TN-13-2_Spatially2DPlasmaModelIncorporatingVelocitySpaceEffects
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Report,2047356-TN-13,(D5.1,+,D5.2):,Spatially,2-D,plasma,model,incorporating,velocity,space,effects,David,Moxey,,King’s,College,London,Ben,Dudson,,Peter,Hill,,Ed,Higgins,,David,Dickinson,&,Steven,Wright,,University,of,York,March,31,,2022,Contents,1,Executive,summary,2,Introduction,3,Test,cases,(D5.1),4,Path,to,implementation,within,Nektar++,(D5.2),4.1,Discretisation,choice,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,4.2,Explicit,vs.,implicit,solutions,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,4.3,Particle,coupling,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,4.4,Practical,considerations,5,Conclusions,1,Executive,summary,1,1,2,3,3,3,4,4,4,This,report,briefly,outlines,potential,routes,for,implementation,of,the,system,2-6,equations,within,the,Nektar++,framework.,Owing,to,the,practicality,of,implementing,such,a,system,within,the,limited,resources,of,this,project,,we,focus,on,the,steps,required,to,implement,such,a,system,,highlighting,where,improvements,should,be,made,within,Nektar++,to,enable,future,implementation.,2,Introduction,As,taken,from,the,equations,document,supplied,with,NEPTUNE,[1],,system,2-6,is,a,spatially,2D,plasma,model,incorporating,velocity,space,effects,and,is,expressed,in,the,following,manner,for,the,electron,number,density,ne,,vorticity,∇,·,E+,,electron,energy,Ee,,ion,energy,Ei,and,neutral,number,density,nn:,∂ne,∂t,+,∇,·,(neue),=,Sne,−,(cid:18),ne,τne,(∇,·,E+),+,∇,·,(cid:0)∇,·,(cid:0)ui,⊗,E+(cid:1)(cid:1),=,∇,·,ni,(u∇Bi,+,ucx),−,∂,∂t,(cid:19),n,−,eu∇Be,1,Zi,+,∇,·,(ν∇⊥(∇,·,E+)),1,Zi,ne,τne,−,ni,τni,+,1,∂Ee,∂t,∂Ei,∂t,+,∇,·,(Eeue,+,peue),=,SEe,−,+,∇,·,(Eiui,+,piui),=,SEi,−,Ee,τEe,Ei,τEi,+,Qie,+,∇,·,(χ⊥ene∇⊥Te),+,Qie,+,∇,·,(χ⊥ini∇⊥Ti),∂nn,∂t,=,Snn,+,∇,·,(Dn∇⊥pn),where,pα,is,pressure,,Tα,is,temperature,,Zi,is,charge,state.,We,omit,the,definitions,of,the,various,sub-terms,and,constants,to,the,equation,document,[1],as,the,equation,system,itself,provides,sufficient,motivation,for,the,remainder,of,this,document.,3,Test,cases,(D5.1),D5.1,focuses,on,the,development,of,adequate,test,cases,for,this,set,of,equations.,Given,the,inexperience,of,the,primary,author,of,this,deliverable,with,the,physics,exhibited,in,these,sys-,tems,,we,will,not,discuss,here,the,consideration,of,simplified,versions,of,this,system,under,the,selection,of,simplified,parameters,(although,such,a,case,may,be,possible).,In,lieu,of,this,,and,also,considering,that,this,approach,would,not,consider,every,term’s,correct-,ness,by,definition,,a,sensible,alternative,strategy,in,this,case,would,to,be,consider,a,functional,test,of,the,solver,via,the,method,of,manufactured,solutions,(MMS),[2],,which,is,an,approach,previously,used,successfully,in,e.g.,BOUT++,[3],,within,the,test,suite,of,Nektar++,,and,in,many,other,codes,where,analytic,solutions,are,unavailable.,In,this,setting,,a,solution,field,is,chosen,a,priori,,and,the,equations,evaluated,analytically,using,this,manufactured,solution.,A,source,term,is,then,added,to,the,equations,with,these,calculated,terms,,which,then,permits,verification,of,the,computed,discretised,equations.,This,permits,a,number,of,benefits,,namely,that:,•,In,the,limit,of,small,element,size,and,increasing,polynomial,order,,this,should,therefore,serve,as,a,mechanism,to,test,the,formal,accuracy,of,the,solver.,•,Given,an,appropriate,choice,of,manufactured,solution,,this,yields,a,functional,unit,test,of,all,of,the,terms,from,the,equations.,•,This,removes,the,limitation,of,the,complexity,of,the,equations,or,the,need,for,exact,analytic,solutions.,The,MMS,approach,does,,however,,come,with,a,number,of,limitations.,•,The,choice,of,domain,and,solution,is,important,in,order,to,ensure,correctness,is,tested,adequately.,For,example,,a,zero,solution,field,is,technically,a,‘correct’,solution,,but,unlikely,to,yield,any,useful,validation,of,the,solver.,•,For,higher-order,methods,,the,use,of,non-polynomial,manufactured,solutions,is,also,crit-,ical,in,ensuring,that,all,modes,of,the,solution,are,adequately,energised,,and,sometimes,this,can,be,difficult,to,ensure,across,the,entire,domain,of,interest.,•,In,parallel,with,the,above,,the,complexity,of,the,system,likely,necessitates,the,use,of,e.g.,a,computer-algebra,system,to,compute,desired,source,terms,accurately,and,without,human,error.,•,Issues,due,to,timestepping,problems,will,very,likely,not,be,highlighted,in,this,approach,(at,least,in,the,experience,of,this,author).,2,4,Path,to,implementation,within,Nektar++,(D5.2),The,complexity,of,system,2-6,poses,some,challenges,in,terms,of,implementation,,particularly,considering,the,existing,infrastructure,within,Nektar++.,In,this,section,we,highlight,a,few,of,the,expected,bottlenecks,and,potential,mitigations,for,these.,4.1,Discretisation,choice,Given,the,conservative,form,of,the,equations,as,written,and,need,to,ensure,accuracy,in,terms,of,mass,conservation,,the,anticipated,discretisation,would,be,a,discontinuous,Galerkin,(DG),formulation,,which,should,map,naturally,onto,the,equations,at,hand.,With,this,said,,there,are,some,terms,(particularly,for,e.g.,the,vorticity),which,will,need,careful,consideration,in,terms,of,the,weak,formulation.,Additionally,,certain,terms,such,as,the,electrostatic,potential,could,potentially,be,solved,in,a,continuous,manner,for,efficiency,and,accuracy,purposes.,This,joint-discretisation,methodology,has,been,shown,to,work,quite,successfully,in,Nektar++,for,e.g.,the,implementation,of,the,Hasegawa-Wakatani,proxyapp,,whereby,the,DG,method,acts,to,stabilise,the,equations,(thereby,meaning,that,hyperelliptic,terms,are,not,required,for,stability),,whilst,the,CG,method,provides,a,route,towards,obtaining,the,elliptic,solution,for,φ,robustly,and,efficiently.,4.2,Explicit,vs.,implicit,solutions,An,initial,implementation,of,these,terms,within,Nektar++,will,need,to,be,explicit,in,time.,There,are,a,number,of,reasons,to,motivate,this:,•,From,a,practical,perspective,this,is,the,path,of,least,resistance,and,the,infrastructure,for,this,is,relatively,robust,,with,several,solvers,(such,as,the,compressible,Navier-Stokes,equations),implemented,in,this,manner.,•,The,dependence,of,terms,as,a,nonlinear,function,of,plasma,properties,(e.g.,eqs,110-113,in,[1]),makes,a,fully,nonlinear,solve,arguably,difficult,in,the,first,instance,,bearing,in,mind,the,likely,time,constraints,of,the,development.,•,For,implicit-in-time,problems,,much,of,the,infrastructure,around,Nektar++,is,designed,for,a,series,of,scalar,solutions,vs.,the,solution,of,a,large,coupled,system.,For,example,in,the,incompressible,Navier-Stokes,equations,the,use,of,the,velocity,correction,scheme,of,[4],involves,the,solution,of,a,pressure,Poisson,and,three,Helmholtz,equations,for,velocity.,There,are,however,exceptions,to,this,,as,we,discuss,in,the,below.,However,,given,an,initial,explicit-in-time,implementation,,there,is,the,potential,to,consider,implicit,implementations,based,on,the,techniques,proposed,in,[5].,This,work,has,extended,the,existing,explicit,implementation,of,the,compressible,Navier-Stokes,equations,through,the,Jacobi-free,Newton-Krylov,method,(JFNK).,In,this,approach,,the,explicit,solver,is,evaluated,twice,in,order,to,determine,an,approximate,Jacobian,based,on,a,simple,first-order,approxima-,tion,,and,the,framework,in,Nektar++,has,been,designed,to,be,reasonably,generic,in,its,use,for,other,systems.,Theoretically,the,JFNK,approach,enables,the,implicit,solution,of,any,sys-,tem,,given,an,appropriate,explicit,solver;,however,,in,practical,terms,,to,address,conditioning,issues,in,the,solution,of,the,resultant,system,,it,is,often,necessary,to,precondition,the,system,by,using,appropriate,parts,of,the,linearised,equations.,Such,an,approach,could,be,adopted,here,,although,deriving,appropriate,preconditioner,terms,and,their,implementation,complexity,would,likely,be,considerable.,3,4.3,Particle,coupling,As,noted,in,[1],,there,is,no,explicit,coupling,of,this,system,with,particle,models,within,the,interior,of,the,domain.,However,,the,sheath,boundary,condition,is,more,accurately,represented,as,particles.,Appropriate,interfaces,must,therefore,be,made,in,order,to,enable,this,integration.,These,are,presently,the,subject,of,discussion,between,UKAEA,and,the,authors,,and,conceptu-,ally,the,process,has,been,established,to,e.g.,determine,cell,information,and,query,for,solution,values,given,a,position,in,space;,however,more,detailed,technical,development,is,required,to,establish,the,parallel,execution,strategy.,4.4,Practical,considerations,System,2-5,as,stated,is,highly,complex,involving,not,only,a,large,number,of,parameters,,vari-,ables,and,equations,,but,a,large,amount,of,domain-specific,knowledge,in,terms,of,the,underlying,physics.,A,successful,solver,development,will,therefore,require,a,balance,of,efforts,between,an,interdisciplinary,team,comprising,experts,in,the,numerical,implementation,as,well,as,assistance,in,interpretation,and,translation,from,the,plasma,physics,&,fusion,domains.,For,example:,•,Initial,assistance,will,be,required,in,developing,the,appropriate,solution,pathway,and,time,integration,scheme,in,a,more,complete,form.,The,lack,of,a,DSL,within,Nektar++,necessitates,more,direct,implementation,,and,is,perhaps,an,area,for,future,development,(e.g.,with,the,provision,of,a,lightweight,DSL,in,the,setting,of,an,explicit,solver,,the,specification,of,these,equations,would,be,more,straightforward).,•,Some,terms,(e.g.,τ·,terms,,eqs,110-113,in,[1]),are,written,as,either,constants,or,nonlinear,functions,of,the,plasma;,guidance,as,to,the,most,appropriate,route,as,the,code,is,developed,is,critical.,•,Even,after,initial,development,,instabilities,and,errors,will,be,inevitable:,tracking,down,the,source,of,these,instabilities,and,determining,their,nature,(as,either,implementation-,specific,or,physics-based),requires,input,from,both,‘teams’,in,this,setting.,5,Conclusions,This,deliverable,has,briefly,outlined,the,likely,development,route,for,a,solver,based,around,system,2-6.,The,final,phase,of,this,project,will,be,to,start,in,the,development,of,some,aspects,of,this,solver;,for,example,in,discussion,with,partners,at,York,and,UKAEA,,to,map,out,the,solver,structure,and,order,of,operations,given,the,system.,This,progress,will,be,reported,in,the,final,deliverable,,D5.3,at,the,conclusion,of,this,project,,along,with,any,initial,development,efforts,that,are,possible.,References,[1],Wayne,Arter.,Equations,for,NEPTUNE,Proxyapps.,Technical,report,,UK,Atomic,Energy,Authority,,2020.,[2],Kambiz,Salari,and,Patrick,Knupp.,Code,verification,by,the,method,of,manufactured,solu-,tions.,Technical,report,,Sandia,National,Lab.(SNL-NM),,Albuquerque,,NM,(United,States),,2000.,4,[3],Benjamin,Daniel,Dudson,,Jens,Madsen,,John,Omotani,,Peter,Hill,,Luke,Easy,,and,Michael,Løiten.,Verification,of,BOUT++,by,the,method,of,manufactured,solutions.,Physics,of,Plasmas,,23(6):062303,,2016.,[4],George,Em,Karniadakis,,Moshe,Israeli,,and,Steven,A,Orszag.,High-order,splitting,methods,for,the,incompressible,navier-stokes,equations.,Journal,of,computational,physics,,97(2):414–,443,,1991.,[5],Z.,Yan,,Y.,Pan,,G.,Castiglioni,,K.,Hillewaert,,J.,Peir´o,,D.,Moxey,,and,S.,J.,Sherwin.,Nektar++:,Design,and,implementation,of,an,implicit,spectral/hp,element,compressible,flow,solver,using,a,jacobian-free,newton,krylov,approach.,Comput.,Math.,Appl.,,81:351–,372,,2021.,5

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