CD-EXCALIBUR-FMS0011-1.00-M1.2.1_YearOneSummaryReport
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   :description: technical note

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ExCALIBUR,NEPTUNE:,Year,1,Summary,M1.2.1,Abstract,This,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Milestone,1.2.1.,This,End,of,Year,One,(Y1),summary,report,outlines,the,reasoning,behind,the,Y2,plan,,listing,questions,to,be,addressed,(especially,by,3rd,parties),,the,proxyapps,that,will,be,developed,,and,how,they,will,be,connected,to,each,other,and,exploited,to,steer,thinking,around,the,development,of,a,next,generation,edge,boundary,code.,Commercial,in,Confidence,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0011,Issue:,Date:,1.00,10,March,2020,Project,Name:,ExCALIBUR,Fusion,Modelling,System.,Version,1.00,Prepared,By:,Name,and,Department,Wayne,Arter,Lucian,Anton,Debasmita,Samaddar,Reviewed,By:,Rob,Akers,Modified,By:,Rob,Akers,Approved,By:,MSSC,Rob,Akers,MSSC,Signature,Date,N/A,N/A,N/A,N/A,10/03/2020,10/03/2020,10/03/2020,10/03/2020,10/03/2020,12/03/2020,1,Commercial,in,Confidence,Chapter,1,Introduction,This,section,begins,with,a,subsection,Section,Error!,Reference,source,not,found.,which,places,the,NEPTUNE,project,in,the,global,context.,The,following,subsections,Section,Error!,Reference,source,not,found.,and,Section,Error!,Reference,source,not,found.,review,more,local,UKAEA,activities,prior,to,commencement,of,NEPTUNE,,and,its,concluding,subsection,brings,history,up-to-date,with,a,summary,of,NEPTUNE,Y1,activities.,Much,of,the,material,in,this,Section,1,bears,on,the,choices,made,for,the,Y2,programme,discussed,in,Section,2,and,summarised,in,Section,3.,1.1,Background,In,the,past,few,years,,the,focus,of,the,UKAEA,fusion,programme,has,been,increasingly,directed,towards,the,design,of,power-generating,reactors,,see,e.g.,papers,presented,at,a,Royal,Sociey,meeting,in,2018,“Fusion,energy,using,tokamaks:,can,development,be,accelerated?’,,notably,refs,[1,,2].,At,the,start,of,November,2018,,two,documents,of,worldwide,importance,for,the,development,of,fusion,appeared,,namely,the,ITER,Research,Plan,[3],and,the,updated,European,Research,Roadmap,[4].,They,correctly,emphasise,the,importance,of,modelling,in-silico,but,are,lacking,in,detail,as,to,how,accurate,and,reliable,models,are,to,be,developed,in,a,timely,fashion,for,the,designers,and,operators,of,the,ITER,and,DEMO,devices.,For,example,there,is,lack,of,usage,of,the,word,“software",,implying,absence,of,the,term,“software,engineering",(SWEng).,However,,the,increased,stress,on,model,reliability,for,“digital,twinning",might,a,priori,be,expected,to,require,careful,management,of,software,throughout,its,entire,life-cycle.,There,are,special,features,for,the,software,engineer,to,account,for,in,a,leading-edge,technology,programme,such,as,fusion,reactor,development,,which,is,nonetheless,long-term.,Digitally,twinned,software,must,be,flexible,,to,deal,with,issues,which,perhaps,were,not,initially,regarded,as,important,and,may,require,rapid,resolution,,for,example,in,the,light,of,new,experimental,data,or,during,construction,work.,This,amounts,to,a,requirement,that,the,software,,based,upon,current,fusion,timelines,needs,to,remain,not,only,usable,,but,also,modifiable,,perhaps,drastically,,over,at,least,the,next,30,years,needed,to,see,DEMO,into,power,production.,Concerning,existing,software,,especially,for,plasma,physics,modelling,,it,can,be,argued,that,the,finite,difference,software,(fd),presently,underlying,much,design,work,is,obsolescent,in,the,above,special,software,engineering,terms,,being,expensive,and/or,error-prone,to,maintain,as,the,products,of,an,era,when,SWEng,was,in,its,infancy.,Specifically,here,is,being,considered,the,2,Commercial,in,Confidence,Patankar,scheme,[5],that,underlies,the,SOLPS,fluid,code,known,as,B2,[6].,SOLPS,is,for,many,people,the,‘gold,standard’,edge,modelling,code,because,it,incorporates,more,physical,effects,than,more,recent,developments.,From,a,practical,point-of-view,however,,fd,is,unsatisfactory,for,design,work,because,it,fails,to,give,robust,error,estimates,,and,inefficient,because,of,21st,Century,trends,in,computer,architecture.,Typically,these,schemes,are,not,very,accurate,with,errors,of,1,%,rising,locally,to,10,%.,Recent,developments,of,the,chebfun,approach,by,Trefethen,and,co-workers,(eg.,ref,[Error!,Reference,source,not,found.,,Error!,Reference,source,not,found.],are,producing,commodity,software,capable,of,solving,PDEs,to,‘spectral’,accuracy,in,double,precision.,Sustained,progress,here,will,render,these,plasma,physics,low-order-accurate,fd,schemes,an,embarrassment.,There,is,perhaps,an,even,more,significant,issue,for,the,use,of,the,particle-based,methods,described,in,say,the,review,[7],,since,these,are,associated,with,relatively,large,amplitude,sampling,‘noise’.,1.2,Spectral/hp,Element,Method,Two,student,projects,involving,the,spectral/hp,element,method,for,modelling,the,tokamak,edge,were,commissioned,by,UKAEA,from,2015,onwards,,see,ref,[8,,9],,and,a,number,of,other,interactions,took,place,,notably,presentations,by,Sherwin,the,joint,author,of,the,textbook,[10],,and,in,2018,by,both,Sherwin,and,his,former,post-doc,Moxey.,Just,prior,to,NEPTUNE,commencement,,Arter,[11],gave,a,presentation,at,the,Isaac,Newton,Institute,explaining,the,desirability,of,using,spectral,elements,for,tokamak,edge,modelling.,Spectral,element,methods,[12],are,capable,of,providing,usable,error,estimates,and,of,exploiting,the,developing,architecture,to,ensure,maximum,use,of,complex,2-D,,3-D,and,time-dependent,physical,models,expected,to,be,more,robust,under,extrapolation,to,new,situations,and,regimes.,Their,extreme,accuracy,much,reduces,the,need,for,special,physics-dependent,coding,,suggesting,that,a,near-complete,replacement,of,code,essential,for,reactor,design,is,feasible,,and,if,carefully,managed,,such,a,new,development,can,meet,the,demanding,SWEng,requirements,of,the,fusion,programme.,Amongst,spectral,element,approaches,,none,has,the,maturity,and,the,full,range,of,advantages,presented,by,spectral/hp,elements.,The,spectral/hp,element,method,(see,650-page,textbook,[10]),was,developed,jointly,by,Karniadakis,and,Sherwin,around,the,turn,of,the,Century.,In,the,name,,the,“h",implies,variable,element,size,,and,“p",implies,use,of,a,pth-order,accurate,approximation,,where,p,is,an,arbitrarily,large,order.,“Spectral",reinforces,the,notion,that,as,p⟶∞,,spectral,accuracy,,ie.,error,smaller,than,any,power,of,element,size,,is,achievable.,This,is,to,be,compared,with,commercial,packages,for,comparable,problems,that,use,low,order,finite,elements,,often,restricted,to,first,or,second,order.,The,name,spectral/hp,does,not,exclude,use,3,Commercial,in,Confidence,also,of,r-elements,,ie.,additional,elements,introduced,dynamically,to,represent,evolving,features,of,say,a,fluid,flow-field.,Figure,1.1,indicates,the,increasing,application,of,the,method,in,published,papers,to,a,wide,range,of,problems.,In,addition,,there,has,been,commercially,driven,application,of,spectral/hp,elements,to,the,design,of,F1,cars.,Other,21st,Century,developments,have,seen,the,the,incorporation,of,special,,geometrically,accurate,meshing,techniques,[13],into,the,NekMesh,library,,which,overcome,the,often,large,errors,associated,with,common,triangulations,of,geometry,[14],,and,production,of,the,Nektar++,library,[15].,Further,investigation,of,geometry,representation,for,NEPTUNE,application,is,to,be,found,in,the,Milestone,Report,[16].,Figure,1.1,indicates,that,the,latter,has,facilitated,widespread,uptake,of,the,method,,giving,evidence,for,a,growing,community,with,a,large,range,of,applications,,many,of,direct,interest,to,fusion.,Figure,1.1:,Citation,counts,for,applications,of,the,spectral/hp,element,textbook,and,Nektar++,library.,Search,performed,on,google,scholar(1/5/18),for,keywords,indicated.,Spectral/hp,element,textbook,from,4,Commercial,in,Confidence,1997/2003,has,2300,citations,,Nektar++,library,description,paper,from,July,2015,has,120,citations,(151,as,of,1/11/18).,The,controllable,accuracy,of,spectral,element,schemes,is,important,despite,the,inevitable,errors,in,any,experimental,confrontation,with,reality,,because,of,great,advances,in,techniques,for,fitting,models,to,data,,in,fields,such,as,optimisation,,uncertainty,quantification,and,data,assimilation,,now,often,bracketed,as,Machine,Learning,-,see,recent,work,by,Arter,and,students,[17,,18].,These,advances,in,VVUQ,techniques,give,more,robust,fitting,with,improved,error,estimates.,One,advantage,of,extreme,accuracy,that,is,highlighted,here,,because,of,its,relevance,to,the,serious,problem,of,heat,deposition,to,the,first,wall,,occurs,when,there,is,very,anisotropic,heat,transport,,found,in,some,layered,materials,or,as,here,,because,of,a,large-scale,magnetic,field.,If,the,numerical,grid,is,not,aligned,with,the,field,,then,heat,leaks,in,the,normal,direction,at,every,timestep,at,a,rate,proportional,to,cell-size,h,,leading,to,an,effective,heat,diffusivity,⊥,num∝h2/Δt,(Δt,is,time-step,size),which,may,greatly,exceed,the,intended,perpendicular,D,transport.,Figure,1.2:,Spectral,element,methods,can,model,the,entire,range,of,misalignment,angles,φ,without,any,5,Commercial,in,Confidence,special,treatment,,from,ref,[19].,Figure,at,top,indicates,the,field,direction,relative,to,the,fe,mesh,,in,the,graph,below,np,corresponds,to,the,order,p,and,dof,to,the,number,of,degrees,of,freedom,per,unit,length.,Even,if,the,grid,is,field-aligned,,either,by,special,local,construction,or,by,use,of,an,unstructured,,finite,element,representation,,since,a,good,first,wall,power,distribution,requires,almost,tangential,incidence,,there,will,be,elements,with,very,small,angles,leading,to,poor,convergence,properties,for,most,solvers.,However,,as,Figure,1.2,shows,,for,spectral,elements,,the,effective,relative,heat,diffusivity,remains,small,over,the,whole,range,of,φ,,even,having,a,local,minimum,at,the,angle,of,maximum,misalignment.,1.3,High,Level,Integration,Work,in,[20],examined,how,by,use,of,appropriate,software,technology,,spectral,element,software,could,be,integrated,into,an,object-oriented,approach,permitting,easy,integration,into,the,overall,reactor,design,package.,This,is,a,two-way,process,,in,that,design,can,only,proceed,at,an,acceptable,speed,using,surrogates,for,full,3-D,to,6-D,models,even,on,multi-core,desktops,,whereas,as,in,critical,cases,,3-D,to,6-D,models,may,need,to,be,invoked,to,reduce,uncertainty.,Preliminary,indications,were,that,this,should,be,achievable,using,a,graph-based,approach,,implemented,in,matrix,form,-,see,Figure,1.3.,Figure,1.3:,Schematic,of,top-level,algorithm.,A,bundled,capability,for,producing,surrogates,needs,to,be,included,,and,emphasis,needs,to,be,placed,upon,generating,well-documented,arms-length,interfaces,that,can,be,used,by,external,packages.,The,programming,style,described,in,[21],is,particularly,appropriate,to,this,purpose,,and,should,be,applicable,in,any,object-oriented,language.,There,will,also,be,a,need,for,an,easy,6,Commercial,in,Confidence,way,to,specify,scalar,and,vector,differentiation,operators.,1.4,UKAEA,Activities,to,date,Work,to-date,on,the,NEPTUNE,project,has,been,driven,by,the,above,considerations.,Of,all,the,tokamak,modelling,codes,,SOLPS,is,the,one,of,which,development,has,been,most,weighed,down,by,its,history,,making,edge,modelling,the,prime,candidate,area,for,producing,new,software.,The,EUROfusion,Theory,and,Advanced,Simulation,Coordination,(E-TASC),programme,has,also,recognised,this,,in,that,two,of,the,first,TSVV,(Theory,,Simulation,,Verification,,and,Validation),tasks,are,to,pilot,writing,of,the,European,Boundary,Code,(EBC),,and,to,launch,a,major,upgrade,of,the,SOLPS,particle,code,known,as,EIRENE.,These,pilot,tasks,are,due,to,finish,at,the,end,of,2020,,and,particularly,the,EBC,project,has,a,challenging,target,to,produce,significant,novel,physics,results,,in,order,to,secure,funding,for,the,proposed,5-year,follow-up.,It,has,been,decided,that,NEPTUNE,will,coordinate,primarily,with,the,TSVV,EBC,project,(which,necessarily,has,to,coordinate,with,the,particle,code,project).,UKAEA,staff,working,on,NEPTUNE,have,attended,the,two,major,meetings,of,the,EBC,pilot,phase,,and,relevant,outcomes,are,discussed,in,the,Milestone,Report,[22].,As,pointed,out,at,the,2019,internal,meeting,by,UKAEA,staff,,some,decisions,taken,to,ensure,the,EBC,targets,are,met,are,sub-optimal,for,a,longer,term,project,such,as,NEPTUNE,,,never,mind,a,thirty-year,horizon.,NEPTUNE,also,has,to,meet,other,UK,SPF,aims,,based,around,the,four,pillars.,However,,it,seems,NEPTUNE,can,coordinate,with,EBC,to,the,extent,that,it,looks,to,examine,a,common,2-,D,fluid,model,in,Y3,,so,that,where,NEPTUNE,Y2,research,tasks,are,directed,towards,this,same,model,,NEPTUNE,should,be,able,to,assist,EBC,with,any,problems,encountered,related,to,those,tasks.,Most,other,research,,such,as,interface,design,,integration,with,particle,methods,etc.,to,be,performed,under,NEPTUNE,should,be,complementary,to,the,EBC,pilot,phase,development,,and,of,immense,value,to,a,5-year,follow-on,programme.,Other,UKAEA,activities,are,based,around,findings,from,the,2019,internal,workshop,for,requirements,capture,and,an,external,,UK,wide,workshop,held,in,Birmingham,on,05-02-2020;,these,are,reported,upon,in,ref,[23].,The,above,activities,have,been,accompanied,by,and,to,some,extent,driven,by,identification,and,preliminary,investigation,of,the,critical,research,needed,into,the,physical,model,,numerical,algorithms,and,software,engineering,through,extensive,community,engagements,and,literature,review.,Further,investigation,of,particle,methods,for,NEPTUNE,application,is,to,be,found,in,the,7,Commercial,in,Confidence,Milestone,Report,[24].,Chapter,2,Task,Work,Subsection,Section,2.1,represents,‘outreach’,to,the,Computational,Fluid,Dynamics,(CFD),community,,to,give,them,‘some,feel’,for,the,basic,physical,issues,at,stake,,similarly,Section,Error!,Reference,source,not,found.,tries,to,illustrate,the,sheer,power,of,Exascale.,The,following,subsection,Section,2.3,discusses,approaches,that,were,raised,during,Y1,by,a,range,of,potential,collaborators,,and,the,last,subsection,Section,2.4,elaborates,a,potential,addition,to,the,NEPTUNE,tasks.,2.1,SOL,and,PFC,as,convection,problem,Figure,2.1:,Relationship,between,the,tokamak,edge,and,classical,fluid,dynamics.,The,plot,at,far,left,is,courtesy,of,EUROfusion,(eurofusion.org).,To,give,the,specialists,in,Computational,Fluid,Dynamics,(CFD),a,feel,for,the,problem,,Figure,2.1,shows,how,the,outer,part,of,the,midplane,(green,box),can,be,approximated,as,a,classical,Rayleigh-Benard,convection,(RBC),problem,-,see,the,recent,work,[25].,Plasma,wants,to,flow,along,fieldlines,,but,there,is,no,escape,from,the,effects,of,momentum,conservation,,and,the,net,result,is,that,plasma,experiences,an,inward,acceleration,g.,Rotating,the,picture,through,90o,so,this,acceleration,looks,vertical,makes,clearer,the,relation.,The,hot,central,plasma,represents,a,heated,floor,capable,of,driving,convection,in,the,SOL.,For,RBC,,Reynolds,numbers,and,boundary,layer,thicknesses,have,been,fitted,by,Grossmann,&,Lohse,[26].,Estimates,of,the,8,Commercial,in,Confidence,dimensionless,pressure/temperature,difference,or,Rayleigh,number,Ra,range,in,the,tokamak,edge,[25],plausibly,from,105−1012.,Assuming,a,Prandtl,number,Pr≈1,,where,Pr,is,the,ratio,of,viscous,to,thermal,diffusion,,the,Reynolds,Re,number,varies,from,Re=10−4×104.,Such,values,imply,relatively,narrow,boundary,layers,,thickness∝1/,Re,and,turbulence,for,most,of,the,range.,Figure,2.2:,Ten,regimes,of,RBC,depending,on,the,parameters,Ra,and,Pr.,Reproduction,of,Fig.,2,from,ref,[26].,9,Commercial,in,Confidence,Figure,2.3:,Sketch,at,top,of,the,so-called,Rayleigh-Jeffreys,problem,,where,a,solid,of,thermal,conductivity,K',overlies,a,fluid,with,thermal,conductivity,K.,Below,is,a,plot,of,normalised,critical,Ra,as,the,ratio,of,conductivities,changes.,Data,from,ref,[27].,Figure,2.4:,Amplitude,plotted,against,Ra,at,fixed,Pr,showing,the,interaction,between,the,‘double-glazing’,solution,branch,which,becomes,unstable,at,small,Ra,and,the,classical,RBC,branch,which,appears,beyond,Ra≈1900.,Reproduction,of,Fig.,2,from,ref,[28].,It,is,worth,remembering,that,the,Rayleigh-Benard,problem,,while,simple,to,pose,,is,hard,to,understand,fully,,and,sensitive,to,details,of,both,boundary,conditions,and,geometry.,For,example:,1.,For,the,scaling,of,dimensionless,heat,flux,Nu,with,convection,,there,are,ten,regimes,of,RBC,depending,on,the,temperature,difference,measured,by,Rayleigh,number,Ra,and,Prandtl,number,Pr=Ku/K,,see,Figure,2.2.,The,scaling,law,exponents,γ,,,γ,1,which,are,2,defined,so,that,Nu∝Raγ1Prγ2,range,over,1/4<γ,1,<4/7,and,−5/6<γ,2,<1/8.,10,Commercial,in,Confidence,2.,There,is,an,approx.,70,%,reduction,in,heat,flux,if,the,boundary,condition,on,the,flow,is,changed,from,free-slip,to,stagnation.,This,follows,because,the,critical,Ra,,Ra,≈657,C,increases,to,Ra,C,≈1708,[29],,and,from,assuming,that,heat,flux,Nu∝(Ra−Ra,)1/2.,C,3.,There,is,an,approx.,10,%,increase,in,Nu,possible,due,to,a,change,outside,the,fluid,region,,namely,the,replacement,of,perfectly,conducting,boundaries,by,poor,insulators,,see,Figure,2.3,,understood,from,the,fact,that,a,poorly,conducting,boundary,constrains,the,convection,less.,4.,In,a,heated,annulus,,which,Figure,2.1,suggests,is,a,more,accurate,model,of,the,edge,,there,is,a,complex,bifurcation,structure,possible,,see,Figure,2.4.,5.,There,are,indications,that,spectral,(and,probably,other,numerical),schemes,converge,non-,monotonically,as,spatial,resolution,increases,,e.g.,the,reduction,of,RBC,to,the,Lorenz,equations,[30],does,not,permit,smaller-scale,instability,of,thermal,boundary,layers,because,it,represents,them,too,crudely.,However,,more,detailed,,but,unresolved,models,of,these,layers,may,exhibit,time-dependence,when,the,full,solution,with,thinner,layers,does,not,,because,layers,of,thickness,d,are,destabilised,by,a,term,∝d3.,2.2,Capabilities,of,Exascale,In,the,tokamak,edge,,following,[25],,suppose,plasma,number,density,n≈1018,m−3.,Order,of,magnitude,dimensions,are,L≈0.1,m,for,SOL,thickness,,reactor,minor,and,major,radii,say,a=3,m,and,R,0,=10,m,,so,volume,of,SOL,≈4π2aR,0,L≈100,m3.,Hence,total,number,of,electrons,≈1020.,Shortest,timescale,is,inverse,eB/m,e,,,the,electron,cyclotron,frequency,,where,e/m,e,=1.76×1011,C,kg−1,,and,B≈10,T,,so,τ,Be,≈10−12,s.,Hence,number,of,particle-steps,to,evolve,1,s,of,physical,time,is,1020+12+1,,and,assuming,1000,flop,per,update,,we,would,need,1036,Flops.,So,to,complete,in,1,s,on,an,Exascale,machine,,one,could,only,sample,1,particle-step,in,1018.,(Unlikely,to,be,adequate,because,of,electrostatic,and,other,effects).,Suppose,then,that,we,adopt,a,fluid,approach,instead,,ie.,representing,the,electron,distribution,by,the,first,3,moments.,Electron,temperature,T,e,≈10,eV,,thermal,speed,V,Te,≈106,m,s−1.,Sample,SOL,at,uniform,1,mm,interval,,number,of,sample-points,≈1011,,timestep,for,explicit,scheme,≈0.1×(10−3/106),so,number,of,sample-point,updates,is,1011+10;,assuming,1000,flop,each,update,,we,would,need,1024,Flops,,6,orders,of,magnitude,greater,than,exascale.,Another,way,,suppose,the,numerical,problem,is,D-dimensional;,we,would,need,say,1000,Flops,11,Commercial,in,Confidence,each,sample,update,and,allowed,N,2,samples,per,spatial,dimension,and,N,D,in,time.,Then,to,D,update,in,1,s,,we,have,N,D+2,D,≈1015.,Thus,if,D=3,,N,3,≈1,000,,and,N,5,≈100.,In,conclusion,,accuracy,controlled,,unstructured,,implicit,fluid,models,should,be,possible.,Similarly,,estimates,of,the,maximum,frequency,at,which,neutral,atom,particle-steps,can,be,sampled,,assuming,n,=n,and,T,0,=T,e,0,,,indicates,that,useful,results,may,be,obtained,from,the,particle,approach.,Moreover,,repeating,the,above,calculation,for,electrons,but,considering,that,only,plasma,ions,have,to,be,simulated,on,a,drift,type,timescale≈L/V,,,V,Ti,Ti,≈3×104,m,s−1,gives,much,more,encouraging,results.,2.3,Alternative,Approaches,During,the,requirements,capture,exercise,,several,other,possible,approaches,were,mooted.,Discounting,special,pleading,,those,felt,worth,of,examining,were,1.,Discontinuous,Galerkin,(DG),with,usual,(polynomial),finite,element,bases,[Error!,Reference,source,not,found.],2.,Isogeometric,finite,element,bases,3.,Lattice,Boltzmann,(LB),4.,Smoothed,Particle,Hydrodynamics,(SPH),5.,Sparse,grids,[Error!,Reference,source,not,found.,,Error!,Reference,source,not,found.],6.,Asymptotic-Preserving,Methods,7.,Dedalus,software,[Error!,Reference,source,not,found.],The,work,of,Nikiforakis,and,coworkers,,eg.,[31],may,be,regarded,as,special,pleading,,but,since,it,also,has,implications,for,geometry,representation,,it,is,discussed,in,the,Milestone,Report,[16].,2.4,Potential,Additions,to,NEPTUNE,2.4.1,Adjoint,or,‘dual’,approach,The,adjoint,approach,to,sensitivity,analysis,exploits,the,fact,that,the,sensitivity,of,only,one,or,perhaps,a,small,number,of,quantities,such,as,Q,,each,a,linear,combination,of,the,N,i,is,needed.,The,technique,is,best,illustrated,[32],by,a,simple,,linear,example.,Imagine,that,the,problem,being,solved,is,the,(‘primal’),set,of,linear,equations,Ax=b,to,find,one,quantity,(the,’objective,function’),gTx,,where,the,vectors,b,and,g,are,given.,It,happens,that,the,12,Commercial,in,Confidence,same,result,may,be,achieved,by,solving,instead,the,(‘dual’),equation,set,ATv=g,and,forming,vTb.,This,may,be,demonstrated,by,one,line,of,matrix-vector,algebra:,vTb=vTAx=(ATv)Tx=gTx,(2.1),It,should,be,apparent,that,if,the,objective,function,is,to,be,evaluated,for,K,different,vectors,b,,but,only,one,g,,only,one,dual,solution,is,needed,,not,K,primal,solutions.,Provided,one,dual,solution,is,not,significantly,more,costly,than,one,primal,one,,a,significant,saving,may,be,realised.,The,main,tension,arises,when,the,model,is,not,self-adjoint,,in,that,the,dual,problem,is,then,different,from,the,one,that,the,physicist,would,like,to,understand.,This,can,be,an,additional,reason,for,preferring,a,variational,approach.,Chapter,3,Summary,Due,to,a,significant,amount,of,preparatory,work,,the,requirements,capture,exercise,from,late,2019,to,end,of,Y1,did,not,raise,any,significant,issues,that,had,not,been,anticipated,in,the,Science,Plan,[33].,Concerning,modelling,and,software,,there,was,remarkably,little,dissension.,The,question,of,the,physics,to,be,included,has,been,settled,in,the,short,term,by,the,need,to,align,with,the,E-TASC,projects,on,edge,modelling.,Longer,term,there,are,questions,still,to,be,resolved,concerning,the,details,of,the,gyrokinetic,(or,kinetic),model,to,be,employed,,the,inclusion,of,special,relativistic,effects,,plasma,chemistry,,and,for,example,whether,the,PFC,boundaries,should,be,allowed,to,‘melt’,,and,whether,particle,dynamics,within,the,top,layers,of,PFCs,should,be,included.,There,is,still,a,debate,about,how,best,to,deal,with,situations,when,the,software,even,at,Exascale,is,incapable,of,resolving,boundary,or,internal,layers.,A,feature,of,the,lower,order,(Patankar),fd,scheme,is,that,it,can,be,formulated,so,that,in,an,implicit,time,advance,of,a,field,advection-,diffusion,equation,,it,acts,a,contraction,mapping,upon,the,field,,ie.,it,converges,to,a,single,,finite,solution,,regardless,of,lack,of,layer,resolution,or,of,size,of,timestep.,Under,such,circumstances,,a,spectral,scheme,may,fail,completely,cf.,the,dispersion,analyses,of,say,Ainsworth,et,al,[34],and,more,recently,for,spectral/hp,element,schemes,in,the,Galerkin,[35],and,discontinuous,Galerkin,[36],contexts,-,users,generally,prefer,an,answer,to,none,at,all.,Of,course,,it,may,be,argued,that,a,manifestly,erroneous,result,without,an,accuracy,estimate,will,not,be,used,to,action,large,procurements,,but,in,any,event,it,would,be,better,for,the,spectral,scheme,to,produce,a,result,at,increasingly,extreme,parameters.,13,Commercial,in,Confidence,There,are,a,number,of,ways,to,achieve,this,,the,most,obvious,being,the,use,of,explicit,artifical,viscosity,,to,broaden,thinner,layers,or,small,features,to,the,point,where,full,spectral,accuracy,is,achievable.,Different,options,have,been,explored,,of,which,the,most,robust,appears,to,be,‘DG-,mimicking,spectral,vanishing,viscosity’,(SVV),,see,ref,[37].,Fernandez,et,al,[38],specifically,addresses,robustness,when,mesh,resolution,becomes,poor.,A,better,approach,from,the,point-of-view,of,accuracy,is,to,insert,more,resolution,(more,elements,or,increased,order,polynomials),where,aliasing,error,has,been,detected.,This,has,its,limitations,in,terms,of,cost,,but,there,are,also,practical,issues,concerning,refinement,that,still,need,investigation.,Some,of,these,latter,points,might,be,more,easily,addressed,by,reformulating,the,problem,in,terms,of,a,variational,approach,and/or,using,the,Lie,derivative,formulation,[39].,It,was,anticipated,that,these,could,be,addressed,in,the,cross-cutting,programme.,Interaction,with,a,reactor,design,framework,awaits,a,better,definition,of,data,structures,for,such,tools.,There,seems,no,reason,however,why,this,should,not,be,addressed,inside-out,,as,regardless,,techniques,have,to,be,produced,for,moving,data,at,Exascale.,Subject,to,the,above,qualifications,,the,production,of,proxyapps,should,proceed,in,Y2,as,indicated,in,the,Activities,Plan,[40],,viz.,a,larger,task,:,•,to,define,a,referent,physics,model,for,the,tokamak,edge,region,,accounting,for,magnetised,plasma,behaviour,in,the,presence,of,significant,numbers,of,neutral,atoms,and,molecules,,allowing,for,radiation,and,chemical,reactions,,and,identifying,important,wall,interactions,such,as,sheath,formation.,This,task,will,be,expected,to,interact,with,other,tasks,to,ensure,a,feasible,implementation.,It,is,desirable,that,theroretical,support,be,provided,for,the,EBC,pilot,code,development,,which,is,a,2-D,fluid,model,,as,well,as,for,the,1-D,fluid,cases,with,kinetic,effects,explicitly,listed,below.,The,candidate,algorithms,are,expected,to,employ,spectral,finite,element,and,particle,representations.,There,are,5,tasks,for,which,advanced,mathematical,skills,will,be,important:,1.,to,assess,performance,of,spectral,elements,for,NEPTUNE,,2.,to,examine,the,optimal,replacement,of,plasma,species,properties,represented,on,high-,order,spatially,accurate,meshes,by,a,particle,representation,and,vice,versa,,3.,to,study,uncertainty,quantification,(UQ),techniques,for,NEPTUNE,,4.,to,study,model,order,reduction,techniques,for,NEPTUNE,,5.,to,investigate,matrix-preconditioning,techniques,for,NEPTUNE.,There,are,4,tasks,to,develop,proxy-apps,for,NEPTUNE,of,demonstrable,,high,accuracy,in,challenging,test-,cases,,namely:,14,Commercial,in,Confidence,a),a,2-D,model,of,anisotropic,heat,transport,,b),a,2-D,elliptic,solver,in,complex,geometry,,c),a,1-D,fluid,solver,with,simplified,physics,but,with,UQ,and,realistic,boundary,conditions,,d),a,1-D,plasma,model,incorporating,velocity,space,effects.,There,are,2,tasks,concerning,software,engineering,for,NEPTUNE:,1.,to,investigate,DSL,and,code,generation,techniques,for,NEPTUNE,and,2.,to,investigate,,in,collaboration,with,UKAEA,staff,,data,structures,and,design,patterns,for,NEPTUNE.,The,Science,Plan,[33],shows,how,the,proxyapps,should,feed,into,the,5-year,development.,All,the,tasks,may,use,any,identified,software,packages,including,commercial,software,as,part,of,the,demonstration,process,,provided,a,feasible,route,to,producing,code,freely,usable,by,NEPTUNE,is,clearly,indicated.,It,will,obviously,be,better,if,a,task,is,linked,to,the,delivery,of,a,proxy-app.,Acknowledgement,The,support,of,the,UK,Meteorological,Office,and,Strategic,Priorities,Fund,is,acknowledged.,15,Commercial,in,Confidence,Bibliography,[1],H.,Wilson.,The,impact,of,plasma,physics,on,the,timescale,to,a,tokamak,fusion,power,plant.,Philosophical,Transactions,of,the,Royal,Society,A,,377(2141):20170435,(14,pages),,2019.,[2],E.,Surrey.,Engineering,challenges,for,accelerated,fusion,demonstrators.,Philosophical,Transactions,of,the,Royal,Society,A,,377(2141):20170442,(14,pages),,2019.,[3],Iter,Organisation.,ITER,Research,Plan,within,the,Staged,Approach.,Technical,Report,ITR-18-003,,Iter,Organisation,,2018.,https://www.iter.org/doc/www/content/com/Lists/,ITER\%20Technical\%20Reports/Attachments/9/ITER-Research-,Plan\_final\_ITR\_FINAL-Cover\_High-Res.pdf.,[4],Eurofusion.,European,Research,Roadmap,to,the,Realisation,of,Fusion,Energy.,Technical,Report,SOFT,2018,Version,,Eurofusion,,2018.,https://www.euro-fusion.org/fileadmin/user\_upload/,EUROfusion/Documents/2018\_Research\_roadmap\_long\_version.pdf.,[5],S.V.,Patankar.,Numerical,heat,transfer,and,fluid,flow.,Hemisphere,Publishing,,Washington,,1980.,[6],B.J.,Braams.,Computational,studies,in,tokamak,equilibrium,and,transport.,PhD,thesis,,Rijksuniversiteit,Utrecht,,Netherlands,,1986.,[7],W.,Arter.,Numerical,simulation,of,magnetic,fusion,plasmas.,Reports,on,Progress,in,Physics,,58:1–59,,1995.,http://dx.doi.org/10.1088/0034-4885/58/1/001.,[8],Z.,Carter.,An,Application,of,the,Spectral/hp,Element,Method,to,Plasma,Fluid,Modelling,in,BOUT++.,Technical,Report,MSc,Thesis,Report,,University,of,York,,2015.,[9],M.,Bhojwani.,Simulating,edge,localised,modes,within,Tokamak,plasmas,using,hp/spectral,finite,elements,in,Nektar++.,Technical,report,,Imperial,College,London,,Department,of,Aeronautics,,2015.,[10],G.,Karniadakis,and,S.,Sherwin.,Spectral/hp,element,methods,for,computational,fluid,dynamics,2nd,Ed.,Oxford,University,Press,,2005.,[11],W.,Arter.,Challenges,for,modelling,fusion,plasmas.,Technical,Report,Presentation,at,Isaac,Newton,Institute,Plasma,Physics,Day,,UKAEA,,2019.,https://www.newton.ac.uk/seminar/20191028140014452.,[12],C.,Pozrikidis.,Introduction,to,finite,and,spectral,element,methods,using,MATLAB.,CRC,Press,,2005.,[13],M.,Turner,,D.,Moxey,,J.,Peiró,,M.,Gammon,,C.R.,Pollard,,and,H.,Bucklow.,A,framework,for,the,generation,of,high-order,curvilinear,hybrid,meshes,for,CFD,simulations.,Procedia,Engineering,,203:206–218.,http://dx.doi.org/10.1016/j.proeng.2017.09.808.,[14],L.,Botti.,Influence,of,reference-to-physical,frame,mappings,on,approximation,properties,of,discontinuous,piecewise,polynomial,spaces.,Journal,of,Scientific,Computing,,52(3):675–703,,2012.,http://dx.doi.org/10.1007/s10915-011-9566-3.,[15],C.D.,Cantwell,,D.,Moxey,,A.,Comerford,,A.,Bolis,,G.,Rocco,,G.,Mengaldo,,D.,De,Grazia,,S.,Yakovlev,,J.-,E.,Lombard,,D.,Ekelschot,,B.,Jordi,,H.,Xu,,Y.,Mohamied,,C.,Eskilsson,,B.,Nelson,,P.,Vos,,C.,Biot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