,needs,to,be,given,to,creating,a,smooth,representation,of,ﬁelds,or,indirectly,evaluating,derivatives.,•,Thought,needs,to,be,given,to,co-designing,timestepping,schemes,for,com-,bined,ﬂuid-particle,codes;,particle,methods,can,use,e.g.,linear,multistep,methods,,or,special,purpose,integrators,(Boris,algorithm),but,the,choice,depends,somewhat,on,the,diﬃculty,of,evaluating,ﬁelds,at,intermediate,timepoints.,•,A,dedicated,study,would,be,needed,to,examine,particle-tracking,in,piece-,wise,curvilinear,grids,if,the,Neptune,software,is,to,use,particle,methods.,There,is,infrastructure,already,existing,to,perform,this,work,with,Nek-,tar++,,both,for,conventional,ﬂuid,problems,or,coupled,to,plasma-speciﬁc,problems.,13,References,[1],Michael,Barnes,Felix,I.,Parra,and,Michael,Hardman.,1d,drift,kinetic,models,with,periodic,boundary,conditions.,Technical,report,,Oxford,University,,2021.,[2],S.,Brunner,,E.,Valeo,,and,J.,A.,Krommes.,Collisional,delta-f,scheme,with,evolving,background,for,transport,time,scale,simulations.,Physics,of,Plas-,mas,,6(12):4504–4521,,1999.,[3],A.Y.,Aydemir.,A,uniﬁed,Monte,Carlo,interpretation,of,particle,simulations,and,applications,to,non-neutral,plasmas.,Physics,of,Plasmas,,1:822–831,,1994.,[4],Roman,Hatzky,,Trach,Minh,Tran,,Axel,Knies,,Ralf,Kleiber,,and,Simon,J.,Allfrey.,Energy,conservation,in,a,nonlinear,gyrokinetic,particle-in-cell,code,for,ion-temperature-gradient-driven,modes,in,theta-pinch,geometry.,Physics,of,Plasmas,,9(3):898–912,,2002.,[5],S.,Ku,,C.,S.,Chang,,R.,Hager,,R.,M.,Churchill,,G.,R.,Tynan,,I.,Cziegler,,M.,Greenwald,,J.,Hughes,,S.,E.,Parker,,M.,F.,Adams,,E.,D’Azevedo,,and,P.,Worley.,A,fast,low-to-high,conﬁnement,mode,bifurcation,dynamics,in,the,boundary-plasma,gyrokinetic,code,xgc1.,Physics,of,Plasmas,,25(5):056107,,2018.,[6],E.,Lanti,,N.,Ohana,,N.,Tronko,,T.,Hayward-Schneider,,A.,Bottino,,B.F.,McMillan,,A.,Mishchenko,,A.,Scheinberg,,A.,Biancalani,,P.,Angelino,,S.,Brunner,,J.,Dominski,,P.,Donnel,,C.,Gheller,,R.,Hatzky,,A.,Jocksch,,S.,Jolliet,,Z.X.,Lu,,J.P.,Martin,Collar,,I.,Novikau,,E.,Sonnendrcker,,T.,Ver-,nay,,and,L.,Villard.,Orb5:,A,global,electromagnetic,gyrokinetic,code,using,the,pic,approach,in,toroidal,geometry.,Computer,Physics,Communications,,page,107072,,2019.,[7],Manuel,F.,Mej´ıa,,Douglas,Serson,,Rodrigo,C.,Moura,,Bruno,S.,Carmo,,Jorge,Escobar-Vargas,,and,Andr´es,Gonz´alez-Mancera.,Erosion,wear,evaluation,using,nektar+,+.,In,Spencer,J.,Sherwin,,David,Moxey,,Joaquim,Peir´o,,Peter,E.,Vincent,,and,Christoph,Schwab,,editors,,Spectral,and,High,Order,Methods,for,Partial,Diﬀerential,Equations,ICOSAHOM,2018,,pages,419–,428,,Cham,,2020.,Springer,International,Publishing.,14,New,features,in,minepoch,for,advanced,particle,tracing,and,noise,reduction,B.F.,McMillan,and,T.,Goﬀrey,September,15,,2021,This,describes,certain,new,features,added,to,the,minepoch,code,as,part,of,the,NEPTUNE,project,(Milestones,4,and,5,of,the,Particles,subproject).,The,minepoch,code,is,designed,as,a,proxy,app,for,computational,perfor-,mance,testing,with,a,minimal,codebase,,and,therefore,does,not,contain,a,full,in-,put,deck,parser.,Certain,parameters,may,be,adjusted,via,the,input,deck,(FOR-,TRAN,namelist),,but,more,complex,modiﬁcations,require,hard-coded,setup,subroutines;,there,is,a,string-valued,namelist,parameter,’problem’,that,selects,which,setup,subroutine,is,used.,See,the,README,ﬁle,for,a,description,of,how,to,run,relevant,testcases.,We,describe,the,foundations,for,the,new,algorithms,,and,in,the,ﬁnal,section,note,some,practical,implementation,aspects.,1,Advanced,particle,tracing,(M4),1.1,Substepping,The,normal,PIC,algorithm[1],may,be,conceptually,represented,as,a,leapfrog,method,of,the,form:,•,Advance,particle,momentum,from,pt−3h/2,to,pt−h/2,using,Et−h,,Bt−h.,•,Advance,particle,position,from,xt−h,to,xh,using,velocity,calculated,from,momentum,pt−h/2.,•,Deposit,(time-integrated),current,based,on,the,previous,and,current,po-,sition,(this,is,the,charge-conserving,Esirkepov,step[2].,•,Advance,Electric,and,Magnetic,ﬁelds,from,Et−h,,Bt−h,to,Et,,Bt,using,the,currents,calculated,in,the,previous,step.,In,cases,where,the,timestep,limitation,is,due,to,particle,gyration,,a,simple,substepping,method,consists,of,repeating,the,ﬁrst,two,of,these,operations,N,times,with,a,reduced,timestep,H,=,h/N,.,Each,substep,n,∈,[1,,N,],is,of,the,form,(pt+(2n−3)H/2,,xt+(2n−2)H,),→,(pt+(2n−1)H/2,,xt−(2n−2)H/2),(1),1,with,overall,eﬀect,of,N,substeps,(pt+H/2,,xt),→,(pt+h+H/2,,xt+h).,(2),In,the,simple,substepping,currently,implemented,in,minepoch,,the,current,is,calculated,using,these,endpoints,,so,that,the,ﬁeld,is,no,longer,time-centred.,This,reduces,the,time-accuracy,of,the,scheme,to,ﬁrst,order,for,time-varying,E,and,B,ﬁelds,(for,particle,tracing,in,static,ﬁelds,the,algorithm,is,second,order).,Although,correcting,the,currents,to,restore,second-order,accuracy,in,time,is,possible,,the,main,purpose,of,substepping,is,in,conjunction,with,an,implicit,ﬁeld,solver.,1.2,Drift-kinetic,species,We,implement,a,minimal,version,of,Drift-kinetics,based,on,the,long-wavelength,general,gyrokinetic,formalism,(the,Lagrangian,used,to,derive,the,Euler-Lagrange,equations,is,eq.,9.,of,Ref.,[3],,with,terms,of,order,(cid:15),neglected).,Users,are,expected,to,make,sure,(and,possibly,post-verify),that,the,drift-kinetic,ordering,is,valid.,For,example,,if,the,magnetic,ﬁeld,strength,is,zero,somewhere,in,the,domain,the,ordering,will,break,down,and,division-by-nearly-zero,will,occur;,of,course,,for,tokamak,applications,,this,is,not,a,problem.,Drift-kinetic,species,with,phase-space,coordinates,Z,=,(R,,v||,,µ),follow,the,Euler-Lagrange,equations,dR,dt,=,bv||,+,mv2,||,qB2,B,×,(b.∇b),+,µB,2qB3,B,×,∇B,,E,×,B,B2,+,dv||,dt,m,=,−b.(e∇φ,+,µ∇B),and,Here,,b,=,B/B.,dµ,dt,=,0.,(3),(4),(5),Free,and,bound,(magnetisation,and,polarisation),currents,are,associated,with,a,drift-kinetic,particle.,The,free,currents,are,found,by,using,the,particle,displacement,over,the,computational,timestep,using,Esirkepov,current,deposi-,tion.,That,is,,a,current,Jn+1/2,is,found,consistent,with,the,particle,displacement,from,Rn,to,Rn−1/2.,Bound,currents,are,not,calculated,in,the,existing,imple-,mentation,of,minepoch,(for,electromagnetic,problems,,or,where,the,drift-kinetic,species,is,not,electrons,,the,bound,currents,are,not,generally,negligible).,The,second,order,drift,kinetic,solve,is,an,explicit,midpoint,method,of,the,form,•,Z0,•,B0,n+1,is,found,via,an,Euler,step,using,the,ﬁelds,Bn,,En.,n+1,,E0,n+1,are,found,by,stepping,Maxwell’s,laws,,evaluating,the,current,n+1,using,the,Esirkepov,due,to,the,particles,displacing,from,Rn,to,R0,method,(as,per,standard,PIC).,2,•,Zn+1,is,found,by,evaluating,the,RHS,of,the,Euler-Lagrange,equations,,es-,n+1)/2,and,likewise,the,midpoint,timating,the,midpoint,Zn+1/2,=,(Zn,+,Z0,ﬁeld,values.,•,Bn+1,,En+1,are,found,by,stepping,Maxwell’s,laws,,evaluating,the,current,due,to,the,particles,displacing,from,Rn,to,Rn+1.,Note,that,unlike,in,the,standard-PIC,method,,all,the,particle,and,ﬁeld,at-,tributes,are,stored,at,the,same,time-point,,so,the,combination,of,drift-kinetics,and,substepping,is,second-order,in,time,without,further,modiﬁcation.,2,Low-noise,PIC,(M5),The,low-noise,PIC,method,is,documented,in,report,2047355-TN-01,and,is,not,further,discussed,here.,3,Technical,details,The,initial,version,of,the,minepoch,code,implements,the,standard,charge-conserving,fully-electromagnetic,PIC,method.,Some,preparatory,work,was,required,to,modularise,the,code,in,preparation,for,‘advanced,algorithm’,implementation.,The,main,work,was,to,split,the,particle,push,loop,into,several,subroutines.,In,particular,,the,ﬁeld,evaluation,and,current,deposition,were,extracted,from,the,push,routine.,The,ﬁeld,interface,is,relatively,simple,,and,consists,of,a,pair,of,subroutines;,a,ﬁeld,(and,derivative),evaluation,at,speciﬁc,position,,and,a,current,deposition,subroutine,that,take,the,particle,initial,and,ﬁnal,positions,as,arguments.,We,also,pass,a,structure,to,allow,temporary,storage.,This,also,allows,simple,implementation,of,other,particle,evolution,equations,(e.g.,drift-kinetic),as,well,as,the,use,of,external,ﬁeld,solvers.,The,drift-kinetic,solver,uses,the,ﬁrst,and,second,element,in,the,momentum,array,to,store,v||,and,µ,respectively;,the,transformation,to,these,velocity-space,coordinates,is,performed,after,the,particle,loading.,3.1,Code,releases,and,availability,There,are,two,releases,associated,with,milestones,4,and,5.,Version,1.0,contains,the,initial,development,of,the,methods,described,in,this,document.,Version,1.1,includes,some,improvements,and,ﬁxes,to,v1.0,,as,well,as,code,documenta-,tion.,The,release(s),associated,with,milestones,are,available,from,the,GitHub,repository.,References,[1],T,D,Arber,,K,Bennett,,C,S,Brady,,A,Lawrence-Douglas,,M,G,Ramsay,,N,J,Sircombe,,P,Gillies,,R,G,Evans,,H,Schmitz,,A,R,Bell,,and,C,P,Ridgers.,3,Contemporary,particle-in-cell,approach,to,laser-plasma,modelling.,Plasma,Physics,and,Controlled,Fusion,,57(11):113001,,2015.,[2],T.Zh.,Esirkepov.,Exact,charge,conservation,scheme,for,particle-in-cell,sim-,ulation,with,an,arbitrary,form-factor.,Computer,Physics,Communications,,135(2):144,–,153,,2001.,[3],B.,F.,McMillan,and,A.,Sharma.,A,very,general,electromagnetic,gyrokinetic,formalism.,Physics,of,Plasmas,,23(9):092504,,2016.,4

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