CD-EXCALIBUR-FMS0075-M7c.2_and_M8c.1_ExcaliburFusionModellingSystem
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   :description: technical note

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ExCALIBUR,Complementary,actions.,Code-coupling,,physics,databases,and,benchmarking,techniques,2.,M7c.2,Version,1.00,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Mile-,stone,M7c.2.,Complementary,actions,are,described,which,assist,code-,coupling,and,the,use,of,benchmarking,techniques,and,physics,databases,,generally,providing,support,to,more,than,one,Y4/Y5,deliverable.,Code-,coupling,activities,have,extended,to,the,provision,of,information,needed,to,ensure,successful,collaboration,between,different,sites,,such,as,intro-,ductory,technical,material,,consistent,use,of,physical,units,,text,processing,tools,designed,to,enforce,uniformity,of,document,production,and,improve-,ments,to,the,developers’,website.,Work,by,the,grantees,at,York,,on,the,de-,velopment,of,benchmark,problems,,DSLs,and,multi-species,physics,,and,at,STFC,on,relevant,preconditioning,and,timestepping,numerical,techniques,,is,summarised.,Internal,work,on,atomic,physics,databases,is,annexed,as,a,separate,report.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0075,Issue:,Date:,1.00,23,March,2023,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Wayne,Arter,Matthew,Barton,Owen,Parry,Signature,N/A,N/A,N/A,Date,23,March,2023,23,March,2023,23,March,2023,BD,Reviewed,By:,Wayne,Arter,23,March,2023,Project,Technical,Lead,1,1,Introduction,This,work,for,Deliverable,7c,is,part,of,a,programme,of,work,items,designed,to,provide,an,unin-,terrupted,programme,of,coordination,work,providing,support,to,more,than,one,Y4/Y5,deliverable.,The,procurement,of,software,support,to,ensure,smooth,continuation,of,external,work,was,de-,scribed,in,the,preceding,M7c.1,Report,[1].,Work,on,the,production,of,classes,and,interfaces,to,atomic,physics,data,for,collisions,and,radiation,formed,a,separate,Deliverable,8c,,the,note,on,which,appears,as,an,annex,to,the,present,report.,Work,conducted,internally,by,UKAEA,is,described,in,Section,2.,Aside,from,routine,maintenance,tasks,such,as,posting,and,indexing,reports,received,from,grantees,after,acceptance,,this,is,effi-,ciently,described,under,four,subsection,headings,,namely,(1),educational,material,in,Section,2.1,,(2),technical,material,in,Section,2.2,,(3),text,processing,tools,in,Section,2.3,,and,(4),improvements,to,the,developers’,website,Section,2.4.,Work,described,in,21,reports,accepted,from,the,external,grantees,during,the,year,Y4,,is,summarised,in,Section,3.,2,2,Task,Work,by,UKAEA,2.1,Educational,Material,Introductory,material,has,been,drafted,to,demonstrate,important,features,of,timestepping,and,preconditioning,algorithms,relevant,to,Project,NEPTUNE.,In,due,course,it,is,intended,that,this,“SIAM,Review”,style,material,will,be,made,public,along,with,reports,already,available,internally,to,UKAEA,(such,as,the,guide,to,finite,elements,aimed,at,people,who,have,formerly,only,used,finite,difference,schemes).,The,timestepping,material,shows,how,different,adaptive,strategies,perform,on,a,model,problem,of,simple,harmonic,motion,,for,third,and,fourth,order,accurate,updating.,The,effects,of,adaptation,using,absolute,and,relative,error,are,compared.,It,is,pointed,out,that,the,cancellations,from,step-to-,step,occurring,in,quadrature-like,schemes,when,fixed,timestep,is,used,,may,make,these,schemes,unexpectedly,competitive.,The,preconditioning,material,examines,the,structure,of,the,inverses,of,matrices,arising,in,1-D,models,of,diffusion,,advection,and,advection-diffusion.,Advection,is,here,supposed,treated,both,by,centred,and,upwind,schemes,,especially,the,Patankar,scheme,used,by,state-of-the-art,finite,difference,codes,for,the,tokamak,edge.,Since,the,aim,of,preconditioning,matrices,is,approximately,to,apply,the,matrix,inverse,,the,properties,of,the,inverse,matrix,are,plausibly,very,important,for,the,behaviour,of,the,preconditioned,iterations.,It,emerges,that,for,matrices,representing,advection,,there,is,strong,dependence,of,the,properties,of,the,inverse,on,stagnation,point,location.,There,may,also,be,a,large,difference,in,condition,number,between,very,similar,formulations,,remarkably,indeed,whether,the,discretisation,uses,an,odd,or,an,even,number,of,grid,points.,The,structure,of,inverse,matrices,for,diffusion,is,much,less,variable,,highlighting,the,much,greater,challenge,presented,by,advection-dominated,compared,to,diffusion,problems.,2.2,Technical,Material,2.2.1,Physical,Units,and,Internal,Scalings,As,far,as,is,compatible,with,numerical,stability,and,accuracy,,NEPTUNE,will,employ,SI,units,,with,the,main,exception,that,temperature,may,be,specified,in,electron-Volts,eV,.,Algorithmically,the,ex-,ception,is,easily,handled,by,always,writing,kT,for,temperature,T,where,the,dimensional,quantity,k,ensures,that,the,product,kT,has,units,of,energy.,(So,k,=,|e|,if,T,is,given,in,eV,or,Boltzmann’s,constant,if,T,is,measured,in,Kelvin.),A,further,practical,exemption,arises,from,the,need,to,interface,with,CAD,systems,,which,typically,work,in,millimetres,mm,,where,the,recommendation,is,to,scale,geometry,to,metres,as,soon,as,is,practicable.,NEPTUNE,will,not,accept,models,expressed,in,dimensionless,units,because,of,the,potential,for,confusion,when,a,range,of,different,physical,time,and,length-scales,are,involved.,It,is,of,course,,entirely,reasonable,for,theorists,to,pick,scalings,appropriate,to,their,particular,problem,,and,indeed,to,incorporate,purely,numerical,factors,in,the,scalings,to,simplify,the,coefficients,of,the,equations,,and,hence,their,analytic,work.,Why,this,causes,problems,for,NEPTUNE,can,be,understood,by,3,considering,the,situation,when,a,user,or,developer,wants,to,introduce,an,additional,physical,effect,not,considered,by,the,original,theorist,,for,then,the,newcomer,needs,to,know,precisely,what,was,done,before,,and,then,has,to,scale,terms,representing,the,new,effect,before,their,introduction,into,the,software.,It,is,worthwhile,emphasising,the,word,‘precisely’,-,the,inclusion,of,purely,numerical,factors,in,the,scaling,may,well,introduce,ambiguity.,Moreover,,there,are,only,a,small,number,of,fundamental,dimensional,quantities,,namely,time,,length,,charge,and,mass,,so,a,limited,number,of,ways,in,which,terms,can,be,non-dimensionalised,unambiguously.,(Temperature,in,Kelvin,and,angular,quantities,are,dimensionless,fundamental,quantities.),An,example,of,the,problem,of,ambiguity,in,the,scaling,,occurs,when,augmenting,Magnetohydro-,dynamics,(MHD),with,electrostatic,phenomena.,In,MHD,,scaling,magnetic,field,by,a,reference,value,B0,causes,no,difficulty,,but,as,soon,as,electric,charge,is,introduced,and,scaled,,typically,by,the,charge,on,the,electron,|e|,,since,in,SI,units,magnetic,induction,has,unit,1,T=,kgs−1C−1,,there,may,arise,the,ambiguity,as,to,whether,B0,or,say,mu/(|e|ts),is,appropriate,for,non-dimensionalising,a,new,magnetic,field,term.,Internally,,however,,NEPTUNE,needs,to,scale,fields,so,that,their,values,are,expressible,efficiently,in,floating,point,,meaning,that,large,exponents,of,either,sign,are,to,be,avoided.,The,suggested,limit,is,to,exponents,smaller,than,38,in,absolute,value,,a,criterion,set,by,the,standard,single,pre-,cision,representation,of,floating-point,numbers.,This,is,expected,to,be,best,achieved,by,non-,dimensionalising,,but,‘under,the,hood’,,ie.,with,minimal,user,interaction.,There,will,be,allowed,one,permissible,exception,to,the,rule,that,no,purely,numerical,factors,should,appear,in,scalings,,namely,to,introduce,a,factor,Nref,to,reduce,number,densities,to,more,reasonable,magnitudes,,and,a,factor,wref,to,allow,for,the,representation,of,many,plasma,particles,by,a,single,computational,super-particle,,in,effect,to,allow,for,the,smallness,of,an,individual,particle’s,charge,and,mass.,With,that,,the,key,scalings,are,1.,time,in,units,of,ts,,2.,length,in,units,of,Ls,,3.,mass,in,units,of,mu,,the,atomic,mass,unit,4.,electric,charge,in,units,of,|e|,=,|qe|,,the,absolute,value,of,the,charge,on,the,electron,5.,number,densities,such,as,ne,made,dimensionless,with,respect,to,an,additional,factor,of,Nref,as,well,as,L−D,s,where,D,is,the,spatial,dimensionality,of,the,problem,6.,superparticles,carry,weight,wref,,,eg.,a,charge,wref,qe,is,carried,by,each,‘super-’electron,The,NEPTUNE,website,has,a,section,on,“Physical,properties,of,the,edge,plasma”,which,gives,likely,values,for,ts,and,Ls.,Equivalently,Ls,and,the,velocity,scale,Us,may,be,specified,instead,,thus,depending,on,the,relevant,physics,Us,=,103,−,107,ms−1,,Ls,=,10−5,−,10,m.,These,two,scalings,could,be,set,by,the,user,(together,with,Nref,and,wref,),,once,for,a,given,simulation.,In,keeping,with,the,application,to,the,Exascale,,the,pure,number,Nref,=,1018,,whilst,taking,wref,=,1010,seems,most,convenient.,In,practice,,the,ratio,of,absolute,electron,charge,to,atomic,mass,unit,(≃,proton,mass),appears,frequently,enough,to,warrant,its,own,symbol,KM,=,|e|,≈,108,,eg.,the,electric,field,mu,unit,becomes,U,2,s,LsKM,and,the,magnetic,field,unit,becomes,Us,LsKM,.,4,The,next,subsections,Section,2.2.2,,Section,2.2.3,and,Section,2.2.4,illustrate,the,application,of,the,proposed,approach,to,the,treatment,of,the,Braginskii,transport,coefficients,and,to,charged,particle,motion,in,a,magnetic,field.,2.2.2,Introduction,to,Scaling,The,Branginskii,coefficients,might,be,used,in,a,multispecies,context,over,a,range,of,different,timescales.,Ultimately,finite,element,codes,,indeed,most,numerical,schemes,,depend,on,solution,of,a,linear,algebra,problem,Ax,=,b,(1),where,x,are,field,values,to,be,determined,,and,A,and,b,follow,from,the,model.,Take,the,simplest,case,of,a,2,×,2,model,A,=,(cid:19),(cid:18)a11,a12,a21,a22,,,b,=,(b1,b2)T,,,x,=,(x1,x2)T,(2),where,the,demands,of,multispecies,and,multiscale,may,lead,to,widely,different,values,of,all,coef-,ficients,and,unknowns.,To,make,these,values,comparable,,which,is,desirable,from,the,numerical,standpoint,,exploit,the,fact,that,it,is,possible,to,multiply,each,each,equation,separately,without,changing,the,value,of,the,solution,x.,Further,,provided,the,coefficients,aij,are,appropriately,ad-,justed,,each,component,xj,of,x,may,be,scaled,separately.,Thus,and,where,(cid:18)a11,a12,a21,a22,(cid:19),(cid:18)x1,x2,(cid:19),=,(cid:19),(cid:18)b1,b2,(cid:18)s1a11t1,s2a21t1,s1a12t2,s2a22t2,(cid:19),(cid:18)˜x1,˜x2,(cid:19),=,(cid:19),(cid:18)˜b1,˜b2,˜x,=,(x1/t1,x2/t2)T,,,˜b,=,(s1b1,s2b2)T,(3),(4),(5),give,the,same,x,,although,likely,by,solving,equations,with,very,different,sized,coefficients,for,very,different,sized,unknowns,˜x.,This,approach,is,written,more,elegantly,in,matrix,notation,as,where,S,and,T,are,diagonal,matrices,SAT,−1T,˜x,=,S,˜b,S,=,(cid:18)s1,0,(cid:19),0,s2,,,T,=,(cid:18)1/t1,0,(cid:19),0,1/t2,(6),(7),(so,that,T,may,be,viewed,as,an,elementary,preconditioner).,Scaling,each,row,of,the,matrix,and,each,unknown,becomes,very,expensive,for,a,nonlinear,and/or,timestepping,problem,where,A,and,b,are,large,and,continually,changing.,Thus,it,is,helpful,on,this,account,only,to,scale,by,equation,subsystem,and,by,field,,where,by,subsystem,is,meant,eg.,momentum,balance,for,species,X,,and,field,is,meant,say,number,density,of,the,coupled,species,Y.,Hence,it,is,convenient,to,regard,x1,as,corresponding,to,an,entire,vector,of,discrete,values,of,the,5,number,density,of,species,X,and,x2,similarly,to,represent,the,density,of,species,Y,,so,that,the,coefficients,aij,for,each,i,and,each,j,separately,represent,a,possibly,very,large,matrix.,Such,a,grouping,is,convenient,for,making,the,equations,dimensionless,,in,that,s1,may,compensate,for,the,dimensions,of,the,equation,and,s2,for,the,dimensions,of,a11x1,+,a12x2,=,b1,a21x1,+,a22x2,=,b2,(8),(9),whereas,tj,may,serve,make,dimensionless,xj,for,each,j,=,1,,2.,There,should,be,a,reduced,,hopefully,avoidable,,need,to,rescale,equations,as,fields,evolve,,and,properly,dimensioned,values,should,be,easily,restored,as,needed.,2.2.3,Fluid,Models,In,the,above,context,,there,is,no,need,,having,evaluated,the,Braginskii,coefficients,to,convert,them,to,non-dimensional,form.,Selected,terms,from,the,energy,evolution,equation,demonstrate,how,the,above,scalings,are,to,implemented,in,practice.,Most,likely,,an,equation,for,energy,evolution,would,be,considered,instead,of,one,divided,by,mi,,and,for,demonstration,purposes,,the,factor,of,B,can,be,omitted,from,System,2-3,[2].,The,following,terms,in,the,energy,evolution,equation,are,representative,∂,∂t,(cid:18),3,2,(cid:19),(N,kTd),+,.,.,.,=,N,∂,∂s∥,κe∥,∂kTe,∂s∥,+,.,.,.,(10),·,L3,The,overall,scaling,of,this,equation,to,render,its,terms,dimensionless,is,evidently,ts,s,Nref,kTs,the,unknowns,to,be,solved,for,,Td,and,N,,,have,respectively,dimensions/scalings,Ts,and,L3,s,Nref,time,dependent,term,in,Equation,(10),involves,both,unknowns,,hence,needs,re-expressing,as,and,.,The,∂,∂t,(cid:18),3,2,(cid:19),(N,kTd),=,3,2,N,k,∂Td,∂t,+,3,2,kTd,∂N,∂t,(11),2,N,k,and,3,so,that,3,2,kTd,are,equivalents,of,coefficients,aT,and,aN,of,matrix,A.,The,partial,deriva-,tive,∂/∂t,leads,to,discrete,operators,˜∆t/∆t,where,˜∆t,is,a,(dimensionless),difference,operator,so,the,scaling,factorisations,have,and,3,2,N,k,∆t,3,2,kTd,∆t,˜∆t(T,),→,L3,sts,Nref,kTs,·,3,2,N,k,∆t,·,Ts,˜∆t,(cid:19),(cid:18),T,Ts,˜∆t(N,),→,L3,sts,Nref,kTs,·,3,2,kTd,∆t,·,Nref,L3,s,˜∆t(n),(12),(13),where,˜∆t,operates,on,unknowns,of,dimensionless,temperature,T,/Ts,and,density,n,=,L3,respectively.,The,partial,derivative,∂2/∂s2,(dimensionless),difference,operator,so,the,scaling,factorisation,for,aκ,is,∥,leads,to,a,discrete,operator,˜∆2,sN/Nref,s/(∆s∥)2,where,˜∆s,is,a,L3,sts,Nref,kTs,·,N,kκe∥,(∆s∥)2,·,Ts,˜∆2,s,(cid:19),(cid:18),T,Ts,6,(14),where,˜∆2,s,operates,on,the,unknown,of,dimensionless,temperature,T,/Ts.,The,algorithm,above,producing,the,scaled,matrix,coefficients,involves,multiplying,3,factors,to-,gether.,If,T,is,measured,in,eV,,,then,k,=,|e|.,Suppose,Ls,=,10,m,and,ts,=,Ls/105,ms−1,=,10−4,s,,N,=,1016,m−3,,T,=,10,eV,,,and,approximating,|e|,→,10−19,,3/2,→,1,,taking,∆t,=,0.1ts,and,∆s∥,=,0.1Ls,,then,numerically,L3,sts,Nref,kTs,kTd,∆t,·,L3,sts,Nref,kTs,L3,sts,Nref,kTs,·,·,·,Ts,→,103−4−18+19−2,·,10−19+16+5,·,102,N,k,∆t,Nref,L3,s,N,kκe∥,(∆s∥)2,·,Ts,→,103−4−18+19−2,·,1016−19κe∥,·,102,→,103−4−18+19−2,·,10−19+1+5,·,1018−3,·,(15),(16),(17),so,that,each,factor,is,not,unmanageably,different,from,unity,when,using,single,precision,arithmetic.,(The,largest,numerical,value,is,1015,if,κe∥,is,estimated,as,109,m2s−1.),Ultimately,,the,scaled,values,of,the,coefficients,aT,,,aN,and,aκ,are,easily,shown,to,be,equal,to,respectively,˜aT,=,tsn,∆t,,,˜aN,=,ts,∆t,Td,Ts,,,˜aκ,=,which,are,all,evidently,of,order,unity.,tsκe∥,(∆s∥)2,n,(18),If,it,is,necessary,to,iterate,to,a,solution,for,Te,or,Ti,,then,the,coefficient,should,be,evaluated,for,T,=,Ts.,For,example,κe∥,,which,varies,proportional,to,T,5/2,,should,be,evaluated,as,κe∥(T,),=,κe∥(Ts),(cid:19),5,2,(cid:18),T,Ts,(19),and,then,any,further,work,,for,example,linearisation,,may,proceed,using,the,dimensionless,group,T,/Ts.,2.2.4,Scalings,for,Particle,Models,The,above,considerations,also,apply,in,the,case,of,particle,evolution.,Given,the,choice,of,units,,if,the,equations,of,motion,are,posed,in,terms,of,superparticles,of,weight,wp,,wpmα,dv,dt,=,wpqα(E,+,v,×,B),(20),It,is,convenient,to,introduce,a,reference,particle,weight,wref,,,cf.,Nref,,,to,make,scalings,more,manageable,numerically.,The,overall,equation,scaling,to,render,the,terms,dimensionless,is,then,and,the,unknown,to,be,solved,for,,namely,v,,has,dimensions,Ls,evidently,.,ts,Consider,the,acceleration,in,Equation,(20),,which,leads,to,discrete,coefficient,terms,of,the,form,mα,∆t,,,then,the,scaling,factorisation,is,t2,s,wref,Lsmu,ts,wref,mu,·,ts,Ls,=,wpmα,∆t,v,→,t2,s,Lswref,mu,·,wpmα,∆t,·,Ls,ts,(21),7,to,be,applied,to,dimensionless,velocity,v,→,ts,Ls,v,(22),As,explained,above,,the,numerical,values,of,the,3,factors,should,be,estimated.,For,definiteness,,suppose,|v|,=,105,ms−1,,B,=,1,T,,then,the,gyro-radius,is,approx.,10−3,m,,so,take,Ls,=,10−3,m,as,a,typical,lengthscale,,then,ts,=,10−3−5,=,10−8,s,,and,a,timestep,size,∆t,of,order,10−9,is,indicated.,Approximate,the,amu→,10−27,kg,,and,if,A,=,Z,=,1,and,suppose,wp,=,wref,=,1010,,the,resulting,factors,follow,as,t2,s,Lswref,mu,·,wpmα,∆t,·,Ls,ts,→,10−16+3−10+27,·,1010−27+9,·,10−3+8,(23),so,that,each,factor,is,not,unmanageably,different,from,unity,when,using,single,precision,arithmetic.,(The,smallest,numerical,value,is,10−8.),The,scaled,coefficient,is,easily,shown,to,simplify,as,˜a,=,t2,s,wref,muLs,·,wpmα,∆t,·,Ls,ts,=,tsA,∆t,(24),if,wref,=,wp,,which,is,clearly,of,order,unity.,Evidently,˜a,has,a,numerical,value,of,order,the,number,of,timesteps,chosen,to,discretise,a,gyro-period,,typically,a,number,of,order,unity.,The,magnetic,field,term,may,similarly,be,treated,wp|qαv,×,B|,→,t2,s,Lswref,mu,·,wp|qα|B,·,Ls,ts,whence,Ls,ts,ie.,the,smallest,numerical,factor,is,10−9,,and,t2,s,Lswref,mu,·,wp|qα|B,·,→,10−16+3−10+27,·,1010−19,·,105,˜a,=,tsZKM,|B|,≈,10−8+8,=,100,=,1,(25),(26),(27),2.3,Text,processing,tools,The,tools,are,only,available,on,the,private,CCFE,gitlab,site.,Their,wider,publication,awaits,user,testing,and,feedback.,For,UKAEA,staff,with,access,to,CCFE,gitlab,,the,tools,are,to,be,found,either,in,directory,tex,or,in,its,subdirectory,importools.,They,are,largely,based,on,Linux,sed,or,vim,editors.,The,more,mature,tools,appear,as,bash,shell,scripts,and,are,as,follows:,1.,In,directory,tex,•,skeltorp1.bash,produces,a,report,directory,skeleton,from,arguments,<milestone>,<FMS,report,number>,<title,string>,•,pub.bash,is,transmitted,by,skeltorp1.bash,into,subdirectories,as,pubN,.bash,script,that,produces,final,versions,of,pbN,as,a,PDF,file,with,name,appropriate,for,web,posting.,8,2.,In,subdirectory,tex/importools,•,fromed.bash,converts,text,file,produced,by,Windows,editor,to,Linux-friendly,ASCII,,en-,forcing,many,of,the,NEPTUNE,text,conventions,•,getsymb.bash,looks,for,definitions,of,mathematics,variables,and,produces,file,allp.var.srt,for,user,inspection,and,intervantion,•,md2tex.bash,converts,Markdown,produced,by,PANDOC,to,LATEX,•,sear.bash,finds,all,the,files,in,the,list,in,file,tex.lis,which,contain,a,specified,string,•,toctortf.bash,produces,WYSIWYG,version,of,a,report’s,table,of,contents,from,LATEX,.toc,file,output,,useful,for,.rtf,conversion,Subdirectory,tex/importools,also,contains,.ed,files,containing,vim,edit,commands,,many,of,which,are,invoked,by,the,bash,scripts.,Two,other,of,these,files,help,produce,.bib,files.,If,XXXbib.inp,contains,the,essential,bibliographic,information,,then,cp,XXXbib.inp,XXX.bib,vim,XXX.bib,-S,YYYbib.ed,will,over-write,XXX.bib,with,Bibtex-readable,references.,The,string,“YYY”,may,be,either,•,reportplus,when,XXX.bib,will,contain,bibliographic,entries,corresponding,to,a,grantee’s,report,(see,file,examplebib.inp,for,an,example,of,the,required,input,format),•,ukaeareport,when,XXX.bib,will,contain,bibliographic,entries,corresponding,to,a,UKAEA,report,(see,file,ukaeareportbib.inp,for,an,example,of,the,required,input,format),Other,.ed,files,are,briefly,described,in,the,README,,beware,that,the,input,is,invariably,overwritten,so,should,be,copied,before,editing.,2.4,Improvements,to,the,Developers’,Website,Consistent,with,the,conclusions,of,the,grantees’,report,[3],about,the,inadequacy,of,ReadThe-,Docs,,work,has,continued,to,produce,a,website,capable,of,aiding,and,encouraging,development,of,NEPTUNE,software,by,a,community,spread,across,UKAEA,and,UKRI.,During,the,year,,the,developers’,website,has,been,subject,to,a,re-examination,as,to,how,best,the,pages,might,be,produced,from,LATEX,source,,resulting,in,the,replacement,of,LATEX2HTML,by,LWARP,,since,the,latter,package,appears,to,be,maintained,far,more,actively,,supports,cascading,style,sheets,(css),so,websites,look,more,attractive,,enables,download,of,LATEX,source,of,mathematics,via,MathJax,,and,offers,an,option,to,produce,a,single,PDF,document,of,the,entire,website.,The,webpages,have,now,been,made,publicly,available,on,GitHub,[4],,and,can,be,viewed,on,mobile,phones,and,tablets,as,well,as,larger,screens.,Webpage,source,material,has,been,separated,from,the,private,CCFE,gitlab,repository,and,has,,like,the,webpages,,been,posted,on,GitHub,,specifically,as,one,of,the,repos,forming,the,ExCALIBUR-,NEPTUNE,organisation,[5].,A,feature,has,been,added,whereby,a,named,git,branch,can,be,used,to,produce,a,correspondingly,named,set,of,developers’,webpages,,see,Section,2.4.1.,Biblio-,graphic,.bib,files,,and,files,describing,mathematical,notation,(symbols),and,listing,acronyms,have,9,been,frequently,updated,throughout,the,year,,and,need,to,be,kept,synchronised,as,explained,in,Section,2.4.2.,Many,minor,improvements,have,been,made,to,the,site,,notably,the,mathematical,notation,used,in,the,developers’,webpages,has,been,changed,to,conform,with,the,list,of,symbols.,2.4.1,Website,branches,The,instructions,to,produce,a,separate,set,of,developers’,webpages,corresponding,to,a,particular,branch,of,repo,[5],are,1.,Create,a,branch,which,is,copy,of,main.,2.,Commit,changes,to,that,branch.,3.,This,branch,of,the,developer,website,should,become,available,online,within,about,5,minutes,assuming,a,successful,build,process,by,the,Github,actions,workflow,‘Deploy,static,content,to,Pages’,,,by,accessing,the,repo,branch,described,as,Github-Pages.,4.,After,testing,,the,new,branch,should,be,deployed,by,merging,the,branch,with,the,main,branch,2.4.2,Synchronising,key,webpages,Key,pages,(as,.tex,files),common,to,the,CCFE,gitlab,ExCALIBUR,repo,may,be,checked,for,syn-,chrony,using,the,script,checksync.bash,(currently,only,available,on,branch,wa,of,the,developers’,website).,Successful,working,of,the,script,relies,on,the,simultaneous,presence,in,the,user’s,disc-,space,of,the,CCFE,gitlab,repo,as,directory,excalibur-wa,and,the,webpages’,source,repo,as,directory,devweb.,The,script,pre-processes,the,gitlab,files,to,conform,to,the,format,required,for,web-posting,then,performs,a,simple,comparison.,It,is,up,to,the,user,to,ensure,that,any,differences,are,reconciled,,eg.,by,use,of,Linux,meld.,10,3,Task,Work,by,Grantees,Each,section,summarises,reports,received,from,a,grantee,,one,report,per,subsection,,with,a,link,in,the,title,to,the,location,of,the,accepted,report.,Accepted,reports,are,held,on,the,Documents,repository,of,the,ExCALIBUR-NEPTUNE,organisation,on,GitHub,,to,which,belong,other,repos,containing,software,developed,under,project,NEPTUNE.,Note,that,some,of,these,repos,may,be,access-controlled,,please,email,neptune@ukaea.uk,if,difficulties,are,encountered,in,seeing,the,material.,3.1,Reports,received,under,Grant,T/NA083/20,,PO,2047356,The,York,grantee,produced,the,following,reports,in,association,with,KCL,•,2047356-TN-06-1,-,Task,1.4,Preconditioning,2D,elliptic,solvers,[6],•,2047356-TN-07-1,-,Task,2.2,BOUT++,1D,fluid,solver,with,realistic,boundary,conditions,:,implementation,[7],•,2047356-TN-08-1,-,Task,2.4,Non-intrusive,UQ,with,BOUT++,1D,fluid,solver,[8],•,2047356-TN-09-1,-,Task,4.1,Test,cases,for,1D,multispecies,plasma,model,[9],•,2047356-TN-10-1,-,Task,0.2,Development,of,a,performance,testing,and,aggregation,tool,for,NEPTUNE,proxy-apps,[10],•,2047356-TN-11-1,-,Task,3,Considerations,for,1D1V,models,Report,covering,tasks,3.1,,3.2,and,3.3,[11],•,2047356-TN-12-2,-,(D1.3):,Elliptic,solvers,within,Nektar++,[12],•,2047356-TN-13,-,Spatially,2-D,plasma,model,incorporating,velocity,space,effeects,[13],•,2047356-TN-14-1,-,DSL,and,code,design,pattern,for,NEPTUNE,[14],•,2047356-TN-15-2,-,Task,2.5,Performance,considerations,for,non-intrusive,uncertainty,quan-,tification,and,the,influence,of,the,boundary,[15],•,2047356-TN-16-1,-,Task,0.3,Community,and,environment,[3],•,2047356-TN-17,-,Progress,on,implementation,of,system,2-6,equations,[16],3.1.1,Report,2047356-TN-06,[6],This,report,simply,indicates,that,the,work,was,done,in,conjunction,with,STFC,and,appears,in,STFC’s,reports,refs,[17,,18,,19,,20].,11,3.1.2,Report,2047356-TN-07,[7],This,report,summarises,the,implementation,of,system,2-3,from,the,Equations,document,[2],using,the,1-D,model,SD1D,including,fluid,neutrals,and,realistic,boundary,conditions.,The,model,and,approach,is,described,in,the,SD1D,manual,which,forms,an,appendix,,providing,a,reference,imple-,mentation,of,a,1-D,fluid,solver,with,realistic,boundary,conditions,for,use,in,later,work,,see,ref,[8].,The,report,also,discusses,test,cases,selected,in,earlier,work,,for,which,carefully,commented,input,files,are,included,with,the,source,code,in,the,SD1D,repository,[21].,The,timing,results,presented,demonstrate,a,factor,10,or,more,reduction,in,time,to,solution,when,using,preconditioning,tech-,niques.,3.1.3,Report,2047356-TN-08,[8],The,focus,of,UQ,in,NEPTUNE,has,shifted,in,favour,of,non-intrusive,uncertainty,quantification,(UQ),since,the,first,date,of,writing,the,equations,document,[22],,with,which,shift,this,report,duly,conforms.,It,summarises,experience,gained,in,the,application,of,non-intrusive,UQ,to,a,sample,test,case,from,the,SD1D,website,[21],described,in,ref,[23].,It,showcases,the,EASYVVUQ,package,to,facilitate,the,construction,of,UQ,workflows.,The,work,provides,a,testing,ground,for,the,practical,application,of,UQ,to,more,expensive,realistic,plasma,simulations.,Useful,,clear,example,scripts,are,included,,as,are,links,to,relevant,repositories,detailing,the,work,of,the,BOUT++,team,at,the,recent,VECMA,hackathons,[24].,This,report,opens,with,an,executive,summary,stating,the,scope:,non-intrusive,UQ,using,EASYVVUQ,for,a,one-dimensional,SD1D,plasma,simulation,with,sheath,boundary,conditions.,An,introductory,section,contains,brief,motivation,for,sensitivity,analysis,and,UQ,,then,introduces,the,EASYVVUQ,toolkit,and,the,major,object,types,(Encoder,,Decoder,,Campaign,,Sampler).,Section,3,details,the,specific,use,case,of,SD1D,within,the,EASYVVUQ,framework.,Section,4,presents,details,of,the,SD1D,test,problems,and,how,the,EASYVVUQ,framework,was,applied,,with,simulations,done,on,a,single,node,of,Archer2.,Useful,cautionary,notes,are,included,eg.,inefficient,resource,usage,with,local,execute,if,some,parts,of,sampled,parameter,space,result,in,outlying,long-duration,simula-,tions.,Example,Python,scripts,for,specifying,a,UQ,/,sensitivity,analysis,campaign,using,polynomial,chaos,(PCE),on,a,simple,neutral-free,problem,are,exhibited,,which,should,be,very,useful,for,ped-,agogical,purposes,(as,should,be,the,scripts,in,Appendix,A,for,the,work,outlined,in,Section,5).,Section,5,presents,UQ,for,a,more,realistic,plasma,model,problem,including,neutrals.,Subsection,5.1,includes,discussion,of,a,PCE-sampler,approach,and,Subsection,5.2,a,stochastic,collocation,approach,which,has,the,advantage,of,being,able,to,sample,adaptively,based,on,which,input,has,the,most,influence,on,the,uncertainty,,and,also,the,ability,to,terminate,the,campaign,based,on,convergence,of,Sobol,indices,(it,is,mentioned,that,this,approach,may,not,always,be,appropriate),,though,at,the,cost,of,not,being,easily,parallelizable.,Section,6,concludes,the,report,,noting,that,extensions,to,the,examples,provided,are,necessary,in,order,to,manage,campaigns,involving,more,demanding,HPC,simulations,but,that,suitable,machinery,is,becoming,increasingly,available,among,the,outputs,of,the,VECMA,project,,notably,EASYVVUQ.,Overall,,this,report,provides,an,instruc-,tive,comparison,of,some,of,the,merits,and,shortcomings,of,the,PCE,and,stochastic,collocation,approaches,to,UQ,embodied,in,EASYVVUQ.,12,3.1.4,Report,2047356-TN-09,[9],This,very,brief,report,is,primarily,a,link,to,places,where,work,performed,on,Hermes-3,may,be,found,,namely,its,manual,(readthedocs),pages,[25],,and,the,Hermes-3,[26],and,SD1D,[21],repos,on,GitHub.,The,work,has,extended,the,1-D,single,plasma,model,to,treat,(1),separate,ion,and,elec-,tron,temperatures,,(2),neutral,gas,models,including,different,excited,states,of,Hydrogen,,and,(3),impurity,species,,including,both,low,atomic,number,,for,which,each,charge,state,forms,a,separate,species,,and,high,atomic,number,where,different,states,are,bundled,together,to,be,represented,by,a,single,fluid.,3.1.5,Report,2047356-TN-10,[10],This,report,outlines,the,requirements,for,,and,subsequent,design,of,,a,software,framework,for,the,automated,testing,of,performance,including,performance,regression,analysis,,of,software,un-,der,git,management,,such,as,the,proxyapps,developed,as,part,of,the,ExCALIBUR,NEPTUNE,project.,The,requirements,are,(1),to,allow,easy,tracking,of,performance,improvements,during,the,development,of,the,proxyapps,,(2),to,identify,if,and,when,any,code,changes,negatively,af-,fect,the,performance,of,such,applications,across,a,range,of,hardware,and,compiler,combinations.,The,design,for,the,framework,,called,‘RoundTable’,is,based,on,GitHub,runners,and,uses,TOML.,RoundTable,will,allow,acceptance,tests,and,performance,benchmarks,to,be,run,automatically,us-,ing,GitHub,Actions,on,self-hosted,runners,at,a,range,of,local,and,HPC,facilities,such,as,Bede,or,Archer2.,The,results,from,these,runs,will,then,be,collated,and,displayed,in,a,web,dashboard,along,with,results,from,other,applications,,sites,and,code,versions.,3.1.6,Report,2047356-TN-11,[11],This,report,documents,preliminary,work,to,develop,a,1d1v,model,,ie.,a,1-D,plasma,model,with,1-D,velocity,space,effects,,to,the,specifications,of,System,2-4,in,ref,[2].,The,1d1v,target,model,consists,of,the,Vlasov-Ampere,system,in,a,doubly,periodic,domain,,although,the,use,of,wall,boundary,conditions,is,important,for,the,edge,,periodicity,is,important,for,testing.,It,is,viewed,as,a,stepping-,stone,exercise,from,which,valuable,lessons,should,be,learnt,,in,the,move,towards,more,realistic,,production,codes.,A,sequence,of,nine,test,cases,is,defined,in,Section,3,,beginning,with,elementary,tests,of,ad-,vection,and,advancing,to,sound,wave,models,for,coupled,species,in,a,self-consistent,electric,field.,Simple,Python,implementations,demonstrate,key,features,of,these,test,problems.,Section,4,compares,different,representations,of,phase-space,(ie.,here,the,velocity,coordinate),using,finite,difference,(FD),,finite,volume,(FV),and,finite,element,(FE),methods,,supporting,the,choice,of,the,spectral/hp,element,for,project,NEPTUNE.,Section,4.3,provides,advice,on,the,use,of,Python,to,prototype,software,,and,Section,4.4,briefly,discusses,schemes,used,in,widely,available,fusion,software,(‘community,codes’),such,as,GS2,and,STELLA,,mentioning,the,latter’s,use,of,a,non-,interpolating,semi-Lagrangian,scheme.,Section,5,discusses,time,integration,schemes,,indicating,how,the,wide,range,of,schemes,implemented,in,the,PETSC,package,might,be,made,available,to,NEPTUNE,,as,well,as,the,other,popular,explicit,SSP-RK,schemes,and,implicit,stiff,CVODE,code,13,from,the,SUNDIALS,package.,Adaptive,and/or,local,timestepping,is,briefly,discussed,,as,is,the,need,for,preconditioning,matrices,arising,in,implicit,time,advance,schemes.,3.1.7,Report,2047356-TN-12,[12],This,report,focuses,on,the,performance,of,the,elliptic,solver,within,NEKTAR++,and,its,scaling,in,the,limit,of,large,processor,counts.,Building,on,the,report,[27],,which,introduces,a,number,of,sample,test,cases,for,the,elliptic,solver,,the,work,tests,both,the,correctness,and,the,functionality,of,the,elliptic,solver.,This,solver,has,been,used,within,the,anisotropic,diffusion,solver,developed,under,project,NEPTUNE,,proxyapp,located,in,the,NEPTUNE,GitHub,repository.,The,report,discusses,the,formulation,of,an,elliptic,solver,within,a,high-order,finite,element,setting,,the,turning,of,this,formulation,into,performant,software,,and,the,preconditioning,strategies,that,may,be,employed,to,solve,the,resulting,systems.,The,report,opens,with,an,executive,summary,indicating,its,scope,:,the,performance,and,scaling,of,the,NEKTAR++,elliptic,solver,,with,reference,to,instances,of,elliptic,solves,in,existing,proxyapps,(viz.,NEKTAR-DRIFTWAVE,and,NEKTAR-DIFFUSION),and,the,main,NEKTAR++,code,eg.,the,incom-,pressible,Navier-Stokes,solver.,Section,2,indicates,that,the,report,is,limited,to,simple,slab-type,geometries,as,treated,in,the,existing,NEKTAR++-based,proxyapps,and,related,CFD,problems.,Insights,into,specific,issues,with,elliptic,spectral/hp,are,indicated,,eg.,matrix-free,operations,for,discretized,operators,and,the,lack,of,effectiveness,of,multigrid,preconditioners,at,higher-order.,Section,3,presents,the,formulation,of,the,matrix,equations,in,the,continuous,Galerkin,approach.,Section,4,presents,specific,numerical,approaches,for,iterative,solvers,,where,the,main,work,com-,prises,repeated,computations,of,the,action,of,the,matrix,on,the,solution,vector.,Specifically,,meth-,ods,for,addressing,the,O(p)3,scaling,of,the,local,element,matrix,size,(in,three,dimensions),are,presented,,eg.,static,condensation,,which,decouples,element,boundary,and,interior,degrees,of,freedom,and,is,appropriate,for,two,of,the,major,basis,sets,in,NEKTAR++,,viz.,the,modified,ba-,sis,of,Karniadakis,and,Sherwin,,and,the,Lagrange,interpolant,basis.,Subsection,4.4,covers,the,matrix-free,operator,implementation,likely,to,be,critical,for,Exascale,exploitation,,where,the,key,idea,is,to,remove,element-specific,information,from,matrices,apart,from,geometric,scale,factors,,and,such,that,the,action,of,the,resulting,matrices,is,computed,by,local,matrix-free,operations.,The,final,Subsection,4.5,gives,a,brief,overview,of,the,global,linear,system,solvers,available,in,NEK-,TAR++.,Section,5,concerns,preconditioners,,with,a,brief,overview,of,the,basic,types,,plus,a,longer,description,of,the,“Low,energy”,preconditioner,of,Sherwin,and,Casarin.,Section,6,includes,a,brief,performance,study,of,the,IterativeStaticCond,solver,for,a,massively-parallel,problem,where,it,is,shown,that,(1),the,implementation,of,the,Laplace,solve,with,a,Jacobi,preconditioner,scales,fairly,well,up,to,131,000,cores,on,the,Argonne,Mira,supercomputer,(results,published,previously,[28]),,and,(2),a,more,realistic,incompressible,Navier-Stokes,problem,involving,an,elliptic,solve,for,pres-,sure,,with,a,low-energy,block,preconditioner,,scales,strongly,to,50,000,cores,on,ARCHER2.,Lastly,,Section,7,outlines,ongoing,development,work,on,the,geometrically,informed,algebraic,multigrid,approach,(GIAMG),which,should,be,a,very,effective,preconditioning,technique,as,it,has,phases,wherein,order,p,and,mesh-spacing,h,are,separately,varied.,The,software,developed,is,to,be,found,as,follows:,•,The,solution,of,electrostatic,potential,for,the,Hasegawa-Wakatani,proxyapp:,https://github.,14,com/ExCALIBUR-NEPTUNE/nektar-driftwave,•,The,solution,of,the,Laplacian,term,as,part,of,the,anisotropic,heat,proxyapp:,https://,github.com/ExCALIBUR-NEPTUNE/nektar-diffusion,•,The,main,NEKTAR++,code,base:,https://gitlab.nektar.info/nektar/nektar,3.1.8,Report,2047356-TN-13,[29],This,report,briefly,outlines,potential,routes,for,implementation,of,the,system,2-6,equations,in,[2],within,the,NEKTAR++,framework,,focussing,on,the,steps,required,to,implement,such,a,system,and,highlighting,where,improvements,should,be,made,within,NEKTAR++,to,enable,future,imple-,mentation.,It,is,suggested,that,testing,should,make,use,of,the,Method,of,Manufactured,Solutions,(MMS),,and,the,value,is,noted,of,the,capability,of,NEKTAR++,to,combine,classical,and,discontin-,uous,Galerkin,schemes,in,one,solver.,3.1.9,Report,2047356-TN-14,[30],This,report,gives,a,brief,overview,of,the,Domain,Specific,Languages,(DSLs),provided,by,BOUT++,[31],and,SD1D,[21],,which,are,both,at,a,high-level,level,intended,for,user,specification,of,a,physical,model,including,parameters.,The,report,treats,the,point,that,a,NEPTUNE,DSL,will,have,to,be,suitable,for,describing,multiple,species,and,their,interactions,in,a,manner,that,is,suitable,for,a,large,number,of,such,species,and,interactions.,Specifically,this,report,covers:,1.,Approaches,to,code,design,in,BOUT++,that,allow,for,single-species,plasma,simulations,to,be,extended,to,multiple,species.,2.,How,these,multi-species,models,might,best,be,expressed,in,a,high-level,DSL.,3.,An,overview,of,the,DSLs,that,exist,already,in,the,BOUT++,and,NEKTAR++,frameworks,and,how,a,common,DSL,could,be,developed,from,the,Unified,Form,Language,(UFL),to,drive,both,BOUT++,and,NEKTAR++,solvers.,3.1.10,Report,2047356-TN-15,[15],This,report,considers,performance,considerations,relevant,to,the,use,of,non-intrusive,uncertainty,It,is,demonstrated,that,for,quantification,(UQ),through,tools,such,as,the,EASYVVUQ,package.,variations,in,the,boundary,conditions,applied,to,the,physical,model,,depending,on,their,precise,nature,,convergence,of,the,measures,of,sensitivity,(first,order,Sobol,indices),needs,between,2nd,and,5th,order,modes,to,be,used,in,the,Polynomial,Chaos,Expansion,(PCE).,Section,3,contains,the,discussion,of,non-intrusive,UQ,,notably,describing,the,benefits,of,non-,intrusive,UQ,,when,it,comes,to,minimising,the,number,of,simulations,and,making,effective,use,of,workflow,management,tools,,both,of,which,are,often,important,routes,to,improving,the,time,to,solution,and,freeing,up,researcher,time.,The,disadvantage,of,adaptive,(non-intrusive),UQ,of,15,making,parallelisation,harder,has,to,be,traded,against,the,likelihood,that,the,required,number,of,simulations,will,be,smaller.,Section,4,describes,the,model,system,,which,one,of,the,most,complicated,described,in,[8],,although,omitting,treatment,of,electric,field,,which,anyway,cannot,apparently,be,treated,self-consistently,using,SD1D,[21].,The,uncertainties,allowed,in,the,boundary,conditions,treated,are,variation,of:,(1),upstream,sources,,(2),neutral,recycling,fraction,for,three,distinct,boundary,conditions,on,the,plasma,,namely,(A),free,at,constant,gradient,,(B),zero,gradient,and,(C),constant,flux.,The,last,case,(C),gives,rise,to,the,apparent,need,for,high,order,PCE,for,accurate,Sobol,index,calculation.,Although,all,three,cases,suffer,from,the,problem,of,unphysical,,negative,values,of,density,,there,is,the,suggestion,that,this,could,be,confounding,the,results,in,particular,in,case,(C),and,thus,that,the,logarithm,of,density,should,be,analysed,in,future,UQ,work.,3.1.11,Report,2047356-TN-16,[3],This,report,discusses,ExCALIBUR-NEPTUNE,community,activities,in,the,October,2021,to,Oc-,tober,2022,period,and,the,state,of,the,community,environment,at,the,date,of,this,report.,Section,2,describes,the,numerous,workshops,and,events,hosted,by,Neptune,project,partners,during,this,period,,and,their,value,in,linking,members,from,the,different,groups.,Section,3,describes,how,the,digital,tools,set,up,at,the,start,of,the,project,to,support,community,interactions,are,still,in,use,,and,suggests,how,to,modify,the,digital,environment,both,so,as,to,increase,engagement,and,to,assist,with,bringing,new,members,up,to,speed.,It,is,pointed,out,that,this,is,likely,to,become,increasingly,important,as,the,project,moves,forwards,and,further,stakeholders,emerge.,Most,of,the,suggestions,require,dedicated,software,admin,support.,The,limitations,of,ReadTheDocs,to,relatively,small,projects,reported,,helps,makes,the,case,for,the,production,in,fact,taking,place,,of,a,developers’,website,that,can,support,the,considerable,complexity,project,NEPTUNE.,3.1.12,Report,2047356-TN-17,[16],This,brief,report,outlines,the,initial,efforts,on,implementing,the,equation,set,System,2-6,[2],within,the,NEKTAR++,framework.,The,set,has,been,changed,from,the,model,adopted,by,the,European,Boundary,Code,project,,equations,for,which,were,described,in,Release,1.00,of,the,Equations,document,[22],,to,the,similar,equations,capable,of,solution,by,the,Hermes-3,software,to,be,found,in,Release,1.26.,The,report,describes,implementation,and,solution,of,a,simplified,set,of,equations,possessing,properties,shared,by,classical,computational,fluid,dynamical,models.,It,indicates,a,future,development,pathway,for,system,2-6,,employing,the,sequence,of,models,of,increasing,complexity,described,on,the,Hermes-3,website,[25].,3.2,Reports,received,under,Grant,T/AW086/21,,PO,2060049,The,STFC,grantee,produced,the,following,reports,•,2060049-TN-02-5,-,Techniques,and,software,relevant,to,the,coupling,of,continuum,(fluid),and,particle,models,of,plasma,for,NEPTUNE,[32],16,•,2060049-TN-03,-,Time,Stepping,Techniques,and,Preconditioning,for,Hyperbolic,and,Anisotropic,Elliptic,Problems:,Numerical,Results,[33],3.2.1,Report,2060049-TN-02,[32],Report,2060049–TN–02,reviews,a,number,of,possible,techniques,and,frameworks,for,coupling,fluid,and,particle,codes,in,project,NEPTUNE.,It,goes,on,to,present,results,from,a,proxyapp,which,demonstrates,the,use,of,the,CWIPI,coupling,library.,In,Section,2,,three,forms,of,coupling,are,discussed,for,fluid-particle,systems:,Spatial,Hybridiza-,tion,(SpH),,the,Micro-Macro,Hybrid,(MMH),approach,and,the,Kinetic,Diffusion,Monte,Carlo,Method,(KDMCM).,SpH,and,MMH,are,both,identified,as,reasonable,strategies,to,use,in,NEPTUNE,,but,KDMCM,is,disfavoured,on,the,grounds,that,it,has,only,been,demonstrated,on,a,few,idealised,physics,prob-,lems,and,is,sufficiently,novel,in,implementation,that,its,use,would,require,significant,changes,to,the,way,NEPTUNE,code,is,being,developed.,Section,3,describes,six,different,open,source,coupling,libaries:,MUI,,OPENPALM,,PRECICE,,PLE,,CWIPI,and,MUSCLE3.,In,each,case,,a,high-level,description,of,the,implementation,is,provided,and,a,number,of,factors,are,discussed,that,could,affect,the,framework’s,suitability,for,use,in,NEPTUNE.,These,include,the,form,of,data,which,may,be,exchanged,,the,types,of,API,that,are,provided,,example,use,cases,in,which,the,coupling,has,already,been,applied,and,the,quality,of,the,documentation.,Of,the,different,frameworks,,MUI,and,CWIPI,are,deemed,most,suitable,,in,that,they,are,minimally,invasive,,provide,good,quality,APIs,and,could,be,used,to,couple,fluid,and,particle,codes,with,no,obvious,negative,impact,on,scalability.,CWIPI,is,slightly,to,be,preferred,as,it,has,already,been,integrated,into,NEKTAR++,to,some,degree.,The,appendix,is,used,to,present,early,results,from,the,coupling,of,a,“particle-style,code”,to,a,NEKTAR++,fluid,solver,using,the,CWIPI,library.,The,former,is,a,simple,executable,that,can,send,and,receive,data,across,the,coupling,interface,,serving,as,a,placeholder,for,an,actual,particle,solver,(eg.,NESO-PARTICLES).,Two,alternative,NEKTAR++,executables,are,discussed,,one,configured,to,solve,an,anisotropic,diffusion,problem,,the,other,a,convection,problem.,In,the,first,two,test,cases,,different,diffusion,coefficients,,first,spatially,uniform,,then,spatially,vary-,ing,,are,sent,to,the,NEKTAR++,solver,and,the,resulting,velocity,fields,are,plotted,in,order,to,check,the,results.,The,third,test,case,demonstrates,both,time-variable,communication,and,two-way,coupling.,The,Navier-Stokes,solver,sends,the,temperature,field,(at,each,timestep),to,the,particle,code,,which,sends,back,a,corresponding,source,term,that,is,added,to,the,fluid,equations.,The,error,associated,with,translating,between,particle,and,continuum,data,can,be,examined,by,comparing,this,to,a,non-coupled,version,of,the,problem,,wherein,the,fluid,solver,computes,the,source,term,directly.,A,number,of,different,mesh,resolutions,were,used,for,the,particle,code,,between,1.25,and,32,times,coarser,than,that,used,by,the,fluid,solver.,The,authors,find,that,the,maximum,error,in,the,temperature,field,does,not,start,to,increase,significantly,until,the,coarseness,ratio,exceeds,a,factor,of,approx.,6,,suggesting,that,some,level,of,mismatch,between,the,fluid,and,particle,resolutions,may,be,tolerated,in,exchange,for,reduced,runtimes.,17,The,authors,conclude,that,the,SpH,coupling,approach,would,be,the,most,natural,fit,for,the,current,NEPTUNEproject,,which,is,reasonable,from,the,stand-point,of,the,software,developer,,although,they,fail,to,consider,the,efficiency,of,the,resulting,code.,The,noise,associated,with,particles,has,to,be,tolerated,when,it,is,too,expensive,to,model,6-D,with,a,continuum,,or,the,physical,processes,involved,do,not,have,a,simple,collective,effect,,but,the,first,justification,almost,never,applies,and,the,second,rarely,applies,,in,3-D.,Thus,NEPTUNE,will,continue,to,use,high,order,elements,to,calculate,3-D,fluid,effects,,and,indeed,software,developed,by,the,fusion,community,increasingly,sees,use,of,continuum,models,for,2d3v,and,3d3v,work.,3.2.2,Report,2060049-TN-03,[33],Report,2060049-TN-03,seeks,answers,to,open,questions,regarding,numerical,solvers,applied,to,the,dynamics,of,magnetised,plasmas,,specifically,by,solving,advection-diffusion,equations,with,a,combination,of,anisotropic,grids,with,highly,anisotropic,flows,and,diffusion,tensors.,The,topics,of,preconditioners,and,time,integrators,are,discussed,in,this,context,,with,comments,on,error,,scalability,and,performance.,The,first,section,discusses,the,machine,on,which,the,tests,are,run,,followed,by,a,second,section,that,describes,how,anisotropy,may,be,incorporated,through,the,equations,themselves,or,via,Ker-,shaw,grids,,which,are,topologically,equivalent,to,regular,(ie.,finite-difference,type),meshes,,but,made,up,of,greatly,distorted,quadrilaterals.,Section,3,concerns,preconditioners,and,begins,with,an,overview,of,algebraic,multigrid,(AMG),,p-,multigrid,,hp-multigrid,,and,preconditioning,of,matrix-free,linear,operators.,p-multigrid,,the,authors,explain,,involves,solving,a,system,with,reduced,polynomial,order,via,,eg.,AMG,and,then,propa-,gating,that,result,as,a,starting,point,for,the,‘full-p’,problem.,One,method,that,the,authors,focus,on,particularly,is,Low-Order,Refined,(LOR),preconditioning.,The,p,=,1,version,of,a,matrix-free,operator,is,assembled,,refined,spatially,in,a,manner,that,is,sensitive,to,the,high-p,origins,of,the,problem,,solved,via,traditional,low-order,AMG,methods,,and,finally,projected,back,up,to,higher-p.,The,Overlapping,Domain,Decomposition,Method,(ODDM),is,dicussed,next,,with,reference,to,its,useful,convergence,properties,,but,the,authors,note,that,the,leading,library,for,ODDM,has,not,yet,reached,maturity.,Sparse,approximate,inverse,preconditioning,(SPAI),is,also,considered.,The,section,concludes,with,a,comparison,of,the,performance,of,the,different,preconditioners,for,vari-,ous,combinations,of,grid,and,physics,anisotropy.,LOR,is,found,to,perform,best,,with,no,obvious,weaknesses.,Section,4,introduces,a,time-stationary,advection-diffusion,problem,with,tunable,Peclet,number.,AMG-LOR,gives,the,best,results,in,terms,of,both,error,achieved,and,runtime.,The,5th,section,compares,backwards,Euler,,Crank-Nicolson,and,a,variety,of,singly-diagonal,im-,plicit,Runge-Kutta,(SDIRK),schemes,by,applying,them,to,an,advection,equation.,For,scales,of,interest,,SDIRK33,is,found,to,give,the,best,performance.,Next,,a,variety,of,SDIRK,,IRK-Gauss,and,IRK-Lobatto,methods,are,compared,for,low,,medium,and,high-order,spatial,discretisations,,with,the,number,of,DOFs,per,stage,held,fixed.,It,is,found,that,the,best,results,are,obtained,using,high-order,methods,both,temporally,and,spatially.,While,the,number,of,iterations,appears,to,be,robust,,the,overall,scaling,is,found,to,be,suboptimal.,The,authors,suggest,that,scaling,could,be,improved,by,combining,the,fully,implicit,RK,methods,with,LOR,and,AMG.,18,3.3,Reports,received,under,Grant,T/AW087/21,,PO,2057699,The,York,grantee,produced,the,following,reports,in,association,with,Warwick,1.,2047358-TN-01-2,-,Approaches,to,Performance,Portable,Applications,for,Fusion,[34],2.,2047358-TN-02-2,-,Identification,of,Testbed,Platforms,and,Applications,[35],3.,2047358-TN-03-2,-,Evaluation,of,Approaches,to,Performance,Portability,[36],4.,2057699-TN-01-5,-,Hardware,at,the,Exascale,Revision,5.0,[37],5.,2057699-TN-02-5,-,Developing,an,Exascale-Ready,Fusion,Simulation,Revision,5.0,[38],6.,2057699-TN-03-3,-,Progress,on,Development,of,an,FEM-PIC,Miniapp,[39],7.,2057699-TN-04-2,-,Proposal,for,a,Particle,DSL,[40],The,first,three,reports,(produced,under,Grant,T/NA086/20,,PO,2047358),were,superseded,by,material,produced,under,the,new,grant,and,are,not,described,below.,3.3.1,Report,2057699-TN-01-5,[37],There,have,been,many,changes,to,the,hardware,makeup,of,supercomputers,as,they,have,reached,ever,higher,levels,of,computing,performance.,Be,it,the,move,towards,to,first,mainstream,multicore,CPUs,in,2004,,to,the,move,from,CPU,only,compute,nodes,to,the,majority,of,machines,now,having,both,a,CPU,and,GPU.,The,UKRI,intends,to,build,the,UK’s,first,exascale,supercomputer,by,2025.,This,report,details,the,hardware,that,Neptune,is,likely,to,encounter,both,pre,and,post,exascale.,This,report,is,split,into,three,parts.,The,first,part,outlines,the,hardware,currently,deployed,or,scheduled,to,be,delivered,in,next,five,years,according,to,each,vendor’s,respective,roadmap.,This,material,includes,details,such,as,the,core,count,range,for,a,particular,CPU,range,,supported,memory,technology,used,,and,peak,double,precision,performance.,The,second,part,outlines,what,machines,are,currently,being,built,or,proposed,within,Europe,and,the,USA,,and,elsewhere.,The,final,part,details,both,homogeneous,and,heterogenous,machines,currently,available,with,which,the,performance,portability,of,codes,developed,as,part,of,NEPTUNE,could,be,evaluated.,3.3.2,Report,2057699-TN-02-5,[38],As,the,Exascale,era,is,entered,,all,machines,,with,the,noticeable,exception,of,Fukgaku,,will,be,made,of,heterogeneous,nodes,containing,both,a,multi-core,CPU,and,a,GPU.,NEPTUNE,software,will,be,required,to,be,performance,portable,,targeting,these,multiple,all,these,CPU,and,GPU,architectures,,while,maintaining,high,levels,of,performance,across,them.,Several,programming,models,exist,to,parallelise,code,,but,as,of,yet,there,is,no,definitive,frontrunner,which,shows,itself,as,the,best,candidate,for,use,at,the,Exascale.,Therefore,,the,objective,of,this,report,is,to,investigate,the,effectiveness,of,different,programming,models,used,for,parallelisation,to,achieve,performance,19,portable,code.,This,was,achieved,by,evaluating,several,fluid,and,particle,proxyapps,using,a,variety,of,performance,benchmarks,,across,different,hardware,architectures,and,compilers.,The,key,findings,of,this,investigation,were,that:,•,The,majority,of,the,NEPTUNE,codebase,should,be,written,in,modern,C++,•,While,vendor-specific,parallelisation,techniques,can,achieve,higher,performance,than,open,standards,,they,suffer,from,the,disadvantage,of,producing,unportable,code.,The,disad-,vantage,of,open,standards,is,that,they,general,take,longer,to,target,newer,hardware,and,hardware,features.,•,NEPTUNE,will,likely,require,both,low,level,and,high,level,DSLs,•,Data,storage,should,be,abstracted,away,from,algorithms,as,much,as,possible,•,Code,such,be,modularised,as,much,as,possible,to,allow,key,kernels,which,are,performance,critical,to,be,re-evaluated,in,isolation,as,new,technologies,emerge,•,Where,possible,code,should,be,reused,including,external,libraries,3.3.3,Report,2057699-TN-03-3,[39],A,finite,element,particle,in,cell,(FEM-PIC),miniapp,which,uses,an,unstructured,grid,was,developed,for,evaluation,when,applied,to,problems,the,Neptune,project,is,likely,to,encounter.,The,particle,pusher,kernel,,the,kernel,for,depositing,charge,and,the,kernel,for,calculating,the,fields,were,identi-,fied,in,a,previous,report,as,key,components,needed,for,performance,evaluation.,The,performance,was,evaluated,by,the,total,time,for,which,the,functions,relating,to,each,particular,kernel,took,to,run.,A,test,case,looking,at,Deuterium,ion,flow,for,a,pipe,was,chosen,for,this,study.,Ten,runs,were,performed,ranging,from,around,10,−,130,particles/element,,using,both,the,Intel,and,GNU,compiler,on,the,Viking,cluster.,For,both,compilers,was,found,that,for,anything,above,approximately,20,ions,per,element,that,the,majority,of,the,time,was,taken,up,by,the,particle,pusher,routines,,which,grows,approximately,linearly,with,the,number,of,particles/element.,The,time,take,for,injecting,ions,onto,the,grid,and,calculating,ϕ,from,Poisson’s,equation,was,found,to,be,approximately,independent,of,the,number,of,particles/element.,The,current,status,of,this,work,can,be,found,at,.,3.3.4,Report,2057699-TN-04-2,[40],Domain,specific,languages,(DSLs),allows,one,to,provide,a,bridge,between,domain,scientists,and,application,developers,,allowing,them,to,write,high,level,abstractions,of,the,problem,they,wish,to,solve,,which,can,then,be,turned,into,low,level,parallelised,code.,Previous,DSLs,for,use,within,plasma,codes,often,are,dedicated,to,describing,only,the,fluid,nature,of,the,plasma.,The,development,of,a,DSL,that,can,also,be,used,for,particle,methods,was,therefore,a,necessity,moving,forward,with,the,NEPTUNE,project.,The,development,of,this,DSL,is,outlined,in,this,report.,The,new,DSL,was,done,by,modifying,the,OP2,DSL,to,accommodate,the,particle,in,cell,(PIC),method.,Since,no,complete,unstructured,mesh,3-D,electromagnetic,FEM,PIC,code,is,available,,20,three,PIC,codes,,namely,SIMPIC,,CABANAPIC,and,FEMPIC,were,selected,for,conversion,to,make,use,of,this,DSL,,to,demonstrate,its,functionality.,The,rewritten,SIMPIC,and,CABANAPIC,codes,were,written,in,C++,for,a,serial,implementation,,and,the,calculated,particle,and,grid,point,data,were,verified,to,be,unchanged.,FEMPIC,was,converted,to,work,both,sequential,and,with,OpenMP.,The,current,status,of,this,work,can,be,found,at,.,21,Acknowledgement,The,support,of,the,UK,Meteorological,Office,and,Strategic,Priorities,Fund,is,acknowledged.,References,[1],J.,Parker,and,W.,Arter.,research.,Numerical,Analy-,sis,Procurement.,Technical,Report,CD/EXCALIBUR-FMS/0068-M7c.1,,UKAEA,,6,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_,reports/CD-EXCALIBUR-FMS0068-M7c.1.pdf.,Management,of,external,[2],W.,Arter,et,al.,Equations,for,EXCALIBUR/NEPTUNE,Proxyapps.,cal,Report,CD/EXCALIBUR-FMS/0021-1.26-M1.2.1,,UKAEA,,1,2023.,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0021-1.26-M1.2.1.pdf.,Techni-,https:,[3],B.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,0.3,Community,and,environment.,Technical,Report,2047356-TN-16-1,,UKAEA,Project,Nep-,tune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2047356/TN-16-1.pdf.,[4],ExCALIBUR,Project,NEPTUNE,website.,https://excalibur-neptune.github.io/,Developers-Website/main/main.html,,2023.,Accessed:,January,2023.,[5],Source,material,for,ExCALIBUR,Project,NEPTUNE,website.,https://github.com/,ExCALIBUR-NEPTUNE/Developers-Website,,2023.,Accessed:,January,2023.,[6],B.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,1.4,Pre-,conditioning,2D,elliptic,solvers.,Technical,Report,2047356-TN-06-1,,UKAEA,Project,Nep-,tune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2047356/TN-06-1.pdf.,[7],B.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,2.2,BOUT++,1D,fluid,solver,with,realistic,boundary,conditions,:,implementation.,Technical,Report,2047356-,TN-07-1,,UKAEA,Project,Neptune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/,Documents/blob/main/reports/2047356/TN-07-1.pdf.,[8],B.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,2.4,Non-,intrusive,UQ,with,BOUT++,1D,fluid,solver.,Technical,Report,2047356-TN-08-1,,UKAEA,Project,Neptune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2047356/TN-08-1.pdf.,[9],B.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,4.1,Test,cases,for,1D,multispecies,plasma,model.,Technical,Report,2047356-TN-09-1,,UKAEA,Project,Neptune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2047356/TN-09-1.pdf.,22,[10],E.,Higgins,and,D.,Dickinson.,Task,0.2,Development,of,a,performance,testing,and,aggregation,tool,for,NEPTUNE,proxy-apps.,Technical,Report,2047356-TN-10-1,,UKAEA,Project,Nep-,tune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2047356/TN-10-1.pdf.,[11],B.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,3,Considerations,for,1D1V,models,Report,covering,tasks,3.1,,3.2,and,3.3.,Technical,Report,2047356-TN-11-,1,,UKAEA,Project,Neptune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/,blob/main/reports/2047356/TN-11-1.pdf.,[12],D.,Moxey,,B.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,and,S.,Wright.,(D1.3):,El-,liptic,solvers,within,Nektar++.,Technical,Report,2047356-TN-12-2,,UKAEA,Project,Nep-,tune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2047356/TN-12-2.pdf.,[13],D.,Moxey,,B.D.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,and,S.,Wright.,Spatially,2-D,plasma,incorporating,velocity,space,effeects.,Technical,Report,2047356-TN-13,,UKAEA,model,Project,Neptune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2047356/TN-13-2.pdf.,[14],B.D.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,DSL,and,code,design,pattern,for,NEPTUNE.,Technical,Report,2047356-TN-14-1,,UKAEA,Project,Nep-,tune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2047356/TN-14-1.pdf.,[15],B.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,2.5,Performance,considerations,for,non-intrusive,uncertainty,quantification,and,the,influence,of,the,boundary.,Technical,Report,2047356-TN-15-2,,UKAEA,Project,Neptune,,2022.,https://github.com/,ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047356/TN-15-2.pdf.,[16],D.,Moxey,,B.D.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,and,S.,Wright.,Progress,on,imple-,mentation,of,system,2-6,equations.,Technical,Report,2047356-TN-17,,UKAEA,Project,Nep-,to,be,posted,at,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/,tune,,2023.,main/reports/2047356/TN-17.pdf.,[17],S.,Thorne.,Priority,Equations,and,Test,Cases.,Technical,Report,2047353-TN-01,,UKAEA,Project,Neptune,,2021.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2047353/TN-01.pdf.,[18],V.,Alexandrov,,A.,Lebedev,,E.,Sahin,,and,S.,Thorne.,and,preconditioners,relating,to,the,NEPTUNE,Programme:,A,brief,overview.,nical,Report,2047353-TN-02,,UKAEA,Project,Neptune,,2021.,ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047353/TN-02.pdf.,Linear,systems,of,equations,Tech-,https://github.com/,[19],M.,Abalenkovs,,V.,Alexandrov,,A.,Lebedev,,E.,Sahin,,and,S.,Thorne.,Implicit-factorization,preconditioners,for,NEPTUNE,Programme.,Technical,Report,2047353-TN-03,,UKAEA,Project,Neptune,,2021.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2047353/TN-03.pdf.,23,[20],M.,Abalenkovs,,V.,Alexandrov,,A.,Lebedev,,E.,Sahin,,and,S.,Thorne.,Implicit-factorization,preconditioners,for,non-symmetric,problems.,Technical,Report,2047353-TN-04,,UKAEA,Project,Neptune,,2021.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2047353/TN-04.pdf.,[21],One-dimensional,plasma-neutral,simulation,for,SOL,and,divertor,studies.,https://github.,com/boutproject/SD1D,,2021.,Accessed:,August,2021.,[22],W.,Arter.,Equations,for,EXCALIBUR/NEPTUNE,Proxyapps.,port,CD/EXCALIBUR-FMS/0021-1.00-M1.2.1,,github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0021-1.00-M1.2.1.pdf.,UKAEA,,2020.,03,Technical,Re-,https://,[23],B.D.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,2.1,1D,fluid,model,tests.,Technical,Report,2047356-TN-04-2,,UKAEA,Project,Neptune,,2021.,https:,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047356/TN-04.pdf.,[24],P.A.,Hill,,J.,T.,Parker,,D.,Dickinson,,and,B.D.,Dudson.,BOUT++,VECMA,hackathon,repository.,https://github.com/boutproject/VECMA-hackathon,,2021.,Accessed:,June,2021.,[25],Hermes-3,2D,axisymmetric,tokamak.,https://hermes3.readthedocs.io/en/latest/,examples.html#d-axisymmetric-tokamak,,2021.,Accessed:,June,2021.,[26],Hermes,plasma,edge,simulation,model:,Hermes-3,,a,hot,ion,multifluid,drift-reduced,model.,https://github.com/bendudson/hermes-3,,2021.,Accessed:,June,2021.,[27],B.D.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Task,1.1,Elliptic,solver,tests.,Technical,Report,2047356-TN-02-2,,UKAEA,Project,Neptune,,2021.,https:,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2047356/TN-02.pdf.,[28],D.,Moxey.,Nektar++,strong,scaling,performance,on,Mira.,nektar-strong-scaling-performance-on-mira/,,2017.,https://www.nektar.info/,[29],D.,Moxey,,B.D.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,Spa-,tially,2-D,plasma,model,incorporating,velocity,space,effects.,Technical,Report,2047356-TN-,13,,UKAEA,Project,Neptune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/,blob/main/reports/2047356/TN-13.pdf.,[30],B.D.,Dudson,,P.A.,Hill,,E.,Higgins,,D.,Dickinson,,S.,Wright,,and,D.,Moxey.,DSL,and,code,design,pattern,for,NEPTUNE.,Technical,Report,2047356-TN-14,,UKAEA,Project,Nep-,tune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2047356/TN-14.pdf.,[31],B.D.,Dudson.,BOUT++,website.,https://boutproject.github.io/,,2020.,Accessed:,June,2020.,[32],H.,Al,Daas,,J.,Williams,,N.,Bootland,,T.,Rees,,P.,Rubin,,S.,Thorne,,and,A.,Sunderland.,Techniques,and,software,relevant,to,the,coupling,of,continuum,(fluid),and,particle,mod-,els,of,plasma,for,NEPTUNE.,Technical,Report,2060049-TN-02-5,,UKAEA,Project,Nep-,tune,,2023.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2060049/TN-02-5.pdf.,24,[33],H.,Al,Daas,,N.,Bootland,,T.,Rees,,and,S.,Thorne.,Time,Stepping,Techniques,and,Pre-,conditioning,for,Hyperbolic,and,Anisotropic,Elliptic,Problems:,Numerical,Results.,Tech-,nical,Report,2060049-TN-03,,UKAEA,Project,Neptune,,2022.,https://github.com/,ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2060049/TN-03.pdf.,[34],S.,Wright,,B.,Dudson,,P.A.,Hill,,D.,Dickinson,,and,G.,Mudalige.,Approaches,to,Per-,formance,Portable,Applications,for,Fusion.,Technical,Report,2047358-TN-01-2,,UKAEA,Project,Neptune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2047358/TN-01-2.pdf.,[35],S.,Wright,,B.,Dudson,,P.A.,Hill,,D.,Dickinson,,and,G.,Mudalige.,Identification,of,Testbed,Platforms,and,Applications.,Technical,Report,2047358-TN-02-2,,UKAEA,Project,Nep-,tune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2047358/TN-02-2.pdf.,[36],S.,Wright,,B.,Dudson,,P.A.,Hill,,D.,Dickinson,,and,G.,Mudalige.,Evaluation,of,Approaches,Technical,Report,2047358-TN-03-2,,UKAEA,Project,Nep-,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,to,Performance,Portability.,tune,,2022.,2047358/TN-03-2.pdf.,[37],S.,Wright,,E.,Higgins,,B.D.,Dudson,,P.A.,Hill,,D.,Dickinson,,G.,Mudalige,,B.F.,McMillan,,and,T.,Goffrey.,Hardware,at,the,Exascale,Revision,5.0.,Technical,Report,2057699-TN-01-,5,,UKAEA,Project,Neptune,,2022.,https://github.com/ExCALIBUR-NEPTUNE/Documents/,blob/main/reports/2057699/TN-01-5.pdf.,[38],S.,Wright,,E.,Higgins,,B.D.,Dudson,,P.A.,Hill,,D.,Dickinson,,G.,Mudalige,,B.F.,McMillan,,and,T.,Goffrey.,Developing,an,Exascale-Ready,Fusion,Simulation,Revision,5.0.,Tech-,nical,Report,2057699-TN-02-5,,UKAEA,Project,Neptune,,2022.,https://github.com/,ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2057699/TN-02-5.pdf.,[39],S.,Wright,,E.,Higgins,,G.,Mudalige,,B.F.,McMillan,,and,T.,Goffrey.,Progress,on,Develop-,ment,of,an,FEM-PIC,Miniapp.,Technical,Report,2057699-TN-03-3,,UKAEA,Project,Nep-,tune,,2023.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/,2057699/TN-03-3.pdf.,[40],S.,Wright,,E.,Higgins,,G.,Mudalige,,Z.,Lantra,,B.F.,McMillan,,and,T.,Goffrey.,Proposal,for,a,Particle,DSL.,Technical,Report,2057699-TN-04-2,,UKAEA,Project,Neptune,,2023.,https://,github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2057699/TN-04-2.pdf.,A,Annex:,D8c,Report,25,ExCALIBUR,Complementary,actions.,Atomic,data,usage,1.,M8c.1,Version,1.00,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Mile-,stone,M8c.1.,This,report,describes,NEPTUNE,use,of,analytic,formulae,and,atomic,physics,databases,for,describing,the,rate,of,atomic,collision,processes.,Scripts,have,been,developed,that,make,use,of,the,Atomic,Data,and,Analysis,Structure,(ADAS),database,to,predict,the,electron,impact,ion-,isation,rate,,and,the,charge,exchange,cross,section.,The,analytic,formula,being,used,to,predict,rate,of,ionisation,from,electron,impacts,on,hydrogen,has,found,good,agreement,with,ADAS.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0076,Issue:,Date:,1.00,March,23,,2023,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Matthew,Barton,Signature,N/A,Date,March,23,,2023,BD,Reviewed,By:,Wayne,Arter,March,23,,2023,Project,Technical,Lead,1,1,Introduction,Collisions,in,plasmas,between,electrons,,atoms,and,molecules,can,happen,in,a,variety,of,ways.,Such,processes,include,but,are,not,limited,to,electron,impact,ionisation,,recombination,,electron,impact,excitation,,charge,exchange,,etc.,Accurately,knowing,the,effect,these,collisions,are,having,on,the,abundance,of,various,ions,and,electron,populations,in,the,system,is,necessary,to,make,predictions,about,the,radiative,properties,of,atoms,and,molecules.,The,rate,at,which,these,differ-,ent,processes,occur,can,be,determined,in,a,variety,of,ways,,such,as,atomic,databases,,analytic,formula,or,collisional,radiative,models.,The,former,two,ways,have,been,investigated,for,this,Mile-,stone,M8c.1,,and,then,compared,against,each,other.,With,regard,to,the,use,of,atomic,databases,used,for,this,Milestone,,it,was,decided,that,the,Atomic,Data,and,Analysis,Structure,(ADAS),database,would,be,used.,ADAS,is,made,up,of,two,compo-,nents,,atomic,data,,which,can,be,searched,online,using,OPEN-ADAS,(see,Ref.,[1]),,and,a,Fortran,codebase,to,process,the,aforesaid,data,to,extract,the,atomic,processes,rates,of,interest.,Rather,than,using,the,Fortran,code,directly,,it,was,decided,that,use,would,be,made,of,the,Python,inter-,face,which,is,available.,Scripts,were,made,which,could,utilise,ADAS,to,output,the,electron,impact,ionisation,rate,,and,charge,exchange,cross,section,for,reactions,for,which,the,data,is,available.,These,scripts,can,be,found,within,the,Neso-Reactions,GitHub,repo.,For,the,initial,introduction,of,the,effect,of,electron,collisions,in,a,plasma,,a,formula,was,found,which,can,predict,the,ionisation,rate,for,electron,impact,ionisations.,The,accuracy,of,this,formula,was,tested,by,comparing,the,result,obtained,by,the,ADAS,database,for,the,following,reaction,H,+,e,⇒,H,+,+,e,+,e,(1),to,that,of,the,formula.,Once,the,accuracy,of,the,formula,had,been,tested,it,was,integrated,into,the,SOL2D-neutral,proxyapp.,The,results,of,this,integration,can,be,found,in,the,report,on,Milestone,4c.2,[2].,2,2,Task,Work,The,purpose,of,this,section,is,to,give,a,more,detailed,breakdown,of,the,work,undertaken,for,Milestone,M8c.1.,2.1,ADAS,One,way,to,predict,atomic,transitions,is,to,make,use,of,atomic,databases,.,There,are,numerous,atomic,databases,currently,employed,with,plasma,codes,,such,as,ALADDIN,[3],,AMJUEL,[4],and,ADAS,[5].,For,initial,investigations,ADAS,was,chosen.,The,ADAS,database,considered,consists,of,two,parts.,The,first,part,is,the,atomic,data,which,has,been,collected,from,numerous,experiments,,and,the,second,part,is,a,Fortran,codebase,to,process,said,atomic,data.,The,Python,interface,to,the,underlying,Fortran,code,was,utilised,for,investigating,its,use,by,Neso.,Scripts,were,created,which,could,allow,the,calculation,of,charge,exchange,cross,sections,and,electron,impact,ionisation,rates.,Charge,exchange,is,the,following,process,and,electron,impact,ionisation,is,A(Zi+1)+,p,+,H,⇒,AZi+,p,+,H,+,AZi+,p,+,e,⇒,AZi+,+,e,+,e.,(2),(3),In,the,energy,range,of,concern,for,Neso,these,are,the,2,most,prominent,reactions,which,can,occur,in,a,plasma,for,atomic,Hydrogen,(See,Fig,37,of,[6]).,The,Python,scripts,can,be,found,in,Neso-Reactions,Github,repo.,2.2,Analytical,Formula,The,starting,point,to,defining,a,ionisation,rate,is,the,cross,section.,A,cross,section,can,be,thought,of,as,an,approximate,area,around,a,target,that,an,electron,has,to,be,in,for,an,interaction,to,take,place.,The,following,formula,taken,from,Ref.,[7],is,used,for,the,cross,section,(originally,from,Ref.,[8],),σ,=,N,(cid:88),i=1,aiqi,(cid:17),ln,(cid:16),E,Ei,EEi,=,0,(cid:16),E,Ei,(cid:17),−1,−ci,[1,−,bie,],for,E,≥,Ei,E,<,Ei,(4),where,E,is,the,energy,of,the,impacting,electron,,Ei,is,the,binding,energy,of,electrons,in,the,i-,th,subshell.,qi,is,the,number,of,electrons,in,the,i-th,subshell.,ai,bi,and,ci,are,constants,to,be,determined,from,theory,or,experiment.,N=1,for,hydrogen,and,helium,like,ions.,The,reference,values,taken,from,Ref.,[7],for,hydrogen,are,as,follows,(i,can,only,take,the,value,1),3,Parameter,a1,b1,c1,E1,q1,Value,4x10−14cm2(eV,)2,0.6,0.56,13.6,eV,1,This,cross,section,can,be,folded,in,with,a,Maxwellian,electron,distribution,at,temperature,T,to,give,the,following,rate,(see,Ref.,[7],),S,=,−6.7x107,(cid:40),N,(cid:88),i=1,aiqi,T,3,2,(cid:16),Ei,−,T,Ei,(cid:17),Ei,T,−,bieci,Ei,T,+,ci,(cid:16),Ei,−,Ei,T,(cid:41),(cid:17),−,ci,(5),where,T,is,in,eV,and,S,is,in,units,of,cm3s−1.,In,the,above,formulas,multiple,ionization,,lowering,of,ionization,potential,,or,collision,limit,are,not,taken,into,account.,Ei(x),is,the,exponential,integral,defined,by,−,(cid:90),∞,−x,e−t,t,dt.,(6),Fig.,2,shows,a,comparison,between,the,rate,obtained,from,this,formula,against,the,rate,obtained,by,ADAS,for,H,+,e,⇒,H,+,+,e,+,e.,(7),As,can,be,seen,from,Fig.,2,the,rates,are,in,approximately,agreement,over,the,energy,range,chosen.,When,integrating,this,rate,into,Neso,the,total,rate,is,required.,This,is,found,using,the,following,dNp,dt,=,SNpNe,(8),where,Np,is,the,population,of,state,p,,and,Ne,is,the,electron,density,[9,,10].,This,rate,was,inte-,grated,into,the,SOL2D-neutral,proxyapp,,the,results,of,which,can,be,found,in,the,report,document-,ing,Milestone,4c.2.,4,Figure,1:,The,above,plot,shows,the,cross,section,obtained,from,the,analytic,formula,(solid,line),4,against,those,obtained,Ref.,[6],(crosses).,Figure,2:,The,above,plot,shows,the,ionisation,rate,obtained,from,Eq.,5,(solid,line),against,those,obtained,from,ADAS,(crosses).,5,3,Summary,Investigations,have,been,conducted,to,compare,the,prediction,of,an,analytical,formula,for,the,rate,of,electron,impact,ionisation,,against,atomic,databases,for,use,within,NESO.,Moderate,agreement,was,found,between,the,two,methods,in,terms,of,the,predicted,rate,for,electron,impact,ionisation.,Consequently,the,analytical,formula,for,the,rate,has,been,included,within,the,SOL2D-neutral,prox-,yapp,[2].,Acknowledgement,The,support,of,the,UK,Meteorological,Office,and,Strategic,Priorities,Fund,is,acknowledged.,References,[1],OPEN-ADAS,Atomic,Data,and,Analysis,Structure.,https://open.adas.ac.uk.,[2],J.,Cook,,E.,Threlfall,,,,O.,Parry,,W.,Saunders,,and,W.,Arter.,in-,tegrated,particle,and,continuum,model.,Technical,Report,CD/EXCALIBUR-FMS/0072-,M4c.2,,UKAEA,,3,2023.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/CD-EXCALIBUR-FMS-0072-M4c.2.pdf.,Three-Dimensional,[3],ALADDIN,database.,https://www-amdis.iaea.org/ALADDIN/.,[4],AMJUEL,database.,https://www.eirene.de/old_eirene/html/amjuel.html.,[5],Atomic,Data,and,Analysis,Structure,(ADAS).,https://www.adas.ac.uk/about.php.,[6],Atomic,Collision,Processes,in,Plasma,Physics,Experiments,CLM-R137.,scientific-publications.ukaea.uk/wp-content/uploads/CLM-R137.pdf.,https://,[7],Wolfgang,Lotz.,Electron-impact,ionization,cross,sections,and,ionization,rate,coefficients,for,atoms,and,ions,from,hydrogen,to,calcium.,Z.,Phys.,,216:,241-7(1968).,,1,1968.,[8],Wolfgang,Lotz.,An,empirical,formula,for,the,electron-impact,ionization,cross-section.,Zeitschrift,f,¨ur,Physik,,206(2):205–211,,1967.,[9],G.,J.,Tallents.,An,Introduction,to,Special,Relativity,for,Radiation,and,Plasma,Physics.,Cam-,bridge,University,Press,,2022.,[10],G.,J.,Tallents.,An,Introduction,to,the,Atomic,and,Radiation,Physics,of,Plasmas.,Cambridge,University,Press,,2018.,6

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