CD-EXCALIBUR-FMS0084-M6c.5_ExcaliburComplementaryActionsCodeIntegrationAcceptanceOperation
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   :description: technical note

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ExCALIBUR,Complementary,actions.,Code,integration,,ac-,ceptance,and,operation,4.,M6c.5,Version,1.00,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Mile-,stone,M6c.5.,It,provides,material,related,to,the,implementation,of,3-D,plasma,turbulence,equations,using,finite,element,modelling,(FEM),tech-,niques.,Results,produced,with,the,Firedrake,FEM,sofware,are,presented,,demonstrating,its,usefulness,as,a,tool,for,prototyping,new,solver,implemen-,tations,and,verifying,the,correctness,of,existing,NEPTUNE,proxyapps.,An,implementation,of,the,3-D,Hasegawa-Wakatani,equations,in,the,Neptune,Exploratory,SOfware,(NESO),is,introduced,and,preliminary,results,are,re-,ported.,A,new,code,for,generating,field-aligned,computational,meshes,is,discussed,and,example,output,is,presented,in,2-D,and,3-D,for,realistic,toka-,mak,geometries.,Finally,,a,synopsis,of,related,work,carried,out,by,grantees,over,the,past,year,is,provided,in,the,form,of,technical,report,summaries,and,reviews,of,code,deliverables.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/ExCALIBUR,-FMS/0084,Issue:,Date:,1.01,13,March,2024,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Ed,Threlfall,Owen,Parry,Chris,MacMackin,Signature,N/A,N/A,N/A,Date,13,March,2024,13,March,2024,13,March,2024,CD,Reviewed,By:,James,Cook,15,March,2024,CD,1,1,Introduction,This,report,covers,a,range,of,topics,related,to,the,use,of,finite,element,modelling,frameworks,in,NEPTUNE.,Section,2,focuses,on,the,use,of,the,Firedrake[1],software,as,a,tool,for,rapid,prototyp-,ing,of,new,solvers,and,as,a,means,of,checking,the,correctness,of,existing,proxyapps.,In,addition,to,reimplementing,some,of,those,proxyapps,and,comparing,to,the,Nektar++,versions,directly,,Fire-,drake,is,also,employed,to,investigate,various,key,components,of,a,more,complex,model,,which,represents,the,near-term,goal,of,fluid,plasma,modelling,in,the,NEPTUNE,Exploratory,Software,(NESO).,Section,3,reports,on,an,implementation,of,the,3-D,Hasegawa-Wakatani,equations,in,NESO.,The,solver,extends,an,existing,Nektar++,equation,system,,but,has,required,substantial,code,refactor-,ing,in,order,to,adhere,to,sound,software,design,practices,,avoiding,code,duplication,,preserving,encapsulation,and,allowing,key,functionality,to,be,shared,between,similar,solvers.,An,important,component,of,project,NEPTUNE,is,the,development,of,technology,to,produce,com-,putational,meshes,that,are,suitable,for,modelling,plasma,and,neutrals,in,the,tokamak,edge,region.,A,related,challenge,is,the,high,degree,of,velocity,anisotropy,associated,with,magnetically,confined,plasmas,,which,can,negatively,impact,numerical,accuracy,when,a,mesh,is,not,aligned,with,the,magnetic,field.,To,this,end,,Section,4,describes,a,tool,called,“NESO-fame”,which,has,been,devel-,oped,to,produce,field-aligned,meshes,in,realistic,geometries,(i.e.,conforming,to,the,first,wall,of,a,tokamak).,A,number,of,example,meshes,are,presented,in,2-D,and,3-D.,Finally,,Section,5,summarises,related,technical,reports,and,code,deliverables,produced,by,project,grantees,in,the,past,year.,A,series,of,progress,reports,are,provided,by,the,Oxford,grantee,,de-,scribing,important,updates,to,their,implementation,of,a,finite,element,drift-kinetic,model.,The,KCL,grantee,work,includes,a,report,outlining,improvements,to,their,mesh,generation,tool,,NekMesh,,and,several,code,deliverables,designed,to,increase,the,usability,and,performance,portability,of,Nektar++,,as,well,as,increase,its,utility,for,modelling,fusion-relevant,problems.,2,Firedrake,as,a,tool,to,assist,NEPTUNE,development,The,aim,of,the,work,described,in,this,section,has,been,to,replicate,NEPTUNE,proxyapps,using,Firedrake[1],,for,verification,and,validation,purposes,,to,enable,the,comparison,of,different,frame-,works,for,NEPTUNE,finite-element,aspects,,to,assist,with,finding,bugs,in,existing,NEPTUNE,proxyapps,,and,also,to,speed,up,development.,It,will,be,demonstrated,that,many,of,the,finite-,element-based,proxyapps,developed,under,NEPTUNE,may,straightforwardly,be,implemented,in,the,Firedrake/Irksome[2],framework.,As,explained,in,[1],,Firedrake,is,a,framework,that,automates,the,finite-element,solution,of,par-,tial,differential,equations.,The,software,user,specifies,the,equation,in,a,high-level,,mathematical,domain-specific,language,called,Unified,Form,Language,(UFL),,in,which,PDEs,are,expressed,in,weak,form,,and,then,the,package,generates,C,code,to,solve,the,system,,using,also,the,PETSc,library.,Time-stepping,is,provided,by,the,complementary,Irksome,package,which,offers,a,range,of,time-steppers,within,the,Firedrake,framework,(though,it,is,also,possible,to,implement,time-,[3]).,One,major,attraction,of,Firedrake,is,the,ease,of,im-,steppers,without,Irksome,,see,e.g.,2,plementation,,which,gives,a,favourable,time-to-working-solver,for,many,problems;,it,is,,however,,the,intention,of,the,developers,of,Firedrake,that,it,be,a,competitive,HPC,tool,,for,example,it,has,been,tested,up,to,approximately,25,000,CPU,cores,(cf.,Nektar++,,which,has,been,run,on,up,to,c.100,000,cores).,Another,attraction,is,the,Firedrake,Github,Discussions,forum,,which,the,author,has,found,a,useful,and,responsive,source,of,guidance.,There,are,other,tangible,benefits,from,Firedrake,such,as,the,easy,interchangeability,of,numeri-,cal,methods,e.g.,time-steppers,and,preconditioners,,with,obvious,synergies,with,the,work,done,under,the,numerical,analysis,procurement,(currently,T/AW087/22).,It,is,well-known,that,these,aspects,can,yield,large,increases,in,computational,efficiency,for,a,relatively,small,investment,in,development,time,(see,e.g.,S.3.1.3,of,[4]).,The,investigation,of,numerical,methods,is,to,an,extent,independent,of,the,finite-element,framework,used,-,one,may,as,well,select,one’s,tool,on,the,ba-,sis,of,expediency,-,and,to,an,extent,not,-,why,spend,time,implementing,/,debugging,a,promising,method,in,one,framework,if,it,is,available,already,elsewhere.,Note,that,the,problems,treated,here,are,small,(reflecting,the,state,of,the,current,NEPTUNE,prox-,yapp,ecosystem),,though,they,can,be,extended,to,truly,HPC-grade,cases,e.g.,by,moving,to,three,dimensions,and,higher,numerical,resolution.,Even,the,small,problems,can,benefit,from,HPC,e.g.,via,running,ensembles,for,UQ,or,surrogate,construction,,or,by,a,parallel-in-time,implementation,(the,latter,has,been,implemented,for,the,Nektar-Driftwave,proxyapp,[5]).,Note,also,that,,at,the,time,of,writing,,little,quantitative,testing,has,been,done,on,the,outputs,of,the,codes,in,the,sequel,(again,a,bit,like,the,case,for,many,of,the,existing,NEPTUNE,proxyapps).,One,extremely,important,example,case,is,that,of,the,Large,PLasma,Device,(LAPD),simulation,in,Nektar++.,Major,components,of,the,physics,in,that,problem,are,a,sonic,outflow,of,a,compressible,fluid,,and,drift-wave,turbulence,in,the,transverse,plane.,The,first,few,sections,in,the,sequel,address,these,cases.,The,implementation,of,the,LAPD,simulation,in,the,NESO,framework,,using,Nektar++,and,described,in,[6],,is,currently,still,ongoing.,The,author,of,this,section,believes,that,the,remaining,issues,with,the,implementation,of,the,LAPD,simulation,will,be,resolved,in,the,near,future,with,the,aid,of,Firedrake,prototyping.,Note,that,the,scripts,used,to,generate,the,outputs,in,this,section,can,be,obtained,by,requesting,access,to,the,private,repository,[7].,2.1,One-dimensional,outflow,problem,The,primary,motivation,for,this,section,is,to,provide,an,initial,implementation,of,the,problem,of,sonic,outflow,of,a,compressible,fluid,,in,order,to,facilitate,the,implementation,of,the,LAPD,simulation,within,Nektar++.,2.1.1,The,model,A,very,simple,1D,model,for,the,scrape-off,layer,is,to,take,the,compressible,isothermal,gas,dynam-,ics,of,a,single,species,with,no,electric,field,and,no,transverse,dynamics.,Note,that,the,isothermal,3,approximation,is,used,in,some,plasma,simulations,e.g.,the,Blob2D,example,at,[8],and,for,the,ions,in,the,cold-ion,approximation,,see,e.g.,[9].,In,the,first,instance,for,discussion,,the,equations,are,the,steady-state,of,eqs.10,and,11,from,[10].,Two,ODEs,on,the,domain,x,∈,[−1,,1],,describe,the,fluid,density,n,and,the,flow,velocity,u:,(nu)′,=,n∗,,(nu2)′,=−T,n′.,(1),Prime,is,x-derivative.,n∗,is,a,spatially-constant,density,source,(cf.,the,LAPD,simulation,in,which,there,is,assumed,to,be,a,longitudinally-constant,source,of,electron,density).,The,solution,is,,specifying,sonic,outflow,at,the,boundaries,,n,=,n∗,√,T,√,T,u,=,(cid:16),1,+,(cid:112),√,(cid:32),1,−,1,−,x2(cid:17),(cid:33),1,−,x2,x,,,.,(2),For,the,following,numerics,the,choices,n∗,=,1,,T,=,1,will,be,made.,Note,that,an,attempt,was,made,to,solve,the,same,system,including,diffusion,terms,of,various,types,but,no,analytic,solution,was,found.,2.1.2,Numerical,treatment:,continuous,Galerkin,(CG),The,equation,treated,by,Firedrake,is,input,in,weak,form,using,the,UFL,interface.,For,the,system,Eqs.1,,the,UFL,is,shown,in,Listing,1.,Listing,1:,UFL,code,defining,the,equations,Eqs.1,for,CG,elements.,Note,that,v1,and,v2,are,test,functions.,a,=,(,grad,(,n,*,u,),[,0,],*,v1−,n,s,t,a,r,*,v1+grad,(,n,*,u,*,u,),[,0,],*,v2+T,*,grad,(,n,),[,0,],*,v2,),*,dx,A,naive,treatment,of,the,above,hyperbolic,system,is,prone,to,grid-scale,oscillations,which,arise,from,a,negative,numerical,diffusivity,coefficient,(see,,for,example,,Remark,2.6,of,[11]).,Thus,the,simple,Firedrake,treatment,in,Listing,1,of,the,steady-state,has,the,undesirable,appearance,shown,in,Fig.1.,There,are,two,main,ways,to,fix,the,problem,of,grid-scale,numerical,oscillation,,both,of,which,are,basically,equivalent,to,adding,in,a,positive,numerical,diffusivity.,These,respectively,will,be,the,subject,of,the,next,subsections.,2.1.3,Numerical,treatment:,streamline-upwind,(SU),correction,The,first,method,is,to,add,numerical,diffusivity,by,modification,of,the,basis,functions.,The,modi-,fication,leads,to,additional,terms,in,the,weak,form,to,account,for,added,numerical,damping;,see,4,Figure,1:,Numerical,solution,of,Eqs.1,using,100,first-order,CG,elements.,Note,obvious,grid-scale,oscillation,in,n.,Generated,by,SOL,1D,CG,SU,DG,upwind.py,[7];,CG,output.,5,Figure,2:,Numerical,solution,of,Eqs.1,using,100,first-order,CG,elements,with,streamline-,upwind,(SU),correction.,Generated,by,SOL,1D,CG,SU,DG,upwind.py,[7];,CG,output,with,SU,correction,enabled.,Note,suppression,of,grid-scale,oscillation.,e.g.,S.2.3,of,[11].,The,method,demonstrated,here,will,be,referred,to,as,a,streamline-upwind,(SU),correction.,Listing,2:,UFL,code,defining,the,SOL1D,equations,for,CG,elements,including,the,SU,correction,terms.,Here,h,is,the,grid,scale.,a,=,(,grad,(,n,*,u,),[,0,],*,v1−,n,s,t,a,r,*,v1+grad,(,n,*,u,*,u,),[,0,],*,v2+T*,grad,(,n,),[,0,],*,v2,),*,dx,\,+,(,0,.,5,*,h,*,(,grad,(,n,),[,0,],−,n,s,t,a,r,),*,grad,(,v1,),[,0,],),*,dx,\,+,(,0,.,5,*,h,*,(,grad,(,n,*,u,),[,0,],),*,grad,(,v2,),[,0,],),*,dx,This,method,correctly,suppresses,the,grid-scale,oscillation,,as,is,seen,in,Fig.2.,It,is,obvious,that,this,method,has,the,weakness,that,it,is,accurate,to,the,model,only,to,first-order,in,the,grid-spacing,scale,h,-,this,is,apparent,from,the,O(h),correction,to,the,weak,form.,A,time-dependent,implementation,is,also,possible;,in,this,case,the,relevant,equations,are,˙n,+,(nu)′,=,n∗,,˙(nu),+,(nu2)′,=−T,n′.,6,(3),Figure,3:,Numerical,solution,of,eqs.3,using,100,first-order,CG,elements,with,SU,correction,starting,from,a,Gaussian,n,and,linear,u,initial,condition.,Plots,from,the,left,are,t,=,0,,1,,2,,3.,Generated,by,SOL,1D,SU,irksome.py,[7].,Note,that,the,second,of,these,is,recast,in,the,form,n,˙u,+,un∗,+,nuu′,=,−T,n′,in,order,to,derive,the,weak,form,in,Listing,3.,Listing,3:,UFL,code,(including,Irksome,time-derivatives),defining,the,equation,system,Eqs.3,plus,SU,corrections.,F,=,(,(,Dt,(,n,),*,v1,),*,dx,+,(,n,*,Dt,(,u,),*,v2,),*,dx,),\,+,(,grad,(,n,*,u,),[,0,],*,v1−v1,*,n,s,t,a,r,),*,dx,\,+,(,n,s,t,a,r,*,u,*,v2+n,*,u,*,grad,(,u,),[,0,],*,v2+Temp*,grad,(,n,),[,0,],*,v2,),*,dx,\,+,(,0,.,5,*,h,*,(,grad,(,n,),[,0,],−,n,s,t,a,r,),*,grad,(,v1,),[,0,],),*,dx,\,+,(,0,.,5,*,h,*,(,grad,(,n,*,u,),[,0,],),*,grad,(,v2,),[,0,],),*,dx,Output,from,the,time-dependent,implementation,with,SU,correction,weak,form,is,exhibited,in,Fig.3.,It,does,show,what,looks,like,a,minor,discrepancy,in,n,near,the,boundaries.,2.1.4,Numerical,treatment:,discontinuous,Galerkin,(DG),The,second,method,of,stabilizing,the,flow,equations,is,to,use,a,discontinuous,Galerkin,method,with,a,dissipative,numerical,flux.,Here,an,upwind,flux,is,used,(other,choices,,e.g.,an,interior,penalty,flux,,are,possible).,Listing,4:,UFL,code,defining,the,equations,Eqs.1,for,DG,elements,with,upwind,numerical,flux.,Note,that,the,backslashes,denote,new,lines,in,the,UFL,whereas,carriage,returns,without,backslash,are,formatting,to,keep,the,listing,text,on,the,page.,aD,=,(,nD*,d,o,t,(,uD,,,grad,(,v1D,),),+,v1D,*,n,s,t,a,r,),*,dx,\,+,(,nD*uD,[,0,],*,d,o,t,(,uD,,,grad,(,v2D,[,0,],),),+,T,*nD*,grad,(,v2D,[,0,],),[,0,],),*,dx,\,−,c,o,n,d,i,t,i,o,n,a,l,(,d,o,t,(,uD,,,norm,),>,0,,,v1D,*,d,o,t,(,uD,,,norm,),*,nD,,,0,.,0,),*,ds,\,7,Figure,4:,Numerical,solution,of,eqs.1,using,100,first-order,DG,elements,with,upwind,numerical,flux.,Note,suppression,of,grid-scale,oscillation.,Generated,by,SOL,1D,DG,upwind.py,[7].,−,(,v1D,(,’,+,’,),−,v1D,(,’,−,’,),),*,(,uD,n,(,’,+,’,),*,nD,(,’,+,’,),−,uD,n,(,’,−,’,),*,nD,(,’,−,’,),),*,dS,\,−,(,v2D,(,’,+,’,),[,0,],−,v2D,(,’,−,’,),[,0,],),*,(,nD,(,’,+,’,),*,uD,(,’,+,’,),[,0,],*,uD,[,0,],(,’,+,’,),−,nD,(,’,−,’,),*uD,(,’,−,’,),[,0,],*,uD,[,0,],(,’,−,’,),),*,dS,\,−,(,v2D,(,’,+,’,),[,0,],−,v2D,(,’,−,’,),[,0,],),*,(,nD,(,’,+,’,),−,nD,(,’,−,’,),),*,T*dS,A,disadvantage,of,the,DG,method,is,that,the,degree-of-freedom,count,is,increased,due,to,the,duplication,of,boundary,degrees,of,freedom.,This,problem,is,most,acute,for,low,orders,,where,boundary,degrees,of,freedom,dominate,the,computation.,It,is,noted,that,the,DG,implementations,are,slower,than,the,SU-corrected,CG,ones,for,the,small,codes,described,in,this,section,of,the,report.,The,DG,method,has,the,advantage,that,the,mass,matrix,is,defined,on,a,per-element,basis;,the,coupling,between,elements,is,exclusively,via,the,element,boundary,numerical,flux.,Output,from,the,stationary-state,and,time-dependent,DG,implementations,of,the,weak,form,in,Listing,5,is,shown,in,Figs.4,,5.,These,outputs,look,better-behaved,than,the,SU,version,near,the,boundaries.,Listing,5:,UFL,code,(including,Irksome,time-derivatives),defining,the,equation,system,Eqs.3,for,upwind,DG.,8,Figure,5:,Numerical,solution,of,eqs.3,using,100,first-order,DG,elements,with,an,upwind,numerical,flux,,starting,from,a,Gaussian,n,and,linear,u,initial,condition.,Plots,from,the,left,are,t,=,0,,1,,2,,3.,Generated,by,SOL,1D,DG,upwind,irksome.py,[7].,F,=,−,(,(,Dt,(,n,),*,v1,),*,dx,+,(,n,*,d,o,t,(,Dt,(,u,),,,v2,),),*,dx,),\,+,(,n,*,d,o,t,(,u,,,grad,(,v1,),),+,v1,*,n,s,t,a,r,),*,dx,\,+,(,n,s,t,a,r,*,d,o,t,(,u,,,v2,)+,n,*,u,[,0,],*,grad,(,d,o,t,(,u,,,v2,),),[,0,],+,n,*,u,[,0,],*,d,o,t,(,u,,,grad,(,v2,[,0,],),),+Temp*,n,*,grad,(,v2,[,0,],),[,0,],),*,dx,\,−,(,v1,(,’,+,’,),−,v1,(,’,−,’,),),*,(,u,n,(,’,+,’,),*,n,(,’,+,’,),−,u,n,(,’,−,’,),*,n,(,’,−,’,),),*,dS,\,+,(,u,(,’,+,’,),[,0,],*,v2,(,’,+,’,),[,0,],−,u,(,’,−,’,),[,0,],*,v2,(,’,−,’,),[,0,],),*,(,n,(,’,+,’,),*,u,n,(,’,+,’,),−,n,(,’,−,’,),*,u,n,(,’,−,’,),),*,dS,\,−,c,o,n,d,i,t,i,o,n,a,l,(,d,o,t,(,u,,,norm,),>,0,,,v1,*,d,o,t,(,u,,,norm,),*,n,,,0,.,0,),*,ds,Note,also,that,scripts,have,been,written,to,solve,the,above,problem,on,2D,and,3D,domains,in,anticipation,of,elaborating,this,simple,case,into,a,script,including,the,physics,in,the,3D,LAPD,simulation;,the,density,outputs,are,shown,in,Fig.6.,2.2,Drift-wave,turbulence,in,Firedrake,The,primary,motivation,for,this,section,is,to,provide,an,initial,implementation,of,drift-wave,tur-,bulence,in,the,transverse,plane,that,can,be,integrated,with,the,longitudinal,outflow,dynamics,explained,above,,in,order,to,facilitate,the,implementation,of,the,LAPD,simulation,within,Nektar++.,Firedrake,is,again,used,here,as,an,expedient,framework,in,which,to,implement,test,cases.,A,sec-,ondary,motivation,for,this,work,is,to,supplement,the,existing,NEPTUNE,2D,turbulence,proxyapp,,the,Nektar++-based,Nektar-Driftwave,[5],,with,an,implementation,in,Firedrake,/,Irksome,in,which,the,details,of,the,numerical,method,,e.g.,the,choice,of,finite-element,discretization,and,the,choice,of,time-stepper,can,be,easily,changed.,(Apologia,-,extensive,investigation,of,the,effect,of,varying,the,numerical,method,has,not,at,the,time,of,writing,been,performed,but,it,could,straightforwardly,be,done,,along,with,the,addition,of,quantitative,diagnostics,such,as,energy,,enstrophy,,and,power,spectra,of,the,turbulent,flow.,Also,,non-conservation,of,ω,leads,to,obvious,problems,solving,the,Poisson,equation,in,the,periodic,case,and,so,explicit,conservation,of,ω,is,desirable.,The,author,9,Figure,6:,Numerical,solution,of,eqs.3,in,2D,(top),and,3D,(bottom),,using,first-order,DG,elements,with,starting,from,a,Gaussian,n,and,linear,u,initial,condition.,Plots,from,left,are,t,=,0,,1,,2,,3,in,both,cases.,Generated,by,SOL,2D,DG,upwind,irksome.py,[7],and,SOL,3D,DG,upwind,irksome.py,[7].,thereby,justifies,the,rather,heuristic,and,visually-led,nature,of,the,following,presentation.,Note,also,that,performance,measurements,(wall,times),for,the,codes,do,not,reflect,the,scaling,expected,for,larger,problems,and,are,quoted,in,the,context,of,obtaining,a,workable,code,that,runs,rapidly,in,order,to,speed-up,testing,and,progress,toward,more,realistic,problem,e.g.,the,LAPD,simulation.),Nektar-Driftwave,implements,the,Hasegawa-Wakatani,equations,,describing,drift-wave,turbulence,,for,density,n,,vorticity,ω,,and,potential,ϕ:,∂n,∂t,∂ω,∂t,+,[ϕ,,n],=,α(ϕ,−,n),+,[ϕ,,ω],=α(ϕ,−,n),−,κ,∇2,⊥ϕ,=,∂ϕ,∂y,ω.,(4),In,the,above,,the,‘Poisson,bracket’,denotes,[a,,b],≡,∂a,∂x,∂b,∂y,−,∂b,∂x,∂a,∂y,.,The,model,is,identical,to,that,used,in,the,existing,Nektar-Driftwave,proxyapp.,The,example,sup-,plied,with,Nektar-Driftwave,propagates,the,equations,,starting,from,a,Gaussian-profile,density,perturbation,,and,a,vorticity,perturbation.,The,vorticity,integrates,to,zero,(or,very,nearly,zero,depending,on,the,condition,that,the,vorticity,amplitude,drops,to,very,nearly,zero,at,the,domain,boundaries),,which,is,a,property,that,must,be,preserved,by,periodic,simulations,as,the,vorticity,plays,the,role,of,a,charge,density.,The,simulations,lasts,for,50,time,units,using,a,timestep,of,0.0005,time,units,,which,is,close,to,the,stability,limit,of,the,explicit,time,evolution,method,used,(standard,RK4).,The,small,timestep,means,that,the,simulation,is,quite,slow,,taking,eight,hours,to,run,on,an,eight-core,workstation,(note,that,this,uses,a,64,×,64,mesh,of,square,p,=,3,DG,elements).,Note,that,we,fix,the,parameters,κ,=,1,(strength,of,driving,force,from,background,gradient),and,α,=,1,(called,adiabaticity,and,related,to,the,coupling,to,the,out-of-plane,dynamics,in,the,3D,form,of,the,equations).,The,Firedrake,examples,in,this,section,used,κ,=,2,and,α,=,2,i.e.,a,larger,driving,force;,this,was,done,in,order,to,ensure,that,the,more,strongly-damped,cases,evolved,to,a,10,Figure,7:,Numerical,solution,of,the,vorticity,from,Eqs.4,from,Nektar-Driftwave,using,a,64×64,mesh,of,square,third-order,DG,,starting,from,a,Gaussian,density,perturbation,initial,condition.,Plots,from,left-to-right,then,top-to-bottom,are,t,=,0,,10,,20,,30,,40,,50.,Note,that,these,outputs,are,not,directly,comparable,to,the,subsequent,outputs,from,Firedrake,due,to,the,choice,of,different,parameters;,however,the,appearance,of,these,is,a,useful,guide,to,the,lengthscales,resolved,in,the,turbulent,state.,turbulent,state.,None,of,the,examples,in,this,section,were,found,to,need,a,hyperdiffusion,term,(in,all,cases,there,is,sufficient,numerical,damping,to,keep,the,time-evolution,stable).,The,initial,data,used,in,the,2D,Firedrake,examples,was,the,same,as,that,used,in,Nektar-Driftwave.,The,domain,is,40,×,40,units,in,size,and,it,has,periodic,boundary,conditions,on,all,boundaries.,Note,that,the,equation,system,can,be,solved,in,one,pass,as,a,differential-algebraic,equation,(DAE).,An,alternative,,which,is,the,one,to,which,we,will,devote,the,most,attention,,is,to,update,the,potential,independently,from,the,other,variables,in,a,leap-frogging,fashion.,This,latter,is,the,approach,used,in,the,existing,Nektar-Driftwave,proxyapp,,though,it,is,the,case,that,the,potential,is,solved,at,each,stage,of,the,RK,stepper,,using,the,vorticity,field,from,the,previous,stage.,Solving,the,potential,on,a,per-stage,basis,in,this,way,in,Irksome,would,require,additional,work,beyond,that,presented,here.,A,word,on,the,current,status,of,Nektar-Driftwave:,the,grantee,(T/AW085/22),has,recently,made,great,progress,in,improving,the,efficiency,of,this,solver,by,replacing,the,explicit,RK4,time-stepping,with,an,explicit,method.,There,is,also,a,parallel-in-time,implementation,,which,the,author,under-,stands,was,the,movitation,behind,the,implicit,implementation.,It,is,,however,,not,clear,whether,the,performance,gains,will,translate,well,to,other,plasma,physics,examples,without,careful,consider-,ation,of,the,preconditioning,of,the,system,(the,simple,2D,Hasegawa-Wakatani,implementation,is,found,to,work,well,without,preconditioning).,11,2.2.1,Dirichlet,boundary,conditions,Outputs,of,a,SU-corrected,CG,version,(Listing,6),are,shown,in,Fig.7.,Listing,6:,UFL,code,defining,the,(non-elliptic,part,of,the),equations,for,the,2D,Hasegawa-Wakatani,system,for,CG,elements,with,SU,correction.,F,=,Dt,(w),*,v1,*,dx,+,Dt,(,n,),*,v2,*,dx,\,−,v1,*,d,i,v,(w*,d,r,i,f,t,v,e,l,),*,dx,−,v2,*,d,i,v,(,n,*,d,r,i,f,t,v,e,l,),*,dx,\,−,alpha,*,(,p,h,i,s,−n,),*,(,v1+v2,),*,dx,\,+,kappa,*,grad,(,p,h,i,s,),[,1,],*,v2,*,dx,\,+,0,.,5,*,h,*,(,d,o,t,(,d,r,i,f,t,v,e,l,,,grad,(w),),−,alpha,*,(,p,h,i,s,−n,),),*,d,o,t,(,d,r,i,f,t,v,e,l,,,grad,(,v1,),),*,(,1,/,s,q,r,t,(,(,d,r,i,f,t,v,e,l,[,0,],),*,*,2,+,(,d,r,i,f,t,v,e,l,[,1,],),*,*,2,),),*,dx,\,+,0,.,5,*,h,*,(,d,o,t,(,d,r,i,f,t,v,e,l,,,grad,(,n,),),−,alpha,*,(,p,h,i,s,−n),−,kappa,*,grad,(,p,h,i,s,),[,1,],),*,d,o,t,(,d,r,i,f,t,v,e,l,,,grad,(,v2,),),*,(,1,/,s,q,r,t,(,(,d,r,i,f,t,v,e,l,[,0,],),*,*,2,+,(,d,r,i,f,t,v,e,l,[,1,],),*,*,2,),),*,dx,Note,that,the,size,of,the,SU,term,has,been,reduced,below,the,amount,specified,by,the,textbook,[11],,by,a,factor,of,ten.,It,was,found,that,this,led,to,a,better,resolution,of,the,field,activity,without,showing,a,large,increase,in,grid-scale,oscillation.,Using,p,=,3,elements,for,the,vorticity,and,p,=,2,for,density,,potential,and,drift,velocity,gives,a,simulation,that,takes,approximately,21,minutes,on,one,core,of,the,author’s,laptop.,Outputs,of,a,DG,implementation,(Listing,7),are,shown,in,Fig.9.,Note,that,only,the,advected,fields,are,DG;,CG,was,used,for,ϕ,and,drift,velocity,,as,in,Nektar-Driftwave,and,e.g.,[12].,Listing,7:,UFL,code,defining,the,(non-elliptic),equations,for,the,2D,Hasegawa-Wakatani,system,using,upwind,DG,elements.,norm,=,FacetNormal,(,mesh,),d,r,i,f,t,v,e,l,n,=,0,.,5,*,(,d,o,t,(,d,r,i,f,t,v,e,l,,,norm,)+,abs,(,d,o,t,(,d,r,i,f,t,v,e,l,,,norm,),),),F,=,Dt,(w),*,v1,*,dx,+,Dt,(,n,),*,v2,*,dx,\,−,v1,*,d,i,v,(w*,d,r,i,f,t,v,e,l,),*,dx,−,v2,*,d,i,v,(,n,*,d,r,i,f,t,v,e,l,),*,dx,\,−,alpha,*,(,p,h,i,s,−n,),*,(,v1+v2,),*,dx,\,+,kappa,*,grad,(,p,h,i,s,),[,1,],*,v2,*,dx,\,+,d,r,i,f,t,v,e,l,n,(,’,−,’,),*,(,w(,’,−,’,),−,w,(,’,+,’,),),*,v1,(,’,−,’,),*,dS,\,+,d,r,i,f,t,v,e,l,n,(,’,+,’,),*,(,w,(,’,+,’,),−,w,(,’,−,’,),),*,v1,(,’,+,’,),*,dS,\,+,d,r,i,f,t,v,e,l,n,(,’,−,’,),*,(,n,(,’,−,’,),−,n,(,’,+,’,),),*,v2,(,’,−,’,),*,dS,\,+,d,r,i,f,t,v,e,l,n,(,’,+,’,),*,(,n,(,’,+,’,),−,n,(,’,−,’,),),*,v2,(,’,+,’,),*,dS,Using,p,=,3,elements,for,the,vorticity,and,p,=,2,for,density,,potential,and,drift,velocity,gives,a,simulation,that,takes,approximately,100,minutes,on,one,core,of,the,author’s,laptop.,This,is,significantly,longer,than,the,SU-corrected,CG,implementation,,but,the,results,to,seem,to,indicate,a,higher,resolution,of,the,turbulent,flow.,A,DG-DAE,implementation,was,created,(Listing,8).,This,was,run,for,50,time,units,using,500,timesteps.,The,timestepper,was,RadauIIA(2).,The,output,shows,considerably,more,resolution,12,Figure,8:,Numerical,solution,of,the,vorticity,from,Eqs.4,using,a,64,×,64,mesh,of,square,first-order,CG,elements,with,a,SU,correction,term,,starting,from,a,Gaussian,density,perturbation,initial,con-,dition.,Plots,from,left,are,t,=,0,,10,,20,,30,,40,,50.,Note,that,in,subsequent,plots,,the,first,three,times,are,not,shown,as,they,are,largely,identical,to,those,in,the,above,(the,system,is,in,a,laminar,regime,in,which,the,numerical,methods,give,visually,very,similar,results).,Note,the,boundary,effects,caused,by,the,use,of,a,homogeneous,Dirichlet,boundary,condition,for,the,potential,(and,homoge-,neous,Neumann,for,the,other,variables).,Generated,by,Nektar-Driftwave,port,irksome,SU.py,[7].,Figure,9:,Numerical,solution,of,the,vorticity,from,Eqs.4,using,a,64,×,64,mesh,of,square,DG,ele-,ments,with,upwind,numerical,flux,,starting,from,a,Gaussian,density,perturbation,initial,condition.,Plots,from,left,are,t,=,30,,40,,50.,Generated,by,Nektar-Driftwave,port,irksome,DG.py,[7].,13,Figure,10:,Numerical,solution,of,the,vorticity,from,Eqs.4,using,a,64,×,64,mesh,of,square,DG,elements,with,upwind,numerical,flux,and,solving,the,elliptic,equation,for,ϕ,concurrently,with,the,rest,of,the,system,,starting,from,a,Gaussian,density,perturbation,initial,condition.,Plots,from,the,left,are,t,=,30,,40,,50.,Generated,by,Nektar-Driftwave,port,irksome,DG,DAE.py,[7].,than,the,preceding,examples,,implying,that,the,resolved,scale,depends,on,the,timestep,(and,quite,likely,the,accuracy,implied,by,the,scheme,for,updating,ϕ).,This,implementation,was,found,to,be,rather,slow,,taking,approximately,24,hours,to,execute,using,a,single,core,of,a,laptop.,This,im-,plementation,at,least,means,that,fully,higher-order,time-stepping,can,be,accessed,without,further,doctoring.,Listing,8:,UFL,code,defining,the,equations,for,the,2D,Hasegawa-Wakatani,system,including,the,elliptic,solve,(DG,for,advected,fields,and,CG,for,potential,solve).,norm,=,FacetNormal,(,mesh,),F,=,Dt,(w),*,v1,*,dx,+,Dt,(,n,),*,v2,*,dx,\,−,v1,*,d,i,v,(w*,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),),*,dx,−,v2,*,d,i,v,(,n,*,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),),*,dx,\,−,alpha,*,(,phi,−n,),*,(,v1+v2,),*,dx,\,+,kappa,*,grad,(,p,h,i,),[,1,],*,v2,*,dx,\,+,0,.,5,*,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),,,norm,),+abs,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),,,norm,),),),(,’,−,’,),*,(w(,’,−,’,),−,w,(,’,+,’,),),*,v1,(,’,−,’,),*,dS,\,+,0,.,5,*,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),,,norm,),+abs,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),,,norm,),),),(,’,+,’,),*,(w,(,’,+,’,),−,w,(,’,−,’,),),*,v1,(,’,+,’,),*,dS,\,+,0,.,5,*,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),,,norm,),+abs,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),,,norm,),),),(,’,−,’,),*,(,n,(,’,−,’,),−,n,(,’,+,’,),),*,v2,(,’,−,’,),*,dS,\,+,0,.,5,*,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),,,norm,),+abs,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,),[,1,],,,−,grad,(,p,h,i,),[,0,],],),,,norm,),),),(,’,+,’,),*,(,n,(,’,+,’,),−,n,(,’,−,’,),),*,v2,(,’,+,’,),*,dS,\,+,i,n,n,e,r,(,grad,(,p,h,i,),,,grad,(,v3,),),*,dx,\,+,w*,v3,*,dx,The,first,step,towards,extending,this,model,into,a,three-dimensional,plasma,fluid,simulator,is,to,14,Figure,11:,Illustration,showing,incorrect,output,from,Firedrake,when,running,with,four,MPI,pro-,cesses.,Field,on,surface,of,3D,domain,shown.,put,the,2D,dynamics,in,a,3D,domain.,Note,that,this,may,or,may,not,result,in,a,system,symmetric,in,the,new,dimension,,depending,on,the,boundary,conditions.,A,CG,/,SU,version,was,implemented.,In,the,initial,version,of,this,script,,an,inconsistency,was,found,when,running,with,MPI,(Fig.11).,This,was,resolved,by,dialogue,with,the,developers;,the,problem,is,the,discontinuous,value,of,the,facet-tangential,derivative,of,the,potential:,the,interpolation,was,giving,a,different,answer,in,MPI,depending,on,which,cell,was,last,evaluated.,The,solution,is,to,avoid,interpolating,the,drift,velocity,onto,a,Firedrake,function,but,rather,to,substitute,the,expression,directly,into,where,it,is,needed.,In,addition,,it,was,found,that,a,small,softening,length,needed,to,be,added,into,the,denominator,of,the,SU,stabilization,terms,,else,the,potential,solver,would,crash,immediately.,Outputs,from,the,working,solver,are,shown,in,Figs.12,and,13.,Listing,9:,UFL,code,defining,the,equation,system,for,a,MPI-capable,implementation,of,the,2D,Hasegawa-Wakatani,equations,in,3D.,Note,softening,length,in,denominator.,F,=,Dt,(w),*,v1,*,dx,+,Dt,(,n,),*,v2,*,dx,\,−,v1,*,d,i,v,(w*,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,s,),[,1,],,,−,grad,(,p,h,i,s,),[,0,],,,0,.,0,],),),*,dx,−,v2,*,d,i,v,(,n,*,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,s,),[,1,],,,−,grad,(,p,h,i,s,),[,0,],,,0,.,0,],),),*,dx,\,−,alpha,*,(,p,h,i,s,−n,),*,(,v1+v2,),*,dx,\,+,kappa,*,grad,(,p,h,i,s,),[,1,],*,v2,*,dx,\,+,0,.,5,*,h,*,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,s,),[,1,],,,−,grad,(,p,h,i,s,),[,0,],,,0,.,0,],),,,grad,(w,),),−,alpha,*,(,p,h,i,s,−n,),),*,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,s,),[,1,],,,−,grad,(,p,h,i,s,),[,0,],,,0,.,0,],),,,grad,(,v1,),),*,(,1,/,s,q,r,t,(,(,grad,(,p,h,i,s,),[,1,],),*,*,2,+,(,grad,(,p,h,i,s,),[,0,],),*,*,2,+,0,.,0,0,0,1,),),*,dx,\,+,0,.,5,*,h,*,(,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,s,),[,1,],,,−,grad,(,p,h,i,s,),[,0,],,,0,.,0,],),,,grad,(,n,),),−,alpha,*,(,p,h,i,s,−n),−,kappa,*,grad,(,p,h,i,s,),[,1,],),*,d,o,t,(,a,s,v,e,c,t,o,r,(,[,grad,(,p,h,i,s,),[,1,],,,−,grad,(,p,h,i,s,),[,0,],,,0,.,0,],),,,grad,(,v2,),),*,(,1,/,s,q,r,t,(,(,grad,(,p,h,i,s,),[,1,],),*,*,2,+,(,grad,(,p,h,i,s,),[,0,],),*,*,2,+,0,.,0,0,0,1,),),*,dx,\,2.2.2,Periodic,boundary,conditions,Two-,and,three-,dimensional,implementations,with,periodic,boundary,conditions,have,been,writ-,ten.,The,2D,case,took,approx.,58,minutes,to,run,-,slower,than,the,equivalent,Dirichlet-boundary,simulation.,Outputs,are,shown,in,Figs.14,and,15.,15,Figure,12:,Numerical,solution,of,Eqs.4,,showing,density,,using,a,64,×,64,×,16,mesh,further,subdi-,vided,into,tetrahedral,first-order,CG,elements,with,a,SU,correction,term,,starting,from,a,Gaussian,density,perturbation,initial,condition,modulated,by,a,sinusoid,in,the,longitudinal,direction.,Density,plotted,using,the,Nvidia,IndeX,plugin,in,ParaView,,using,a,custom,non-monotonic,colour,map,to,emphasize,the,shape,of,the,flow,features.,Plots,from,left-to-right,then,top-to-bottom,correspond,to,t,=,0,,10,,20,,30,,40,,50.,Generated,by,Nektar-Driftwave,port,irksome,SU,2Din3D.py,[7].,16,Figure,13:,Numerical,solution,of,Eqs.4,,showing,vorticity,,using,the,same,mesh,as,in,Fig.12,of,first-order,CG,elements,with,a,SU,correction,term,,starting,from,a,Gaussian,density,pertur-,bation,initial,condition,modulated,by,a,sinusoid,in,the,longitudinal,direction.,Plots,from,left,are,t,=,0,,10,,20,,30,,40,,50.,Generated,by,Nektar-Driftwave,port,irksome,SU,2Din3D.py,[7].,Figure,14:,Numerical,solution,of,Eqs.4,,showing,vorticity,,using,a,periodic,64,×,64,mesh,of,square,first-order,CG,elements,with,a,SU,correction,term,,starting,from,a,Gaussian,Plots,from,left,are,t,=,30,,40,,50.,Generated,by,density,perturbation,initial,condition.,Nektar-Driftwave,port,irksome,SU,periodic.py,[7].,17,Figure,15:,Numerical,solution,of,Eqs.4,,showing,density,,using,a,64×64×16,mesh,of,first-order,CG,elements,with,a,SU,correction,term,,starting,from,a,Gaussian,density,perturbation,initial,condition,modulated,by,a,sinusoid,in,the,longitudinal,direction.,Plots,from,left,are,t,=,0,,10,,20,,30,,40,,50.,Note,symmetry.,Generated,by,Nektar-Driftwave,port,irksome,SU,2Din3D,periodic.py,[7].,18,2.2.3,Truly,3D,Hasegawa-Wakatani,A,version,of,the,fully,3D,Haswgawa-Wakatani,system,has,been,implemented,and,preliminary,results,have,been,obtained,from,it.,(See,HW3D,SU,periodic.py),2.3,Blob2D,example,in,Firedrake,The,motivation,for,this,work,is,to,enable,debugging,of,some,of,the,examples,implemented,in,Nektar++,that,do,not,yet,work,correctly.,There,are,two,solvers,representing,the,examples,Blob2D-,Te-Ti,and,2D-drift-plane-turbulence-Te-Ti,at,[13].,In,the,present,work,,only,the,simplest,example,,Blob2D,,is,exhibited.,The,explicit,solver,was,constructed,on,the,basis,of,mirroring,closely,the,numerical,implementation,in,the,Nektar-Driftplane,proxyapp.,A,higher-order,explicit,time-stepping,scheme,was,chosen,,after,the,implementation,in,[3],(note,that,this,time-stepper,does,not,use,the,Irksome,package).,A,mesh,of,64*32,square,fourth-order,DG,elements,was,used,,along,with,a,timestep,of,dt,=,2,×,10−3.,Listing,10:,UFL,code,defining,the,equation,system,for,the,Hermes-3,Blob2D,example.,L1,=,−,d,t,*,(,−,v1,*,d,i,v,(w*,d,r,i,f,t,v,e,l,),*,dx,−,v2,*,d,i,v,(,n,*,d,r,i,f,t,v,e,l,),*,dx,\,−,(,p,h,i,s,*,n,/,L,p,a,r,),*,(,v1+v2,),*,dx,\,+,kappa,*,grad,(,n,),[,1,],*,v1,*,dx,\,+,d,r,i,f,t,v,e,l,n,(,’,−,’,),*,(,w(,’,−,’,),−,w,(,’,+,’,),),*,v1,(,’,−,’,),*,dS,\,+,d,r,i,f,t,v,e,l,n,(,’,+,’,),*,(,w,(,’,+,’,),−,w,(,’,−,’,),),*,v1,(,’,+,’,),*,dS,\,+,d,r,i,f,t,v,e,l,n,(,’,−,’,),*,(,n,(,’,−,’,),−,n,(,’,+,’,),),*,v2,(,’,−,’,),*,dS,\,+,d,r,i,f,t,v,e,l,n,(,’,+,’,),*,(,n,(,’,+,’,),−,n,(,’,−,’,),),*,v2,(,’,+,’,),*,dS,),The,outputs,(Fig.16),can,be,compared,with,those,generated,by,the,Nektar-Driftplane,proxyapp;,see,Fig.2,of,[14].,As,in,that,case,,the,equations,simulated,here,have,no,explicit,diffusive,terms,(hence,the,behaviour,is,rather,similar,to,that,of,an,Euler,fluid,with,the,scale,of,feature,generation,being,cut,off,by,numerical,diffusion).,One,advantage,of,Firedrake,is,that,diffusion,terms,are,easily,added,into,the,equations.,2.4,Plasma,kinetics,in,Firedrake,2.4.1,Two-stream,instability,The,Vlasov-Poisson,system,is,implemented,in,Firedrake.,The,code,simulates,the,same,two-,stream,instability,as,described,in,the,report,[15],in,which,a,prototype,Nektar++,solver,was,imple-,mented.,As,noted,in,that,report,,the,problem,can,be,cast,as,an,advection,coupled,to,an,elliptic,solve,,and,it,can,be,linearized,by,solving,the,elliptic,problem,and,using,that,output,to,compute,the,advection,velocity,for,a,separate,advection,time,step.,The,set-up,involves,two,counter-propagating,beams,of,like-charged,particles,on,a,periodic,domain,,along,with,a,static,neutralizing,background,19,Figure,16:,Numerical,solution,of,the,Hermes-3,Blob2D,system,,showing,density,,using,a,64,×,32,mesh,of,square,fourth-order,DG,elements,,starting,from,a,Gaussian,density,perturbation,ini-,tial,condition.,Plots,show,density,,left-to-right,then,top-to-bottom,t,=,0,,2,3,,,2.,Generated,by,Blob2D,explicit.py,[7].,3,,,4,20,to,make,the,Poisson,problem,well-posed.,The,beams,are,given,an,initial,density,perturbation,in,the,form,of,a,sinusoid,with,one,period,across,the,domain.,The,Vlasov-Poisson,system,is,,for,charged,matter,in,a,uniform,neutralizing,background,,˙f,+,v,d2Φ(x),dx2,=,−ω2,P,+,E(x),∂f,∂x,(cid:18)(cid:90),+∞,∂f,∂v,=,0;,(cid:19),−∞,f,dv,vN,−,1,E,=,−,dΦ,dx,;,;,(5),where,the,constant,vN,is,chosen,such,that,the,domain,contains,no,net,charge.,The,constant,ωp,is,the,plasma,frequency,,which,specifies,the,strength,of,the,coupling,and,is,the,only,parameter,n,for,electron,number,in,the,problem,(excepting,initial,data).,Physically,this,translates,into,∝,e,density,n,for,spatially-homogeneous,solutions.,The,function,f,is,the,distribution,function,(DF),for,the,charged,particles,,ρ,is,the,charge,density,,and,Φ,is,the,electrostatic,potential.,Note,that,the,earlier,references,contain,a,sign,error,in,the,RHS,of,the,second,equation,(the,physical,effect,of,which,is,to,make,like,charges,attract,,which,is,correct,for,the,gravitational,case,,where,of,course,the,neutralizing,background,is,unphysical,,but,not,the,electrostatic).,√,A,brief,note,about,the,physics.,These,beams,repel,one,another,and,so,a,concentration,of,charge,density,is,associated,with,slower-moving,particles.,This,is,easily,seen,to,be,the,case,in,the,simulations.,Note,that,this,problem,is,also,the,subject,of,the,grantee,proxyapp,using,the,TensorRegions,exten-,sion,to,the,Nektar++,library,,developed,under,grant,T/AW084/21,which,had,the,aim,of,exploring,the,use,of,Nektar++,for,this,type,of,phase-space,fluid,problem.,The,two-stream,instability,using,TensorRegions,suffers,from,two,outstanding,issues:,firstly,,the,solution,,while,showing,the,correct,general,appearance,,demonstrates,a,circular,oscillation,as,if,the,centre,of,mass,of,the,system,is,undergoing,simple,harmonic,motion;,secondly,,there,are,visible,artifacts,associated,to,element,edges,that,are,not,seen,in,other,implementations.,The,aspiration,of,the,current,work,is,that,it,may,give,a,way,to,diagnose,and,debug,the,Nektar++,code,,e.g.,if,the,same,artifacts,can,be,reproduced,-,this,work,is,ongoing.,The,system,is,implemented,as,an,advection,problem,coupled,to,an,elliptic,solve.,The,elliptic,solve,is,done,outside,of,the,time-stepping,stages,,which,is,expected,to,limit,the,accuracy,of,the,time-,stepping,-,as,before,,this,can,be,remedied,by,constructing,a,DAE,implementation.,The,evaluation,of,the,right-hand,side,of,the,Poisson,equation,involves,the,integration,of,f,over,the,velocity,axis,(this,sort,of,thing,is,central,to,such,kinetic,problems,and,is,technically,a,section,of,the,radon,trans-,form,of,the,phase,space,density),and,this,is,accomplished,using,the,R-space,functionality,of,Fire-,drake,(this,means,that,a,function,-,in,this,case,the,potential,-,in,a,two-dimensional,ExtrudedMesh,can,be,independent,of,one,of,the,dimensions),and,a,projection,of,f,into,the,R-space.,Note,that,unfortunately,this,method,currently,works,only,in,two-dimensions.,The,author,is,not,certain,that,this,projection,mechanism,is,available,for,DG,elements,since,an,error,is,reported,if,trying,to,ob-,tain,the,facet,normals,on,an,ExtrudedMesh,and,so,a,SU-stabilized,CG,implementation,was,used.,The,SU,term,leads,to,significant,damping,and,stabilizes,the,long-term,behaviour,relative,to,the,implementation,described,in,[15],,which,showed,energy,gain,over,the,course,of,the,simulation.,21,Figure,17:,Numerical,solution,of,Eqs.5,using,a,64×256,mesh,of,square,second-order,CG,elements,with,SU,correction,,starting,from,a,bi-Gaussian-beam,initial,condition,,and,using,ω2,P,=,10.,Plots,,L-R,,are,t,=,0,,8,,16,,24,,which,can,be,compared,to,similar,outputs,in,Fig.5,of,[15].,Generated,by,two-stream,instability,SU,repel.py,[7].,The,UFL,for,the,advection,can,be,seen,in,Listing,11.,Listing,11:,UFL,code,defining,the,equation,system,for,the,non-elliptic,part,of,the,Vlasov-Poisson,system,(h,is,the,mesh,cell,size).,F,=,Dt,(,f,),*,v1,*,dx,\,−,v1,*,d,i,v,(,f,*,d,r,i,f,t,v,e,l,),*,dx,\,+,0,.,5,*,h,*,(,d,o,t,(,d,r,i,f,t,v,e,l,,,grad,(,f,),),),*,d,o,t,(,d,r,i,f,t,v,e,l,,,grad,(,v1,),),*,(,1,/,s,q,r,t,(,(,d,r,i,f,t,v,e,l,[,0,],),*,*,2,+,(,d,r,i,f,t,v,e,l,[,1,],),*,*,2,),),*,dx,Note,that,this,problem,has,interesting,stability,properties,that,were,examined,in,Appendix,A.3,of,[14].,These,properties,could,easily,be,studied,further,using,Firedrake.,As,well,as,the,case,of,two,beams,of,like-charged,particles,in,a,neutralizing,background,,it,is,possible,to,consider,the,situation,where,the,beams,have,opposite,charges,i.e.,they,attract,one,another,and,no,neutralizing,background,is,needed.,This,is,easily,implemented,by,having,two,species,(in,the,simple,case,presented,here,the,two,species,have,the,same,mass,,but,this,is,not,necessary,and,stiffness,can,be,introduced,by,making,one,species,electrons,and,the,other,much,heavier,ions).,The,UFL,for,the,advection,in,this,case,is,shown,in,Listing,12.,Listing,12:,UFL,code,defining,the,equation,system,for,the,Vlasov-Poisson,system,in,the,case,of,oppositely-charged,matter,beams.,F,=,Dt,(,f,a,),*,v1,a,*,dx,+,Dt,(,f,b,),*,v1,b,*,dx,\,−,v1,a,*,d,i,v,(,f,a,*,d,r,i,f,t,v,e,l,a,),*,dx,−,+,0,.,5,*,h,*,(,d,o,t,(,d,r,i,f,t,v,e,l,a,,,grad,(,f,a,),),),*,d,o,t,(,d,r,i,f,t,v,e,l,a,,,grad,(,v1,a,),),*,(,1,/,s,q,r,t,(,(,d,r,i,f,t,v,e,l,a,[,0,],),*,*,2,+,(,d,r,i,f,t,v,e,l,a,[,1,],),*,*,2,),),*,dx,\,+,0,.,5,*,h,*,(,d,o,t,(,d,r,i,f,t,v,e,l,b,,,grad,(,f,b,),),),*,d,o,t,(,d,r,i,f,t,v,e,l,b,,,grad,(,v1,b,),),*,(,1,/,s,q,r,t,(,(,d,r,i,f,t,v,e,l,b,[,0,],),*,*,2,+,(,d,r,i,f,t,v,e,l,b,[,1,],),*,*,2,),),*,dx,v1,b,*,d,i,v,(,f,b,*,d,r,i,f,t,v,e,l,b,),*,dx\,22,Figure,18:,Numerical,solution,of,Eqs.5,using,a,64×256,mesh,of,square,second-order,CG,elements,with,SU,correction,,starting,from,a,bi-Gaussian-beam,initial,condition,,and,using,ω2,P,=,10.,Plots,,L-R,,are,t,=,0,,8,,16,,24.,The,plots,are,done,by,differencing,the,density,of,the,two,species,using,a,ParaView,Calculator;,the,plotted,quantity,then,corresponds,to,the,electrostatic,charge,density,in,phase,space.,Generated,by,two-stream,instability,SU,attract.py,[7].,The,potential,is,similar,to,that,observed,in,the,like-charges,case,-,a,near-sinusoid.,This,time,the,two,species,behave,differently,-,because,they,feel,opposite,potential,gradients,,one,species,is,fast,at,the,same,position,that,the,other,one,is,slow.,2.4.2,An,analytic,attractive,equilbrium,Another,system,that,can,be,tried,in,this,solver,is,the,analytic,stationary,solution,found,in,Appendix,A.1,of,[16].,It,must,be,noted,that,,due,to,the,sign,error,mentioned,earlier,,this,is,not,an,electrostatic,solution,but,rather,one,that,applies,in,the,case,of,a,self-attractive,force.,Nevertheless,,it,can,be,studied,numerically,and,has,some,value,as,a,test,case.,The,solution,,which,satisfies,Eqs.5,(but,with,the,sign,of,ω2,P,flipped),,is,f,(x,,v),=,,,vN√,2π,ρ(x),=,d2Φ,dx2,=,1,−,4π2Φ0,ω2,P,(cid:113),Φ0,−,cos,2πx,−,1,2,v2,8π2(cid:113),Φ0,−,cos,2πx,−,1,2,v2,ω2,P,,,+,(cid:18)(cid:90),−ω2,P,(cid:19),−,1,f,dv,vN,=,4π2,cos,2πx;,−ρ.,The,non-singular,case,(Φ0,=,ω2,P,4π2,),is,f,(x,,v),=,(cid:114),√,4,2πvN,ω2,P,ω2,P,4π2,−,cos,2πx,−,1,2,v2.,23,(6),(7),A,quick,test,was,done,on,this,for,ω2,P,=,5π2,and,the,solution,was,time-evolved,for,10,time,units.,Minimal,change,was,seen,in,the,form,of,the,solution,(Fig.19).,Small,perturbations,were,added,to,the,analytic,equilibrium,and,the,solution,was,seen,to,relax,back,to,something,close,to,the,analytic,solution.,Solutions,with,the,integrable,singularity,might,also,be,tested,-,yet,to,be,done.,3,A,3-D,Hasegawa-Wakatani,solver,Report,[17],presented,a,NESO,solver,for,the,2-D,Hasegawa-Wakatani,(hereafter,HW),equations,in,a,3-D,domain.,This,section,describes,how,the,2D-in-3D,Nektar++,EquationSystem,has,been,extended,to,solve,the,full,3-D,HW,equations.,These,equations,are,of,interest,for,two,main,reasons.,Firstly,,as,a,stepping,stone,on,the,path,to,modelling,the,LAPD,problem,,which,is,the,near-term,goal,for,plasma,turbulence,simulation,in,NESO,,and,second,because,they,have,already,been,studied,using,BOUT++,[18],,offering,an,opportunity,to,verify,the,results,with,the,help,of,grantees,who,are,experienced,users,of,that,code.,The,3-D,equations,are,similar,to,the,2-D,form,,but,with,the,constant,adiabacity,parameter,replaced,by,a,second,order,spatial,derivative,in,the,parallel,direction,,that,is:,∂n,∂t,∂ω,∂t,=,−,[ϕ,,n],−,α∇,·,(cid:2)b(∂∥ϕ,−,∂∥n)(cid:3),−,κ,∂ϕ,∂y,=,−,[ϕ,,ω],−,α∇,·,(cid:2)b(∂∥ϕ,−,∂∥n)(cid:3),(8),(9),Note,that,,as,with,the,(modified),2D,Hasegawa,Wakatani,equations,(see,,e.g.,[19]),,n,represents,a,fluctuations,on,a,fixed,background,,often,represented,as,ˆn.,In,the,following,discussion,,we,omit,the,‘hat’,notation,for,brevity.,∂∥,indicates,a,spatial,derivative,parallel,to,the,magnetic,field,,and,[a,,b],is,the,Poisson,bracket,operator,,defined,as,Note,that,,for,a,scalar,field,f,[a,,b],=,∂a,∂x,∂b,∂y,−,∂a,∂y,∂b,∂x,.,[ϕ,,f,],≡,∇,·,(f,vE×B),,(10),(11),so,in,the,code,,the,Poisson,brackets,are,implemented,via,the,Nektar,advection,API,,having,first,calculated,vE×B,from,Φ,with,a,z-axis-aligned,magnetic,field.,As,in,the,“2Din3DHW”,system,,Φ,is,updated,at,each,stage,of,the,Runge,Kutta,algorithm,by,solving,a,Poisson,equation,∇2,⊥Φ,=,ω,24,(12),Figure,19:,Numerical,solution,of,phase-space,density,f,from,7,using,a,128,×,768,mesh,of,square,P,=,5π2.,Left-hand,plots,show,the,initial,data,second-order,CG,elements,with,SU,correction,,for,ω2,and,right-hand,plots,the,state,after,evolving,for,10,time,units;,top,is,phase-space,density,profile,and,bottom,is,the,profile,at,the,central,x-value,of,the,upper,plots.,Note,that,the,interaction,can,be,seen,to,be,attractive,by,the,fact,that,greater,particle,speeds,are,associated,to,greater,particle,density.,Generated,by,Vlasov-Poisson,analytic,SU.py,[7].,25,where,the,⊥,subscript,signifies,that,the,operator,is,restricted,to,the,two-dimensional,subspace,transverse,to,the,magnetic,field.,Implementations,of,the,3-D,HW,equations,in,the,literature,,e.g.,[20],,typically,include,diffusion,or,hyper-diffusion,(∇4),terms,in,both,the,n,and,ω,equations,to,add,stability.,In,our,solver,this,role,is,played,,to,some,extent,,by,numerical,diffusion,associated,with,the,discretisation.,3.1,Implementation,details,The,solver,provides,for,two,ways,of,specifying,parameters,in,the,Nektar++,configuration,file.,Users,may,either,supply,α,and,κ,directly,,or,may,set,physical,parameters,which,are,then,used,to,calculate,α,and,κ,when,the,Nektar,EquationSystem,is,initialised.,In,this,latter,mode,,the,code,first,computes,the,Coulomb,logarithm,for,electron-ion,interactions,,logΛe,i,,based,on,equation,131,of,[21],,then,the,electron,resistivity,via,the,Spitzer,formula,with,a,singly,charged,ion,,η,=,5.2,×,10−5logΛe,i/(T,3/2,),,[22].,α,then,follows,from:,0,α,=,T0,n0ηeωci,,,(13),where,T0,and,n0,are,the,background,electron,temperature,in,eV,and,number,density,in,SI,units,,respectively.,Recall,that,the,turbulent,drive,term,coefficient,,κ,,controls,the,slope,of,the,exponential,background,density,profile.,The,user,may,set,an,approprate,value,for,the,scale,length,,λn,and,the,code,then,computes,κ,,normalising,to,the,ion,gyro-radius,ρs,=,miT0/eB.,√,κ,=,ρs,λn,,,(14),The,parallel,diffusion,terms,in,Equations(8),and,(9),are,implemented,via,the,Nektar++,diffusion,API.,A,matrix,C,can,be,supplied,such,that,,for,a,scalar,field,f,,,Nektar++,computes,The,matrix,coefficients,are,labelled,according,to,∇,·,(C∇f,),.,,,C,=,d00,d01,d02,d01,d11,d12,d02,d12,d22,,,.,(15),(16),Setting,f,=,(Φ,−,n),,d22,=,1,and,all,other,coefficients,to,zero,,therefore,yields,the,required,term,in,the,3-D,HW,equations:,∇,·,(cid:2)b(∂∥ϕ,−,∂∥n)(cid:3),,,(17),with,b,=,(0,,0,,1).,As,is,the,case,for,the,“2Din3D”,solver,,n,and,ω,are,discretised,using,the,Discontinuous,Galerkin,(DG),method,,while,the,potential,,Φ,,is,represented,as,a,Continuous,Galerkin,(CG),field,that,is,26,Figure,20:,A,PlantUML,diagram,showing,the,inheritance,structure,of,EquationSystem,classes,in,the,H3LAPD,(HERMES-3-LAPD),solver.,Note,that,all,member,functions,are,shown,,but,member,variables,are,excluded,for,the,sake,of,brevity.,Image,generated,from,NESO,header,files,using,the,hpp2plantuml,[24],and,plantuml,[25],Python,packages,recomputed,from,ω,at,each,timestep.,n,and,ω,are,evolved,using,an,explicit,4th,order,Runge,Kutta,scheme,and,the,advection,operations,are,stabilised,using,a,first-order,upwinding,scheme.,Since,several,aspects,of,the,numerical,implementation,described,above,are,common,to,both,the,“2Din3D”,and,3-D,HW,equation,systems,,substantial,refactoring,of,the,equation,system,classes,(over,and,above,that,reported,in,M4c.3,[17]),was,undertaken,in,order,to,avoid,code,duplication.,The,refactoring,is,also,intended,to,properly,encapsulate,functionality,relevant,to,equation,systems,that,are,in,development,,notably,those,used,to,solve,the,LAPD,problem,discussed,in,section,2.3.3,of,report,M4c.3.,Figure,20,shows,a,PlantUML,[23],diagram,illustrating,the,new,inheritance,struc-,ture,of,C++,equation,systems,classes,in,the,H3LAPD,solver.,Note,that,only,member,functions,,and,not,member,variables,,are,shown,for,the,sake,of,brevity.,Public,function,names,are,preceded,by,a,green,circle,,protected,functions,by,a,yellow,diamond,,and,private,functions,by,a,red,square.,All,of,the,systems,inherit,from,an,abstract,base,class,called,“DriftReducedSystem”,,which,itself,inherits,from,Nektar++’s,“AdvectionSystem”.,This,base,class,implements,a,number,of,pure,virtual,functions,which,all,derived,classes,must,implement,,including:,•,explicit,time,int:,called,by,a,Nektar,time,integrator,(of,explicit,type),at,each,step,to,compute,terms,on,the,right-hand,side,of,the,time,evolution,equations.,27,(a),Neutral,particle,coupling,(b),Solver,diagnostic,callback,example,Figure,21:,PlantUML,diagrams,showing,the,coupling,between,the,base,equation,system,class,and,the,neutral,particle,system,(a),and,an,example,of,attaching,callbacks,to,a,solver,in,order,to,run,analysis,or,diagnostics,(b).,•,get,phi,solve,rhs:,defines,the,right-hand,side,of,the,Poisson,equation,used,to,calculate,the,potential,•,load,params:,reads,any,class-specific,parameters,from,the,Nektar,configuration,file,•,v,InitObject:,a,function,inherited,from,the,Nektar++,EquationSystem,class,that,carries,out,class-specific,initialisation,at,startup,,including,allocating,temporary,arrays,,instantiating,ob-,jects,from,the,Nektar++,API,,etc.,The,base,class,also,stores,and,initialises,a,system,of,neutral,particles,(see,panel,a,of,Figure,21),,which,can,be,configured,,or,disabled,entirely,,via,parameters,in,the,Nektar,XML,file.,All,derived,classes,then,call,functions,inherited,from,the,base,class,to,interact,with,the,underlying,NESO-,Particles,objects.,An,example,of,a,coupled,simulation,based,on,the,3-D,HW,solver,,including,ionisation,and,recombination,,can,be,found,in,[26].,Two,diagnostic,classes,are,attached,to,the,base,class,in,order,to,monitor,total,fluid,mass,,particle,mass,,energy,and,enstrophy,(panel,b,of,Figure,21).,These,can,be,used,for,analysis,but,are,also,employed,in,integration,tests,of,the,solver,hierarchy,in,order,to,verify,that,(e.g.),the,process,of,ionisation,conserves,mass,in,coupled,solvers,and,that,the,growth,of,energy,and,enstrophy,matches,analytically,expected,rates,for,the,Hasegawa-Wakatani,equations.,“HWSystem”,includes,data,storage,and,implementation,common,to,both,the,2-D,and,3-D,versions,of,the,Hasegawa-Wakatani,equations,,including,variables,to,store,the,α,and,κ,parameters,(de-,fined,slightly,differently,in,the,two,systems),,an,override,of,the,get,phi,solve,rhs,function,,which,retrieves,the,current,values,of,the,vorticity,field,at,each,quadrature,point,and,an,override,of,the,calc,E,and,adv,vels,function,that,passes,values,of,the,vE×B,(calculated,in,the,base,class,imple-,mentation),to,the,electron,advection,object.,Finally,,the,“HW2Din3DSystem”,and,“HW3DSystem”,classes,implement,different,versions,of,ex-,plicit,time,int,,adding,the,appropriate,terms,to,the,n,and,ω,equations,in,each,case.,28,3.2,Results,and,discussion,The,results,described,in,this,section,are,from,a,simulation,run,in,a,cuboid-shaped,domain,using,a,5x5x10,mesh,with,unit,cube,elements.,Element,basis,functions,were,set,to,be,modified,7th,order,Legendre,polynomials,of,the,form,described,by,Karniadakis,and,Sherwin,[27].,The,parameters,α,and,κ,were,set,to,0.1,and,3.5,respectively.,Note,that,this,configuration,does,not,correspond,to,a,specific,physical,scenario,of,interest,,but,was,instead,chosen,to,demonstrate,evolution,to,a,turbulent,state,and,to,test,the,operation,of,the,new,equation,system,in,the,solver.,The,initial,conditions,were,set,to,be,Gaussian,in,the,perpendicular,direction,,with,a,width,of,0.5,normalised,units,,and,sinusoidal,in,the,parallel,direction,with,a,period,one,fifth,of,the,box,length.,Periodic,boundary,conditions,were,used,in,all,three,dimensions.,The,simulation,is,set,up,as,a,NESO,solver,example,,stored,in,the,GitHub,repository,[28],at,examples/H3LAPD/3D-hw,,at,time,of,writing.,The,example,directory,contains,a,Nektar++,configura-,tion,file,and,a,GMSH,file,that,defines,the,characteristics,of,the,mesh.,Utility,scripts,are,provided,to,generate,a,Nektar++-compatible,mesh,from,the,GMSH,file,and,then,run,the,example,in,a,separate,directory,,using,MPI.,3.2.1,Density,evolution,Figure,22,shows,the,evolution,of,the,density,field,captured,at,six,output,times.,In,general,,one,might,expect,the,relatively,high,value,of,κ,to,drive,linear,growth,of,perturbations,via,the,drift-,wave,instability,,which,then,undergo,secondary,instabilities,that,drive,non-linear,growth,,generat-,ing,shear,and,giving,rise,to,zonal,flows.,While,some,evidence,of,turbulent,structure,is,apparent,in,the,final,output,,at,t=40,,it,appears,to,be,somewhat,less,prevalent,than,in,simulations,run,with,the,’2Din3D’,implementation,of,the,solver,(c.f.,Figure,6,in,[17]),,presumably,due,to,the,addition,of,the,diffusive,term,in,the,field-parallel,direction.,Examples,of,similar,simulations,in,the,literature,(e.g.,[19]),show,that,,in,general,,a,steady,state,is,reached,in,which,the,resistive,dissipation,(controlled,by,α),balances,the,turbulent,drive,term.,Note,,however,,that,results,in,Section,3.2.2,suggest,that,such,a,state,has,not,yet,been,reached,by,t=40.,3.2.2,Energy,and,enstrophy,Two,quantities,of,interest,in,HW,simulations,are,the,total,system,energy,and,enstrophy.,These,can,be,calculated,with,integrals,over,the,domain:,E,=,W,=,(cid:90),(cid:90),1,2,1,2,(n2,+,|∇⊥ϕ|2),dx,(n,−,ζ)2,dx,29,(18),(19),(a),t,=,0,(b),t,=,10,(c),t,=,17.5,(d),t,=,25,(e),t,=,32.5,(f),t,=,40,Figure,22:,Evolution,of,the,electron,density,in,the,3-D,Hasegawa-Wakatani,solver,at,several,output,times.,30,Figure,23:,Rates,of,energy,and,enstrophy,growth,,in,normalised,units,,for,the,first,100,steps,of,the,simulation.,Dashed,lines,indicate,analytic,expected,values,in,each,case,,calculated,based,on,Equations(20),and,(21).,As,explained,in,reference,[19],,the,energy,and,enstrophy,are,expected,to,evolve,at,rates,where,dE,dt,dW,dt,=,Γn,−,Γα,=,Γn,(cid:90),α[,Γα,=,(cid:90),Γn,=,−κ,∂,∂z,(n,−,ϕ)]2,dx,n,∂ϕ,∂y,dx,(20),(21),(22),(23),In,order,to,study,these,quantities,with,NESO,,a,solver,callback,was,added,to,the,HWSystem,class,to,compute,E,,W,,,Γα,and,Γn,at,each,step,and,write,them,to,file.,Figure,23,shows,the,rates,of,change,of,E,and,W,over,the,first,100,steps,of,the,simulation,(in,the,linear,growth,phase),,together,with,the,expected,values,calculated,from,Equations(20),and,(21).,The,former,are,calculated,by,differencing,each,quantity,between,consecutive,steps,and,dividing,by,the,system,timestep.,Both,quantities,grow,at,a,rate,very,close,to,the,expected,values,,verifying,,to,a,degree,the,implementation,of,the,“3DHW”,equation,system.,In,Figure,24,,the,evolution,of,the,total,energy,is,plotted,over,the,full,simulation,duration.,In,a,steady,state,,where,resistive,dissipation,matches,turbulent,drive,from,the,background,density,profile,,one,expects,Γα,=,Γn,,such,that,the,total,energy,saturates.,In,this,simulation,it,is,clear,that,the,system,is,still,in,flux,at,t,=,40,and,has,not,yet,reached,the,saturation,point.,31,Figure,24:,Evolution,of,the,total,system,energy,over,the,full,duration,of,the,simulation.,Time,and,energy,are,plotted,in,normalised,units.,32,3.3,Summary,The,3-D,Hasegawa-Wakatani,equations,have,been,implemented,in,NESO,by,adding,a,new,Nek-,tar++,equation,system,to,the,“H3LAPD”,solver.,Substantial,refactoring,work,has,been,undertaken,to,avoid,code,duplication,and,to,facilitate,coupling,between,the,various,H3LAPD,systems,and,an,ensemble,of,kinetic,neutrals,modelled,with,NESO-Particles,,an,example,of,which,is,demonstrated,in,report,[26].,Analysis,of,the,total,energy,and,enstrophy,in,our,example,simulation,demonstrates,good,agree-,ment,with,the,evolution,predicted,analytically,,and,reveals,that,the,total,duration,may,be,slightly,too,short,to,reach,a,steady,state.,Further,verification,of,our,3-D,HW,implementation,is,already,underway,via,a,comparison,to,the,results,presented,by,the,York,grantees,in,report,[29].,4,Producing,Field-Aligned,Meshes,This,section,describes,the,work,done,to,produce,field-aligned,meshes,for,calculating,anisotropic,diffusion.,In,finite,difference,(and,low-order,finite-element),codes,it,has,been,found,that,anisotropic,diffusion,can,only,be,solved,accurately,if,the,mesh,is,aligned,with,the,magnetic,field.,Otherwise,the,solver,will,converge,to,the,wrong,solution.,This,issue,may,be,less,significant,for,higher-order,finite,differ-,ence,basis,functions,,but,there,could,still,be,performance,benefits,if,a,lower-order,basis,could,be,used,to,achieve,the,same,level,of,accuracy.,While,it,is,possible,to,generate,meshes,where,elements,follow,magnetic,field-lines,,the,problem,that,arises,is,that,,in,general,,field,lines,will,not,form,a,closed,loop,after,a,single,circuit,of,a,tokamak.,This,means,there,would,be,a,discontinuity,at,ϕ,=,0,,2π,where,faces,of,adjacent,elements,would,not,align.,In,finite,difference,codes,the,toroidal,direction,is,not,actually,aligned,with,the,field,but,interpolation,is,used,when,taking,derivatives,along,the,field-lines.,A,similar,approach,can,be,used,for,finite,element,methods.,First,,a,poloidal,mesh,is,generated,in,the,plane,with,ϕ,=,0.,The,field,line,intersecting,each,control,point,in,that,mesh,is,then,followed,to,±∆ϕ/2,,where,∆ϕ,is,the,toroidal,grid,spacing.,These,field,lines,will,make,up,the,edges,of,one,“layer”,of,elements.,Successive,layers,will,be,identical,except,rotated,a,suitable,angle,in,the,toroidal,direction.,In,general,the,poloidal,faces,of,elements,in,adjacent,layers,will,not,align,and,are,said,to,be,“nonconformal”.,There,are,established,algorithms,to,interpolate,values,from,nonconformal,elements,when,using,the,discontinuous,Galerkin,method[30].,DG,is,less,efficient,than,Continuous,Galerkin,for,elliptical,problems,such,as,diffusion.,In,production,,however,,there,will,also,be,hyperbolic,terms,in,the,equations,,at,which,point,it,will,be,necessary,to,use,DG,anyway.,Note,that,it,is,assumed,that,the,equilibrium,magnetic,field,is,axisymmetric,,i.e.,,does,not,depend,on,ϕ.,This,means,that,the,shape,of,elements,only,needs,to,be,computed,for,one,layer;,all,layers,would,be,the,same,except,rotated,in,ϕ.,The,techniques,outlined,in,this,section,could,be,modified,to,work,without,axisemmetry,but,this,would,be,more,computationally,expensive,as,every,layer,would,need,to,be,calculated,individually.,33,4.1,NESO-fame,It,is,necessary,to,generate,a,mesh,that,can,be,used,by,Nektar++.,Gmsh,meshes,can,be,converted,to,the,Nektar++,format,and,,if,necessary,,undergo,further,manipulations,using,the,NekMesh,tool.,This,is,the,workflow,which,is,normally,used.,Unfortunately,,Gmsh,and,most,other,existing,software,for,generating,meshes,does,not,allow,one,to,specify,the,exact,arrangement,and,alignment,of,elements,within,the,domain.,This,meant,it,was,not,suitable,to,ensure,the,elements,are,field-,aligned.,Instead,,new,software,had,to,be,written.,This,was,called,NESO-fame,,where,“fame”,stands,for,“field-aligned,mesh,extrusion”.,Data,on,equilibrium,magnetic,fields,for,tokamaks,are,usually,distributed,using,the,G-EQDSK,format.,To,avoid,needing,to,implement,a,new,reader,for,EQDSK,files,,the,existing,hypnotoad,package,was,used.,This,software,was,designed,to,generates,meshes,for,the,BOUT++,finite,dif-,ference,library[31].,In,addition,to,reading,the,EQDSK,file,,it,can,be,used,to,generate,(most,of),the,2D,poloidal,mesh,from,which,the,3D,mesh,will,be,extruded.,The,only,disadvantage,of,using,hypnotoad,is,that,it,is,released,under,the,GNU,General,Public,License,,meaning,that,NESO-fame,also,had,to,use,this,license.,However,,NESO-fame,is,sufficiently,ancillary,to,the,main,NEPTUNE,codebase,that,this,was,considered,an,acceptable,compromise.,Furthermore,,it,remains,possi-,ble,to,write,a,replacement,for,the,hypnotoad,functions,used,in,fame,in,the,future,,if,it,becomes,necessary,to,distribute,the,latter,under,a,more,permissive,license.,hypnotoad,is,written,in,Python,,so,the,NekPy,Python,bindings,were,used,to,produce,Nektar++,meshes,from,it.,This,required,a,few,additional,class,and,method,bindings,to,be,added,to,NekPy,,particularly,for,nonconformal,interfaces.,4.2,2D,and,Simple,3D,Meshes,Initial,work,focused,on,very,simple,2D,and,3D,meshes.,For,further,simplicity,,these,used,Carte-,sian,coordinates,rather,than,cylindrical.,NESO-fame,would,first,generate,a,structured,1D,or,a,2D,mesh.,It,would,take,the,angle,between,the,magnetic,field,and,the,normal,direction,of,this,mesh,and,extrude,each,element,along,that,angle.,The,user,would,specify,how,many,“layers”,there,would,be,in,the,direction,of,the,field.,Between,each,layer,is,a,nonconformal,interface,(see,Figures,25b,and,26b).,Each,layer,can,be,further,subdivided,into,conformal,elements,in,the,field,direction,,if,desired,(see,Figures,25c,and,26c).,This,makes,it,possible,to,experiment,to,determine,what,frequency,of,conformal,and,nonconformal,interfaces,can,give,the,best,accuracy,for,the,least,computational,cost.,It,was,planned,to,use,the,existing,Nektar++,anisotropic,diffusion,mini-app,[32],to,compare,the,accuracy,of,2D,meshes,with,no,field,alignment,,with,fully-conformal,field,alignment,,and,with,field,alignment,and,non-conformal,interfaces.,The,computational,cost,of,each,of,these,would,also,be,compared.,However,,it,was,discovered,that,the,mini-app,only,worked,with,the,Continuous,Galerkin,method,and,that,Nektar++,did,not,support,Neumann,boundary,conditions,with,the,Discontinuous,Galerkin,method.,These,only,became,available,in,the,final,weeks,of,the,NEPTUNE,project.,Sup-,port,for,3D,meshes,in,the,mini-app,was,not,available,at,the,time,of,writing.,34,(a),Unaligned,,Conformal,(b),Aligned,,Nonconformal,(c),Aligned,,Subdivided,Figure,25:,Examples,of,various,types,of,2D,meshes,which,are,aligned,(or,not),to,a,magnetic,field,with,a,2◦,angle,to,the,x-axis.,4.3,Generating,a,Poloidal,Mesh,While,awaiting,the,changes,to,Nektar++,necessary,to,benchmark,anisotropic,diffusion,in,the,2D,case,,work,continued,to,generate,realistic,tokamak,meshes.,The,first,step,of,this,was,to,produce,a,useful,2D,poloidal,mesh.,The,starting,point,was,a,poloidal,mesh,generated,by,hypnotoad,from,a,G-EQDSK,file.,hypnotoad,provides,a,large,number,of,configurable,parameters,and,a,graphical,user,interface,which,allows,one,to,experiment,with,the,values,of,these,and,preview,the,result.,These,meshes,are,made,up,of,quads.,Two,edges,of,each,quad,follow,flux,surfaces,and,the,other,two,are,(by,default),perpen-,dicular,to,flux,surfaces.,This,mesh,can,be,brought,arbitrarily,close,to,the,core,of,the,tokamak,,but,will,always,have,a,hole,in,the,centre.,The,mesh,will,not,conform,to,the,wall,of,the,tokamak,(the,shape,of,which,will,typically,be,included,in,a,G-EQDSK,file).,Some,portions,of,it,may,protrude,past,the,wall,,while,in,other,regions,there,may,be,a,gap,between,the,mesh,and,the,wall;,compare,Figures,27a,and,27b.,4.4,Filling,the,Gaps,Filling,the,hole,in,the,centre,was,straightforward.,The,quad,edges,closest,to,the,centre,are,used,as,one,edge,of,a,triangular,element,,with,the,other,edges,formed,by,perpendicular,segments,connecting,to,the,O-point.,Due,to,numerical,difficulties,,these,edges,deviate,slightly,from,perpen-,dicular,when,very,near,the,centre,,instead,taking,the,shortest,path,in,Euclidean,space,for,the,rest,of,the,distance.,Note,that,extending,the,mesh,into,the,centre,is,optional;,if,this,region,is,not,of,interest,then,a,boundary,condition,can,be,imposed,on,the,inner,edge,of,the,annulus,instead.,Next,,any,vertices,of,the,hypnotoad,mesh,which,lie,outside,the,tokamak,wall,are,eliminated.,The,user,may,also,specify,some,minimum,distance,between,these,vertices,and,the,wall;,any,vertices,nearer,than,that,will,also,be,removed.,It,is,necessary,to,follow,the,field,lines,starting,at,these,vertices,(see,next,section),,to,ensure,that,at,no,point,will,the,edge,of,the,element,extruded,from,this,node,pass,outside,the,wall,or,within,the,minimum,distance,to,the,wall.,The,meshpy,library,,with,35,(a),Unaligned,,Conformal,(b),Aligned,,Nonconformal,(c),Aligned,,Subdivided,Figure,26:,Examples,of,various,types,of,3D,meshes,which,are,aligned,(or,not),to,a,magnetic,field,with,a,2◦,angle,from,the,x-axis,towards,the,y-axis,and,a,2◦,angle,from,the,x-axis,towards,the,z-axis.,36,(a),Basic,hypnotoad,mesh,(b),Fitted,to,the,tokamak,vessel,(c),Adjusted,wall,resolution,(d),Merged,very,narrow,elements,Figure,27:,Poloidal,meshes,from,which,3D,meshes,can,be,extruded,,demonstrating,how,they,are,adjusted,from,the,output,of,hypnotoad.,(a),The,mesh,generated,by,hypnotoad.,(b),The,previous,mesh,but,restricted,to,inside,the,tokamak,vessel,,with,any,empty,spaces,filled,in,with,triangles.,(c),The,same,,except,with,the,wall,resolution,adjusted,to,more,closely,match,that,of,the,main,mesh.,(d),Very,narrow,elements,above,the,x-point,have,been,merged,,so,this,region,is,not,over-resolved.,37,the,triangle,code,as,a,backend,,was,then,used,to,produce,triangles,to,fill,the,space,between,the,remaining,hypnotoad,vertices,and,the,tokamak,wall.,An,example,of,applying,these,adjustments,can,be,found,in,Figure,27b.,4.4.1,The,Wall,The,information,on,the,shape,of,the,wall,provided,in,the,G-EQDSK,data,consists,of,a,series,of,lin-,ear,segments.,The,length,of,these,segments,can,vary,significantly,and,may,be,much,smaller,than,the,resolution,of,the,mesh,being,generated.,Generating,a,high,quality,mesh,therefore,requires,adjusting,the,shape,and,resolution,of,the,wall,,before,generating,the,triangular,elements,between,it,and,the,hypnotoad,mesh.,The,user,may,specify,a,target,wall,resolution.,This,is,expressed,as,a,multiple,of,the,average,size,of,the,hypnotoad,mesh,edges,that,are,closest,to,the,wall.,The,user,will,also,specify,the,minimum,size,of,any,sharp,angles,which,should,be,preserved,after,regridding.,Adjacent,line,segments,are,put,into,groups,,separated,by,sharp,angles,above,the,user-provided,threshold.,The,total,length,of,each,group,in,space,is,calculated.,If,a,group,has,a,length,less,than,0.1,of,the,target,size,then,it,is,deleted,and,the,ends,of,adjacent,groups,moved,to,its,centre.,Cubic,splines,are,used,to,interpolate,the,shape,of,the,remaining,groups.,A,number,of,curved,segments,,as,close,to,the,target,length,as,possible,,will,be,created,for,each,group.,The,length,of,the,segments,created,from,the,same,group,will,be,equal,,but,in,general,the,length,of,segments,from,different,groups,will,not,be.,An,example,of,a,mesh,with,a,regridded,wall,can,be,found,in,Figure,27c.,4.4.2,Improving,Mesh,Quality,All,of,these,2D,elements,may,be,higher-order,(i.e.,,have,non-linear,edges).,This,allows,the,flux,surfaces,and,perpendiculars,to,be,followed,more,accurately.,The,triangles,near,the,wall,of,the,tokamak,will,mostly,have,straight,edges,,except,where,they,adjoin,the,hypnotoad-generated,mesh,or,the,wall,itself.,As,the,triangle,code,assumes,first-order,elements,it,can,sometimes,produce,shapes,that,are,self-intersecting,once,curved,edges,are,applied.,To,guard,against,this,,the,Jaco-,bian,of,each,triangle,adjacent,to,the,wall,is,checked.,Those,with,negative,Jacobians,(indicating,self-intersection),may,be,merged,with,an,adjacent,triangle,to,produce,a,quad,or,,if,no,such,quad,is,well-formed,,forced,to,be,first-order.,Lines,that,are,perpendicular,to,the,flux,surfaces,tend,to,diverge,near,the,X-point.,Ensuring,there,is,adequate,resolution,in,that,region,will,often,result,in,overly-resolved,areas,towards,the,O-point,,where,these,lines,converge.,Columns,of,elements,radiating,away,from,the,X-point,and,towards,the,O-point,are,checked,to,see,if,elements,become,extremely,narrow.,If,the,ratio,between,their,length,and,width,exceeds,some,threshold,(set,by,the,user),,they,are,merged,with,adjacent,elements.,This,means,computational,resources,will,not,be,wasted,with,unnecessary,resolution.,An,example,of,a,mesh,to,which,this,process,has,been,applied,can,be,found,by,comparing,Figure,27c,to,Figure,27d.,38,4.5,Following,the,Field,The,path,of,a,field,line,is,the,solution,to,the,following,system,of,differential,equations:,dR,dϕ,dZ,dϕ,=,=,BR(R,,Z),Bϕ(R,,Z),BZ(R,,Z),Bϕ(R,,Z),,,.,(24),A,solution,would,have,the,form,R(R0,,Z0,,ϕ),,Z(R0,,Z0,,ϕ),,where,R0,and,Z0,are,the,coordinates,where,the,field,line,intersects,the,poloidal,plane,at,ϕ,=,0.,In,general,there,will,not,be,a,closed,form,for,this,solution,and,it,must,be,integrated,numerically,from,the,G-EQDSK,equilibrium,data.,To,extrude,the,2D,elements,into,the,toroidal,direction,,this,integral,is,evaluated,starting,at,the,control,points,on,each,of,the,edges,of,the,toroidal,mesh.,The,integral,is,taken,from,ϕ,=,0,to,ϕ,=,±∆ϕ/2,,where,∆ϕ,is,the,spacing,of,elements,in,the,toroidal,direction.,The,distance,travelled,along,the,field,line,is,also,calculated.,The,results,for,each,field,line,are,then,interpolated,against,the,normalised,distance,travelled,,to,ensure,that,any,new,control,points,are,equally-spaced,on,the,line.,This,will,not,work,for,elements,falling,between,the,mesh,generated,by,hypnotoad,and,the,tokamak,wall,,as,the,field,line,may,leave,the,tokamak,vessel.,The,edges,of,these,elements,are,therefore,left,unaligned,,with,R(ϕ),=,R0,and,Z(ϕ),=,Z0.,Elements,immediately,outside,the,hypnotoad,mesh,will,have,some,edges,that,are,field-aligned,and,some,that,are,not.,This,can,sometimes,cause,the,element,to,be,self-intersecting.,It,is,also,anticipated,that,the,sudden,transition,might,cause,numerical,artefacts.,To,avoid,this,,the,user,can,specify,the,number,of,steps,to,take,between,elements,that,are,fully,field-aligned,and,fully,unaligned.,Elements,within,the,hypnotoad,mesh,but,near,its,edge,will,have,edges,that,are,the,weighted,average,between,the,aligned,and,unaligned,shape,,with,the,weight,of,the,unaligned,component,rising,as,the,edge,is,approached.,It,is,this,weighted,average,that,is,used,when,evaluating,whether,a,vertex,and,its,corresponding,field,line,lie,within,the,tokamak,vessel,(see,Section,4.4).,An,example,of,the,meshes,generated,by,this,process,can,be,found,in,Figure,28.,5,Grantee,Deliverable,Summaries,Each,section,below,summarises,the,deliverables,received,from,a,grantee,,with,one,report,or,code,deliverable,per,subsection.,A,link,is,provided,in,the,title,to,the,location,of,the,accepted,report,,where,applicable.,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.,5.1,Reports,received,under,Grant,T/AW086/22,,PO,2070839,The,Oxford,grantee,produced,the,following,reports,39,Figure,28:,The,field-aligned,mesh,for,a,quadrant,of,a,tokamak.,On,the,left,is,the,full,mesh,,while,on,the,right,some,of,the,elements,have,been,hidden,to,give,a,view,of,the,mesh,interior.,Edges,and,quads,on,the,surface,of,the,tokamak,are,not,field,aligned,and,are,therefore,conformal.,This,contrasts,with,the,internal,elements.,40,5.1.1,Report,2070839-TN-03,[33],The,report,examines,how,best,to,formulate,the,calculation,of,the,electrostatic,potential,ϕ,when,solving,the,drift,kinetic,equations,,to,ensure,the,potential,is,accurately,and,stably,computed,nu-,merically.,Use,of,the,Boltzmann,response,to,calculate,ϕ,may,give,rise,to,a,significant,error.,The,first,requirement,is,therefore,a,way,of,deciding,when,the,Boltzmann,response,is,inadequate,,which,is,addressed,by,a,theory,that,makes,a,perturbation,to,the,Boltzmann,response,in,the,estimation.,If,the,perturbation,is,small,,then,it,is,used,to,correct,the,Boltzmann,response,,otherwise,a,far,more,complicated,formulation,leading,to,the,electron,fluid,equations,for,electron,density,,velocity,and,temperature,is,argued,to,be,necessary.,One,truncated,variant,is,shown,to,be,equivalent,to,drift-reduced,Braginskii.,The,analytic,results,in,the,report,should,be,valuable,for,ensuring,accurate,and,efficient,numerical,solution,of,drift,kinetic,equations.,5.1.2,Report,2070839-TN-04,[34],The,report,describes,work,designed,to,test,using,the,Method,of,Manufactured,Solutions,(MMS),,correct,implementation,of,the,2-D,drift,kinetic,model,introduced,in,previous,reports,from,Oxford,,and,restated,in,the,current,report.,This,work,has,exposed,limitations,both,of,high,order,finite,elements,and,of,MMS,,one,of,the,latter,being,somewhat,subtle.,Dealing,first,with,the,limitations,of,high,order,,the,peculiar,nature,of,the,boundary,condition,implies,that,the,chosen,solution,becomes,zero,at,off-node,points,,ie.,effectively,discontinuous.,Moreover,,solutions,corresponding,to,realistic,source,terms,are,expected,to,have,the,sheath,singularity,dependence,,ϕ,∝,z,,where,ϕ,is,the,electric,potential.,Spectral,convergence,is,only,to,be,expected,when,approximating,a,smooth,solution.,Otherwise,,two,strategies,to,obtain,better,convergence,are,possible,,the,first,(adopted),is,to,use,lower,order,elements,,but,more,of,them,,and,this,was,successful.,The,second,option,,particularly,applicable,to,a,known,form,of,singularity,,is,to,use,singular,elements,,treated,in,detail,by,Strang,and,Fix,[35],,as,referenced,by,Boyd,[36,,§,16.6].,This,suggestion,is,added,to,the,list,of,options,for,further,exploration,and,testing,of,the,model,implementation,at,higher,accuracy.,√,The,first,limitation,of,MMS,exposed,by,the,study,is,inappropriateness,of,the,function,chosen,,which,does,not,have,the,expected,singular,potential,behaviour.,Secondly,,evidence,is,presented,of,the,more,subtle,issue,that,the,source,implied,by,the,solution,chosen,tends,to,provoke,a,short,wavelength,instability.,Convergence,is,restored,by,appropriately,chosen,numerical,dissipation,,again,of,course,at,the,expense,of,accuracy.,Nonetheless,,it,is,hard,to,doubt,the,correctness,of,coding,and,likely,fidelity,of,simulation,to,be,achieved,using,the,program,,for,which,a,link,to,the,repository,is,given,in,Appendix,A.,5.1.3,Report,2070839-TN-05,[37],Report,2070839–TN–05,continues,the,series,of,high-quality,reports,produced,by,Oxford,for,the,project,,with,clear,explanations,,notably,as,to,why,particular,options,have,been,selected.,The,report,concentrates,on,the,treatment,of,the,electrons,,leaving,description,of,the,kinetic,modelling,of,the,ion,and,neutral,species,to,previous,reports.,A,further,assumption,,made,for,the,purposes,of,the,Oxford,development,,is,that,not,only,are,the,models,1-D,in,space,,but,also,in,velocity,41,space,1d1v,,where,the,relevant,component,is,parallel,to,the,magnetic,field,,which,is,here,taken,to,be,uniform,and,straight,in,the,z-direction.,The,electron,treatment,is,moment-based,,implying,a,set,of,fluid-like,equations,for,mass/charge,,momentum,and,energy.,The,electron,momentum,balance,gives,the,electric,field,which,can,other-,wise,be,hard,to,calculate.,Boundary,conditions,are,chosen,so,that,the,electrons,have,the,same,density,and,velocity,as,the,ions.,Tests,show,that,making,zero,flux,assumptions,,an,electron,Boltz-,mann,response,is,recoverable,,and,that,using,the,Braginskii,closure,a,steady,state,is,achievable,,although,the,E-field,is,noisy.,5.1.4,Report,2070839-TN-06,[38],The,report,2070839–TN–06,continues,the,series,of,high-quality,reports,produced,by,the,Oxford,grantees,for,the,project,,with,clear,explanations,,notably,as,to,why,particular,options,have,been,selected.,The,model,is,1D2V,for,the,ions,,which,are,coupled,to,Boltzmann,electrons,as,in,previous,Oxford,reports,,the,novelty,here,being,the,collision,operator,for,the,ions.,Two,different,operators,-,the,simple,Krook,and,a,more,complicated,gyro-averaged,Fokker-Planck,type,-,are,introduced,as,indicated,in,the,report,title.,The,simple,Krook,operator,is,used,in,the,range,of,Knudsen,numbers,0.1,to,100,where,both,ac-,curate,modelling,and,intuition,about,likely,results,are,hard,to,come,by.,Nonetheless,,there,is,a,satisfactory,test,of,the,code,by,the,Method,of,Manufactured,Solutions,(MMS),,and,physically,plau-,sible,‘smooth’,solutions,are,found,for,a,Maxwellian,particle,source,,in,the,presence,of,a,small,artificial,viscosity.,There,are,numerical,challenges,associated,with,the,latter,Fokker-Planck,opera-,tor,since,it,leads,to,fourth,order,derivatives,,and,its,calculation,involves,integration,over,the,infinite,range,and,an,inverse,square-root,singularity,at,one,integration,limit.,Progress,has,been,made,in,the,understanding,of,these,issues.,5.1.5,Report,2070839-TN-07,[39],The,report,2070839–TN–07,describes,in,sufficient,detail,for,implementation,,the,use,of,a,high,order,basis,to,treat,the,Rosenbluth-MacDonald-Judd,form,of,the,Fokker-Planck,collision,operator,for,charged,particle,species.,The,basis,consists,of,the,direct,product,of,Lagrange,polynomials,in,velocity,coordinatesr,v∥,and,v⊥,defined,on,Lobatto,points,except,for,the,element,including,v⊥,=,0.,Since,the,operator,involved,is,both,nonlinear,and,integro-differential,,and,there,are,chal-,lenging,boundary,conditions,at,infinity,,as,well,as,the,usual,issues,expected,at,the,origin,of,a,radial,coordinate,v⊥,,the,analysis,involves,approximately,100,numbered,equations.,Preliminary,testing,and,timing,of,the,initialisation,and,a,single,step,calculation,using,the,operator,is,success-,fully,performed,,with,results,that,the,subsequent,discussion,concludes,are,encouraging,for,further,implementation.,42,5.1.6,Report,2070839-TN-08,[40],The,report,2070839–TN–08,describes,openly,and,honestly,progress,made,in,pursuit,of,a,challeng-,ing,technical,objective.,The,objective,of,a,more,accurate,electron,distribution,function,is,pursued,further,in,report,2070839–TN–10,,of,which,see,the,summary,below,for,more,technical,details,of,the,objective,and,the,latest,progress.,5.1.7,Report,2070839-TN-09,[41],The,report,2070839–TN–09,describes,the,introduction,into,the,ExCALIBUR,drift-kinetic,(“Ox-,ford”),model,of,a,magnetic,field,of,spatially,varying,strength,(formerly,only,field,direction,varied).,This,z-dependence,of,Bz,leads,to,more,complex,ion,dynamics,,whilst,the,electrons,are,treated,as,adiabatic.,It,is,argued,that,in,the,1-D,and,2-D,situations,considered,,it,is,adequate,to,demonstrate,magnetic,mirroring,without,forcing,the,field,to,be,solenoidal.,The,Method,of,Manufactured,Solu-,tions,is,used,to,test,the,resulting,software,,demonstrating,spectral,convergence,in,both,one-,and,two-,space,dimensions,,where,in,the,latter,case,the,solution,is,assumed,to,be,periodic,in,the,radial,coordinate.,The,success,of,the,tests,bodes,well,for,more,complicated,open,field,geometries.,5.1.8,Report,2070839-TN-10,[42],The,report,2070839–TN–10,returns,to,a,problem,attacked,unsuccessfully,in,previous,work,[43],,viz.,to,find,an,electron,particle,distribution,function,fe,by,replacing,use,of,the,Boltzmann,assump-,tion,with,a,less,prescriptive,formulation.,Indeed,,the,new,model,for,evolution,of,fe,is,almost,as,de-,tailed,as,for,the,ion,species,fi,,the,major,difference,being,the,assumption,that,fe,may,be,evolved,in,pseudo-time,to,find,a,steady-state.,The,new,model,equation,(42),for,ge,∝,fe,is,inevitably,of,a,complicated,integro-differential,form,with,boundary,conditions,that,are,linked,to,the,fi,distribu-,tion,via,the,electric,potential.,The,boundary,conditions,are,reformulated,to,impose,a,zero,parallel,current.,Following,implementation,of,the,new,boundary,conditions,and,other,changes,to,the,coding,,Com-,putations,assuming,up-down,spatial,symmetry,obtain,physically,reasonable,fe,except,very,near,the,domain,boundaries,where,a,finite,grid-scale,oscillation,is,observed.,Thus,,the,work,demon-,strates,significant,progress,towards,more,accurate,calculation,of,electron,behaviour,,but,indicates,that,more,research,is,still,needed.,5.2,Reports,and,code,deliverables,received,under,Grant,T/AW085/22,,PO,2068435,The,KCL,grantee,produced,the,following,reports,and,code,deliverables,43,5.2.1,Report,2068435-TN-01,[44],The,report,2068435-TN-01,presents,in,Section,2,,Nektar++,implementations,of,three,HERMES-3,examples,in,2-D,,viz.,Blob2D,,Blob2D-Te-Ti,,and,Blob2D-Te-Ti-Turb,,the,last,of,these,in,pre-,liminary,form.,All,have,domains,that,are,simple,square,regions,,but,the,equations,become,more,numerous,and,complicated,with,each,example.,Section,3,contains,an,explanation,of,the,various,Nektar++,components,needed,to,implement,a,Discontinuous,Galerkin,(DG),solver,and,their,use,in,the,API.,Section,4.1,contains,implementation,details,including,two,alternative,ways,of,solving,the,Poisson,problem,in,the,Blob2D-Te-Ti,case;,one,method,works,while,the,other,does,not.,Further,work,is,needed,to,understand,the,reason,for,this,,possibly,there,are,problems,with,the,dimen-,sionless,scaling.,Section,4.2,considers,an,issue,with,the,initial,version,of,the,Blob2D,proxyapp,,in,that,the,numerical,flux,implementation,in,the,DG,solver,was,not,fully,conservative,,however,little,difference,is,observed,after,modification.,Section,5,suggests,further,work,,incidentally,mentioning,changes,to,help,developers,deal,with,multi-species,problems.,5.2.2,Report,2068435-TN-02,[45],The,report,2068435-TN-02,discusses,improvements,made,to,the,Nektar++,mesh,generation,It,contains,links,to,the,relevant,merge,requests,in,the,toolkit,that,cover,many,different,aspects.,Nektar++,repository,for,the,new,and,corrected,code.,The,report,describes,the,following:,1.,Completion,of,work,on,r-adaptation,of,mesh,(ie.,the,movement,of,finite,elements,according,to,suitable,criteria,,without,introduction,of,additional,elements),,with,an,associated,figure,showing,anisotropic,refinement,of,a,grid,around,a,circle.,2.,Usability,improvements,with,relation,to,mesh,quality,-,statistics,of,the,values,of,mesh-map,Jacobians,are,printed,out,to,the,screen,including,graphics,showing,the,distribution,of,Jaco-,bians.,3.,Bug,fixes,and,improvements,for,domains,incorporating,periodicity,have,made,periodic,do-,mains,with,pyramidal,cells,more,robust,,which,should,improve,the,situation,where,oscillations,would,emanate,from,periodic,boundaries.,4.,Improvements,to,isoparametric,layer,mesh,generation,in,boundary,layers.,Previously,,face,curvature,was,not,used,,which,led,to,low,quality,meshes,especially,at,low,resolution.,This,has,now,been,fixed.,5.,An,examination,of,tokamak-relevant,first,wall,geometries,,which,discusses,the,generation,of,a,higher,order,mesh,from,simplified,CAD,models,of,a,divertor,concept.,The,bad,elements,which,were,found,,are,attributed,to,areas,of,poor,CAD,parameterisation.,6.,Some,future,directions,for,NekMesh,post-NEPTUNE.,This,section,describes,planned,(and,funded),work,to,make,the,NekMesh,code,more,robust,,extend,the,Python,API,,and,to,provide,a,standalone,GUI,for,use,by,non-experts.,Another,workstream,is,the,development,of,an,Industrial,Pipeline,and,wrapping,that,functionality,back,into,NekMesh,and,Nektar++.,Finally,,44,plans,are,laid,out,to,incorporate,NekMesh,into,Nektar++,so,that,the,mesh,can,be,modified,at,runtime,,enabling,eg.,dynamic,load,balancing.,5.2.3,Code,deliverable,2068435,Deliverable,4,,Task,3.2,This,deliverable,consists,of,modifications,to,the,Nektar++,codebase,,submitted,in,the,form,of,merge,request,!1759,in,the,main,Nektar,GitLab,repository.,It,entails,the,addition,of,the,”MUI”,(Multiscale,Universal,Interface),library,Nektar++,as,an,alternative,coupling,framework,to,”CWIPI”.,MUI,has,been,enabled,via,additions,to,the,MPI,communications,code,,as,is,required,for,coupling,frameworks,over,a,split,MPI,commicator.,Further,alterations,have,been,made,to,the,simulation,configuration,tools,,so,that,coupling,may,be,defined,via,the,standard,user,interface.,In,addition,to,these,source,code,changes,,there,is,a,DummyMui,EquationSystem,,which,serves,as,an,example,of,how,to,couple,codes,(in,this,case,two,solvers,within,Nektar++),in,1,,2,and,3,spatial,dimensions.,All,tests,added,to,the,code,as,part,of,the,merge,request,have,been,confirmed,to,execute,success-,fully,by,a,UKAEA,reviewer.,5.2.4,Code,deliverable,2068435,Deliverable,5,,Task,3.3,This,deliverable,consists,of,modifications,to,the,Nektar++,codebase,,submitted,in,the,form,of,two,merge,requests,,!1379,and,!1752,in,the,main,Nektar++,GitLab,repository.,They,expand,Nektar’s,existing,set,of,Python,bindings,(NekPy),to,include,the,SolverUtils,library,,which,allows,time-evolving,Nektar,solvers,to,be,run,directly,from,Python,for,the,first,time.,Demos,are,provided,for,an,unsteady,implicit,diffusion,solver,and,a,Helmholtz,solver,,both,in,2D,domains.,In,the,context,of,project,NEPTUNE,,these,changes,have,two,important,benefits:,1.,The,potential,for,rapid,prototyping,of,new,equation,systems,in,Python,,reducing,overall,de-,velopment,time,2.,The,removal,of,one,of,the,major,obstacles,to,building,a,Python,DSL,for,Nektar++,solvers,All,tests,added,to,the,code,as,part,of,the,two,merge,requests,have,been,confirmed,to,execute,successfully,by,a,UKAEA,reviewer.,5.2.5,Code,deliverable,2068435,Deliverable,1,,Task,1.3,This,deliverable,consists,of,modifications,to,the,Nektar++,codebase,,submitted,in,the,form,of,merge,request,!1743,in,the,main,Nektar,GitLab,repository.,It,is,the,third,and,final,part,of,deliverable,1,,which,comprises,a,significant,refactoring,of,Nektar’s,core,libraries,,with,the,aim,of,enhancing,the,modularity,and,performance,portability,of,the,code.,45,The,changes,that,have,been,made,decouple,operators,from,the,data,that,they,act,upon,,lifting,a,restriction,that,was,significantly,limiting,the,capacity,of,the,code,to,be,performance,portable.,This,decoupling,also,means,that,developers,can,implement,their,own,operators,and,use,them,with,other,Nektar++,components,,without,having,to,make,changes,to,Nektar,itself.,All,tests,added,to,the,code,as,part,of,the,merge,request,(for,both,CPU,and,CUDA-based,operators),have,been,confirmed,to,execute,successfully,by,a,UKAEA,reviewer.,While,the,grantee,only,committed,to,adding,(refactored),CPU-based,implementations,for,each,op-,erator,,CUDA,versions,,suitable,for,execution,on,NVIDIA,GPUs,,have,also,been,written,in,several,cases.,The,overall,effect,of,these,changes,is,to,make,the,software,more,suitable,for,developing,Exascale-capable,applications.,6,Summary,This,report,has,presented,results,from,a,number,of,workstreams,related,to,the,implementation,of,3-D,plasma,turbulence,equations,in,a,realistic,tokamak,geometry,using,finite,element,techniques.,A,variety,of,test,cases,were,implemented,using,the,Firedrake,finite,element,code,,demonstrat-,ing,its,utility,as,a,tool,for,rapid,prototyping,as,well,as,for,experimenting,with,different,numerical,implementations,and,debugging,existing,NEPTUNE,Proxyapps.,An,implementation,of,the,3-D,Hasegawa-Wakatani,equations,using,Nektar++,was,demonstrated,,intended,as,a,stepping,stone,to,a,more,complex,set,of,fluid,plasma,equations.,The,code,extends,existing,Nektar++,solvers,embedded,in,NESO,whilst,retaining,existing,functionality,(in,particular,,coupling,to,a,kinetic,neutrals,framework),via,a,substantial,code,refactoring,exercise.,The,report,also,described,NESO-fame,,a,tool,for,generating,field-aligned,meshes.,Particular,focuses,of,this,work,were,on,producing,meshes,that,can,conform,to,the,first,wall,of,the,tokamak,and,that,are,of,sufficient,quality,to,use,in,scalable,fluid,plasma,simulations.,Finally,,summaries,were,provided,for,a,number,of,grantee,deliverables,,covering,ongoing,im-,provements,to,the,Oxford,grantee’s,finite,element,drift-kinetic,model,and,various,updates,to,the,Nektar++,codebase,that,enhance,its,suitability,for,modelling,fusion-relevant,problems,on,Exascale,platforms.,Acknowledgement,The,support,of,the,UK,Meteorological,Office,and,Strategic,Priorities,Fund,is,acknowledged.,ET,acknowledges,assistance,from,the,rest,of,the,UKAEA,NEPTUNE,team,,as,well,as,Hussam,al-,Daas,(STFC),,Professor,Patrick,Farrell,(Oxford),and,Professors,Colin,Cotter,and,David,Ham,of,Imperial,College,London.,46,References,[1],Florian,Rathgeber,,David,A.,Ham,,Lawrence,Mitchell,,Michael,Lange,,Fabio,Luporini,,An-,drew,T.,T.,Mcrae,,Gheorghe-Teodor,Bercea,,Graham,R.,Markall,,and,Paul,H.,J.,Kelly.,Fire-,drake:,automating,the,finite,element,method,by,composing,abstractions.,ACM,Trans.,Math.,Softw.,,43(3):24:124:27,,2016.,URL:,http://arxiv.org/abs/1501.01809,,arXiv:1501.01809,,doi:10.1145/2998441.,,2015.,[2],Patrick,E.,Farrell,,Robert,C.,Kirby,,and,Jorge,Marchena-Menendez.,Irksome:,Automating,runge–kutta,time-stepping,for,finite,element,methods,,2020.,[3],DG,advection,with,upwinding.,https://www.firedrakeproject.org/demos/DG_advection.,py.html.,Accessed:,March,2024.,[4],Busch,K.,,Koenig,M.,,and,Niegemann,J.,Discontinuous,Galerkin,methods,in,nanophotonics.,Laser,and,Photonics,Reviews,Vol.5,Issue,6,(Nov,2011).,,pages,773–809,,2011.,[5],Nektar-Driftwave.,https://github.com/ExCALIBUR-NEPTUNE/nektar-driftwave.,Ac-,cessed:,March,2023.,[6],E.,Threlfall,and,W.,Arter.,Finite,Element,Models,Complementary,Actions:,Code,Integration,,Integration,and,Operation,3.,Technical,Report,CD/EXCALIBUR-FMS/0079,,UKAEA,Project,Neptune,,2023.,[7],m6c5,scripts.,https://github.com/ethrelfall/m6c5_phase2/tree/main,,2024.,Ac-,cessed:,March,2024.,[8],Hermes-3.,https://hermes3.readthedocs.io/en/latest/examples.html#,d-drift-plane.,Accessed:,March,2023.,[9],STORM.,https://github.com/boutproject/STORM,,2024.,Accessed:,March,2024.,[10],W.,Arter.,Equations,for,ExCALIBUR,/,NEPTUNE,proxyapps.,port,CD/EXCALIBUR-FMS/0021,,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0021-1.26-M1.2.1.pdf.,UKAEA,Project,Neptune,,2023.,Technical,Re-,https:,[11],Donea,J.,and,Huerta,A.,Finite,Element,Methods,for,Flow,Problems.,Wiley,,2003.,[12],Bernsen,E.,,Bokhove,O.,,and,van,der,Vegt,J.J.W.,A,(Dis)continuous,finite,element,model,for,generalized,2D,vorticity,dynamics.,Memorandum,No.,1787,,Department,of,Applied,Mathe-,matics,,University,of,Twente,,2005.,[13],Nektar-Driftplane.,https://github.com/ExCALIBUR-NEPTUNE/nektar-driftplane.,Ac-,cessed:,March,2023.,[14],E.,Threlfall,,J.,Cook,,M.,Barton,,O.,Parry,,W.,Saunders,,and,W.,Arter.,Complementary,actions.,Code,integration,,acceptance,and,operation,3.,Technical,Report,CD/EXCALIBUR-FMS/0074-,M6c.3,,UKAEA,,3,2023.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/ukaea_reports/CD-EXCALIBUR-FMS0074-M6c.3.pdf.,47,[15],E.,Threlfall,and,W.,Saunders.,Support,high-dimensional,procurement.,cal,Report,CD/EXCALIBUR-FMS/0066,,UKAEA,Project,Neptune,,2022.,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0066-M4.1.pdf.,Techni-,https:,[16],E.,Threlfall,,O.,Parry,,and,W.,Arter.,Finite,Element,Models,Complementary,Actions:,Code,In-,tegration,,Integration,and,Operation,3.,Technical,Report,CD/EXCALIBUR-FMS/0074,,UKAEA,Project,Neptune,,2023.,[17],J.,Cook,,W.,Saunders,,and,O.,Parry.,integrated,particle,and,continuum,model.,Technical,Report,CD/EXCALIBUR-FMS/0079-M4c.3,,UKAEA,,9,2023.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_,reports/CD-EXCALIBUR-FMS0079-M4c.3.pdf,to,be,posted.,Three-Dimensional,[18],B.D.,Dudson.,BOUT++,repository.,https://github.com/boutproject/BOUT-dev,,2020.,Ac-,cessed:,June,2020.,[19],R.,Numata,,R.,Ball,,and,R.L.,Dewar.,Bifurcation,in,electrostatic,resistive,drift,wave,turbulence.,Physics,of,Plasmas,,14:102312,,2007.,[20],B.,Friedman,and,T.,A.,Carter.,A,non-modal,analytical,method,to,predict,turbulent,properties,applied,to,the,Hasegawa-Wakatani,model.,Physics,of,Plasmas,,22(1):012307,,01,2015.,[21],W.,Arter,et,al.,Equations,for,EXCALIBUR/NEPTUNE,Proxyapps.,cal,Report,CD/EXCALIBUR-FMS/0021-1.30-M1.2.1,,UKAEA,,6,2023.,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0021-1.30-M1.2.1.pdf.,Techni-,https:,[22],F.F.,Chen.,2012.,Introduction,to,Plasma,Physics,,3rd,ed.,Springer,Science,&,Business,Media,,[23],A.,Roques,and,open,source,developers.,Plantuml,website.,https://plantuml.com/,,2024.,Accessed:,February,2024.,[24],T.,Marin.,hpp2plantuml,python,package.,https://pypi.org/project/hpp2plantuml/,,2024.,Accessed:,February,2024.,[25],D.,Napoleone,,S.,Marks,,and,E.,Frederich.,plantuml,python,package.,https://pypi.org/,project/plantuml/,,2024.,Accessed:,February,2024.,[26],J.,Cook,,E.,Threlfall,,,,O.,Parry,,W.,Saunders,,and,W.,Arter.,Three-Dimensional,integrated,particle,and,continuum,model,continued.,Technical,Report,CD/EXCALIBUR-FMS/0082-,M4c.4,,UKAEA,,3,2024.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/ukaea_reports/CD-EXCALIBUR-FMS0082-M4c.4.pdf.,[27],G.,Karniadakis,and,S.,Sherwin.,Spectral/hp,element,methods,for,computational,fluid,dy-,namics,2nd,Ed.,Oxford,University,Press,,2005.,https://doi.org/10.1093/acprof:oso/,9780198528692.001.0001.,48,[28],J.,W.,Cook,,J.,T.,Parker,,W.,R.,Saunders,,and,Parry,O.,Neptune,Exploratory,SOftware,(NESO),repository.,https://github.com/ExCALIBUR-NEPTUNE/NESO,,2023.,Accessed:,Jan-,uary,2023.,[29],C.,Ridgers,and,M.,Kryjak.,Benchmarks,for,basic,turbulence,cases,(Report,on,Test,Cases,and,Proxy-App),-,updated,.,Technical,Report,2067270-TN-06,,UKAEA,Project,Neptune,,2024.,to,be,posted,at,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2067270/TN-06.pdf.,[30],Edward,Laughton,,Gavin,Tabor,,and,David,Moxey.,A,comparison,of,interpolation,techniques,for,non-conformal,high-order,discontinuous,galerkin,methods.,Computer,Methods,in,Applied,Mechanics,and,Engineering,,381:113820,,2021.,[31],John,Omotani,,Ben,Dudson,,Peter,Hill,,Vandoo,,and,bshanahan.,boutproject/hypnotoad:,0.5.2,,March,2023.,[32],Nektar-diffusion,proxyapp.,https://github.com/ExCALIBUR-NEPTUNE/nektar-diffusion,,2021.,Accessed:,September,2021.,[33],M.,Barnes,,M.R.,Hardman,,S.L.,Newton,,J.,Omotani,,and,F.I.,Parra.,A,reduced,electron,model,for,testing,plasma,dynamics,for,closed,field,lines.,Technical,Report,2070839-TN-03,,UKAEA,Project,Neptune,,2023.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2070839/TN-03.pdf.,[34],M.R.,Hardman,,J.,Omotani,,M.,Barnes,,S.L.,Newton,,and,F.I.,Parra.,Convergence,of,the,solutions,of,the,2D-1V,ion,drift,kinetic,equation,with,wall,boundary,conditions:,manufactured,solutions,tests.,Technical,Report,2070839-TN-04,,UKAEA,Project,Neptune,,2023.,https:,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2070839/TN-04.pdf.,[35],G.,Strang,and,G.J.,Fix.,An,Analysis,of,the,Finite,Element,Method.,Prentice-Hall,,Englewood,Cliffs,,NJ,,1973.,[36],J.P.,Boyd.,Chebyshev,and,Fourier,Spectral,Methods.,Dover,Publications,,Mineola,,NY,,2001.,http://www-personal.umich.edu/,jpboyd/BOOK,Spectral2000.html.,[37],M.,Barnes,,M.R.,Hardman,,S.L.,Newton,,J.,Omotani,,and,F.I.,Parra.,Numerical,implemen-,tation,of,a,fluid,model,for,electrons,in,a,drift,kinetic,code.,Technical,Report,2070839-TN-,05,,UKAEA,Project,Neptune,,2023.,https://github.com/ExCALIBUR-NEPTUNE/Documents/,blob/main/reports/2070839/TN-05.pdf.,[38],M.R.,Hardman,,J.,Omotani,,M.,Barnes,,S.L.,Newton,,and,F.I.,Parra.,collision,operators:,nical,Report,2070839-TN-06,,UKAEA,Project,Neptune,,2023.,ExCALIBUR-NEPTUNE/Documents/blob/main/reports/2070839/TN-06.pdf.,a,Krook,operator,and,a,model,Fokker-Planck,operator.,Ion-ion,model,Tech-,https://github.com/,[39],M.R.,Hardman,,M.,Abazorius,,J.,Omotani,,M.,Barnes,,S.L.,Newton,,and,F.I.,Parra.,A,higher-,order,finite-element,implementation,of,the,full-F,Landau,Fokker-Planck,collision,operator,for,charged,particle,collisions,in,a,low,density,plasma.,Technical,Report,2070839-TN-07,,UKAEA,Project,Neptune,,2023.,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/,reports/2070839/TN-07.pdf.,49,[40],M.,Barnes,,M.R.,Hardman,,S.L.,Newton,,J.,Omotani,,and,F.I.,Parra.,Numerical,implemen-,tation,of,moment-kinetic,electrons.,Technical,Report,2070839-TN-08,,UKAEA,Project,Nep-,tune,,2023.,to,be,posted,at,https://github.com/ExCALIBUR-NEPTUNE/Documents/blob/,main/reports/2070839/TN-08.pdf.,[41],M.R.,Hardman,,J.,Omotani,,M.,Barnes,,S.L.,Newton,,and,F.I.,Parra.,Aspects,of,magnetic,geometry,i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