CD-EXCALIBUR-FMS0078-M6c.4_ExcaliburFusionModellingSystemComplementaryActionsCodeIntegrationAcceptanceOpera
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ExCALIBUR,Complementary,actions.,Code,integration,,ac-,ceptance,and,operation,4.,M6c.4,Version,1.00,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Mile-,stone,M6c.4.,It,provides,material,concerning,the,implementation,of,plasma,fluid,solvers,in,NEKTAR++.,Restricting,attention,to,finite,element,problems,without,particles,,there,is,a,basic,exploration,of,two-way,code,coupling,,in-,cluding,time-independent,and,time-dependent,problems.,There,are,also,heuristic,discussions,of,the,role,of,dimensionless,parameters,,particularly,in,relation,to,the,numerical,accuracy,and,stability,of,advection,calculations.,Thereafter,are,discussed,numerical,issues,associated,with,solving,highly-,anisotropic,diffusion,problems,,including,a,possible,new,option,for,reducing,the,difficulties,caused,by,anisotropy.,Appendices,report,progress,(A),on,the,application,of,harmonic,functions,to,2-D,and,4-D,fluid,problems,,(B),a,sim-,ple,application,of,the,Nvidia,IndeX,plugin,for,use,with,PARAVIEW,and,(C),a,discovery,of,misleading,1-D,plots,of,data,in,higher,order,representations,by,PARAVIEW.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/ExCALIBUR,-FMS/0078,Issue:,Date:,1.01,29,September,2023,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Ed,Threlfall,Owen,Parry,Signature,N/A,N/A,Date,29,September,2023,29,September,2023,BD,Reviewed,By:,Wayne,Arter,29,September,2023,BD,1,1,Introduction,The,current,phase,of,Project,NEPTUNE,involves,the,construction,of,proxyapps,,implemented,in,the,NEKTAR++,framework,,capable,of,simulating,plasma,fluid,equations.,An,outline,of,current,code,development,work,in,this,direction,,which,is,being,performed,by,UKAEA,in,collaboration,with,some,of,the,grant,holders,,is,presented,in,Sec.2.,Following,on,from,the,realisation,that,NEKTAR++,does,not,support,calculations,where,there,is,a,finite,volume,of,thermally,conducting,solid,in,addition,to,fluid,,it,has,proved,necessary,to,consider,coupling,of,finite,element,code,with,finite,element,code,,(The,coupling,of,finite,element,code,with,particle,code,will,be,which,is,the,topic,of,Section,3.,discussed,in,the,Deliverable,7c,report,due,at,the,end,of,the,FY.),The,computational,results,have,thrown,up,a,range,of,issues,which,have,inspired,correspond-,ing,work,on,code,integration,,acceptance,and,operation.,Often,there,is,difficulty,is,identifying,the,mechanism,responsible,for,the,computational,problem,,which,might,be,as,simple,as,a,mis-,understanding,of,the,NEKTAR++,documentation,,but,could,involve,a,deficiency,in,the,numerical,algorithm,,a,more,fundamental,problem,due,to,the,underlying,nonlinear,physics,,or,a,combination,of,both,the,latter.,Thus,a,number,of,lines,of,work,have,been,pursued,in,parallel,with,the,compu-,tations,,then,re-prioritised,perhaps,in,light,of,new,calculations,indicating,a,different,issue,,so,that,parts,of,this,present,report,inevitably,describe,work,in,progress.,Issues,concerning,stability,of,calculations,,the,need,for,implicit,schemes,and,the,accuracy,of,the,results,,are,capable,of,being,addressed,by,the,dimensionless,parameters,introduced,and,described,in,Section,??.,Section,6,contains,a,presentation,of,a,simplified,anisotropic,diffusion,problem,and,the,numerical,issues,that,arise,in,cases,where,the,anisotropy,is,very,large,,including,an,idea,for,moderating,these,numerical,difficulties,that,could,be,considered,for,implementation.,Appendices,include,preliminary,material,on,other,lines,of,study,which,have,been,paused,in,the,absence,of,any,immediate,application,to,Project,NEPTUNE.,2,Plasma,dynamics,in,NEKTAR++,This,section,contains,a,brief,overview,of,plasma,simulation,codes,implemented,,and,currently,under,implementation,,in,the,NEKTAR++,framework.,In,the,language,of,that,framework,the,codes,herein,implemented,are,“novel,NEKTAR++,solvers”.,2.1,2-D,Proxyapps,The,repository,[1],contains,a,NEKTAR++,implementation,,called,Nektar-Driftwave,,of,the,Hasegawa-,Wakatani,equations,,which,describe,drift-wave,turbulence,in,two,dimensions,under,a,number,of,simplifying,assumptions.,The,outputs,of,this,code,are,described,briefly,in,the,UKAEA,report,[2],,in,which,context,it,should,be,noted,that,the,‘density’,in,the,model,is,not,restricted,to,non-negative,values,because,it,actually,describes,a,perturbation,of,a,fixed,background.,Note,that,the,small,time-step,means,that,the,code,is,quite,slow,-,c.32,core-hours,to,reproduce,the,example,shown,in,Fig.1.,2,Figure,1:,Plots,showing,the,time-evolution,in,Nektar-Driftwave,of,the,electron,density,,starting,from,an,initial,Gaussian-like,density,,vorticity,,and,potential,profile.,The,repository,[3],contains,NEKTAR++,implementations,of,the,Blob2D,and,Blob2D-Te-Ti,examples,taken,from,the,plasma,simulation,framework,HERMES-3,[4],,which,is,a,currently,active,project,led,by,Ben,Dudson,now,of,LLNL,,differing,from,NEPTUNE,in,that,it,uses,finite-difference,methods,as,opposed,to,spectral/hp,element.,Both,these,implementations,were,written,by,the,holder,of,NEP-,TUNE,grant,T/AW085/22,and,were,then,further,developed,in,collaboration,with,UKAEA,NEP-,TUNE,personnel.,Blob2d,,described,in,more,detail,in,[5],,is,an,isothermal,model,of,an,extended,plasma,filament,propagating,in,two,dimensions,,with,the,forcing,provided,by,the,diamagnetic,drift,term.,(The,forcing,arises,as,the,centrifugal,acceleration,experienced,by,a,region,of,increased,density,in,a,curved,magnetic,field).,Blob2D-Te-Ti,is,an,extension,allowing,different,electron,and,ion,temperatures.,There,was,an,issue,with,the,grantee’s,initial,implementation,of,the,above,systems,in,that,the,Discontinuous,Galerkin,(DG),numerical,flux,term,for,the,forcing,term,,eg.,∂n,∂y,,,was,absent,,but,the,codes,seemed,to,work,acceptably,well,,giving,plausible,outputs.,Plausibility,is,likely,explained,by,the,relatively,small,magnitude,of,the,forcing,term,relative,to,pressure,gradients,and,viscous,effects,,in,any,event,this,issue,is,currently,being,addressed,in,conjunction,with,the,grantee.,The,issue,does,not,affect,the,aforementioned,Nektar-Driftwave,as,there,the,forcing,term,is,a,derivative,of,a,continuous,field.,(To,be,more,specific,,the,electrostatic,potential,Φ,is,a,NEKTAR++,ContField,,whereas,the,other,fields,are,of,type,DisContField,as,they,are,used,in,a,DG,scheme).,2.2,3-D,Proxyapps,A,central,goal,of,this,phase,of,Project,NEPTUNE,is,to,implement,a,3-D,plasma,fluid,model,capable,of,scaling,ultimately,to,Exascale.,Given,the,history,of,the,project,,it,is,natural,to,base,this,code,on,the,NEKTAR++,framework.,The,code,is,necessary,in,order,to,meet,upcoming,UKAEA,deliverable,requirements,beginning,with,M4c.3,Integrated,3-D,Particle,and,Continuum,Model,,and,is,also,to,be,supplied,to,grantees,to,focus,eg.,efforts,on,uncertainty,quantification,and,reduced-,order,modelling,during,the,remaining,duration,of,the,project.,Three,possible,models,have,been,identified,,with,the,idea,that,the,simplest,is,a,minimal,system,that,nevertheless,exhibits,3-D,plasma,turbulence,and,which,still,can,be,coupled,to,the,UKAEA-,developed,particle,capability,,whereas,the,most,complex,represents,the,ability,to,address,current,problems,of,research,interest.,All,three,of,these,models,exist,in,the,same,framework,,using,the,same,NEKTAR++,solver,with,different,choices,for,EquationSystem,,meaning,there,is,a,natural,3,upgrade,path,for,any,coupled,code,implemented,using,the,simpler,choices.,All,three,examples,use,a,uniform,magnetic,field,aligned,with,the,z-axis.,Their,initial,implementations,use,purely,explicit,time-evolution,schemes,since,in,many,tokamak,applications,the,size,of,the,timestep,is,likely,to,be,determined,by,the,parallel,dynamics,,whence,implicit,treatment,of,cross-field,diffusion,would,not,be,required.,An,important,note,is,that,one,issue,with,NEKTAR++,affects,all,three,examples:,it,is,not,yet,entirely,clear,how,to,implement,a,purely,transverse,Laplacian,operator,∇2,⊥,,meaning,basically,the,ability,to,invert,∂2,∂y2,in,a,3-D,simulation,,though,the,API,for,this,appears,to,exist.,The,following,applications,currently,use,the,full,Laplacian,inverse,,awaiting,full,implementation,of,∇2,⊥,into,the,library,from,the,NEKTAR++,developers.,∂x2,+,∂2,2.2.1,2-D,Hasegawa-Wakatani,equations,implemented,in,3-D,In,this,case,the,equations,are,those,of,the,Nektar-Driftwave,proxyapp,referenced,earlier,[1].,The,suppressed,dimension,is,the,one,along,the,magnetic,field,direction.,In,this,model,the,proxyapp,website,[1],gives,values,for,all,necessary,values,of,scaling,or,physical,parameters.,There,is,no,im-,plicit,parameter,(see,Section,5.1),because,this,model,typically,exhibits,growth,leading,apparently,to,a,saturated,state,regardless,of,initial,perturbation.,There,is,in,Nektar-Driftwave,a,pre-existing,example,exhibiting,saturated,plasma,turbulence.,In,order,to,realise,turbulence,the,accuracy,of,the,simulation,was,found,to,need,to,be,greater,than,a,certain,threshold,,because,the,growth,of,turbulence,is,suppressed,by,numerical,dissipation,in,low-accuracy,simulations.,This,Nektar-Driftwave,example,has,been,run,successfully,in,a,3-D,configuration,,see,Fig.2.,By,starting,with,an,axially-varying,field,and,using,a,domain,with,a,larger,z-extent,than,this,latter,case,,a,more,demanding,test,of,3-D,numerical,stability,has,also,been,produced,,see,Fig.3.,2.2.2,Fully,3-D,Hasegawa-Wakatani,equations,The,2-D,Hasegawa-Wakatani,system,of,the,preceding,sections,is,a,reduction,under,the,assump-,tion,of,a,single,longitudinal,Fourier,mode,,of,a,3-D,system,that,includes,parallel,dynamics.,The,3-D,equations,are,from,B.Dudson,(unpublished,note,,dated,13/5/2015),∂n,∂t,∂ω,∂t,+,[Φ,,n],+,+,[Φ,,ω],+,ωce,νei,ωce,νei,∇,·,(cid:0)b,(cid:0)∂∥Φ,−,∂∥n(cid:1)(cid:1),=,0,∇,·,(cid:0)b,(cid:0)∂∥Φ,−,∂∥n(cid:1)(cid:1),=,0,Z∇2,⊥Φ,=,ω.,(1),where,in,the,above,,the,‘Poisson,bracket’,[a,,b],≡,∂a,∂x,∂b,∂y,−,∂b,∂x,∂a,∂y,.,One,appealing,aspect,of,this,system,is,that,,in,the,above,dimension-reduced,form,,one,ad-,justable,physical,parameter,,namely,the,ratio,of,the,electron,plasma,oscillation,frequency,ωce,to,the,electron-ion,collision,frequency,νei,,stands,out.,However,,although,there,is,an,implementation,of,this,model,in,HERMES-3,,at,the,time,of,writing,it,appears,that,this,is,simply,a,test,that,the,code,4,Figure,2:,Plots,showing,the,time-evolution,of,an,initial,Gaussian,profile,of,electron,density,in,a,3-D,implementation,of,the,2-D,Hasegawa-Wakatani,equations.,Periodic,conditions,are,used,in,all,dimensions,and,the,initial,data,was,symmetric,in,the,axial,direction.,Note,that,this,output,corresponds,to,a,time,before,the,turbulence,appears.,Figure,3:,Plots,showing,the,time-evolution,of,an,initial,Gaussian,density,profile,modulated,by,a,sinusoid,in,the,axial,direction,,in,a,3-D,implementation,of,the,2-D,Hasegawa-Wakatani,equations.,Periodic,conditions,are,used,in,all,dimensions.,This,output,either,precedes,the,appearance,of,turbulence,or,lacks,the,physical,and,/,or,numerical,conditions,needed,to,generate,turbulence.,5,Figure,4:,At,left,is,output,from,the,HERMES-3,LAPD,example,of,the,electron,density,at,the,end,of,the,simulation.,On,the,right,is,the,computational,mesh,showing,in,particular,the,elongated,aspect,of,the,LAPD.,runs,correctly,on,a,GPU,,hence,its,results,are,not,certain,to,be,physically,relevant.,The,developers,of,HERMES-3,via,the,holders,of,Grant,T/AW088/22,,have,advised,that,a,value,of,approx.,2,×,105,for,ωce,corresponds,to,physically-relevant,values,Te,=,50eV,and,n0,=,1019m−3,for,the,scrape-off,νei,layer,of,a,tokamak,such,as,MAST-U.,Work,to,implement,this,system,in,NEKTAR++,is,ongoing,in,collaboration,with,the,holders,of,Grant,T/AW085/22.,2.2.3,Large,Plasma,Device,(LAPD),simulation,The,aim,here,is,a,NEKTAR++,implementation,of,the,HERMES-3,model,of,the,Large,Plasma,Device,machine.,The,system,of,equations,is,a,model,for,the,physics,of,the,Large,Plasma,Device,[6],located,at,UCLA,,which,differs,from,a,tokamak,in,that,it,has,a,long,straight,geometry.,This,model,system,was,chosen,as,a,relatively,straightforward,3-D,example,of,plasma,turbulence,and,a,prob-,lem,of,current,research,interest,as,agreed,in,discussion,with,representatives,of,UKAEA,Tokamak,Science.,The,HERMES-3,example,executes,in,approximately,eight,core-hours;,its,output,is,shown,in,Fig.4,,along,with,a,computational,mesh,showing,the,basic,geometry,of,the,LAPD.,This,two-fluid,model,has,considerable,additional,complexity,beyond,the,simpler,Hasegawa-Wakatani,treatments,shown,above.,The,equations,,with,subscript,e,used,for,electrons,and,i,for,ions,which,are,assumed,to,be,Deuterium,,are,for,the,evolution,of,respectively,electron,density,,separate,parallel,species,velocities,,and,a,perpendicular,plasma,velocity,in,the,streamfunction-vorticity,for-,6,mulation,me,∂ne,∂t,∂nev∥e,∂t,∂niv∥i,∂t,mi,∂ω,∂t,(cid:19),B2,∇⊥Φ,(cid:18),mn,∇,·,=,−∇,·,(cid:0)ne,(cid:0)vE×B,+,bv∥,(cid:1)(cid:1),;,=,−∇,·,(cid:0)menev∥e,(cid:0)vE×B,+,bv∥,(cid:1)(cid:1),−,∂∥pe,−,eneE∥,+,meneνei,(cid:0)v∥i,−,v∥e,(cid:1),;,=,−∇,·,(cid:0)miniv∥i,(cid:0)vE×B,+,bv∥,(cid:1)(cid:1),−,∂∥pi,−,eniE∥,−,miniνei,(cid:0)v∥i,−,v∥e,(cid:1),;,=,−∇,·,(cid:0)ω,(cid:0)vE×B,+,bv∥,(cid:1)(cid:1),+,∇,(cid:0)niv∥i,−,nev∥e,(cid:1),;,=,ω.,where,mn,is,plasma,mass,density,,and,the,ion,density,and,species,pressures,are,ni,=,ne;,pe,=,kneTe;,pi,=,kniTi;,vE×B,=,b,×,∇Φ,B,.,(2),(3),where,Te,and,Ti,are,user-specified,,and,the,last,equation,shows,how,the,electrostatic,potential,plays,a,role,analogous,to,a,streamfunction.,Fuller,details,appear,in,a,new,release,of,the,Equations,document,[7].,An,important,feature,of,the,boundaries,is,the,sonic,outflows,of,electrons,and,ions,at,the,ends,of,the,device,,and,it,is,worth,noting,the,densities,in,the,equations,(2),are,strictly,non-negative.,Work,to,implement,this,system,in,NEKTAR++,is,ongoing,in,collaboration,with,the,holders,of,Grants,T/AW085/22,and,T/AW088/22.,2.3,Demonstration,of,Kelvin-Helmholtz,instability,in,NEKTAR++,The,physics,of,3-D,plasma,involves,the,generation,of,turbulence,by,several,mechanisms,,including,the,Kelvin-Helmholtz,instability,which,means,that,a,velocity,field,with,sufficient,velocity,gradient,transverse,to,the,main,flow,experiences,a,growing,instability.,In,order,to,study,this,,a,simple,flow,field,driven,by,a,body,force,was,implemented,in,the,incompressible,Navier-Stokes,solver,of,NEKTAR++.,The,simple,functional,form,of,the,body,force,produces,flow,fields,reminiscent,of,the,fluid,convection,studied,in,eg.,[8],but,lacking,the,complexities,of,the,heat,transfer,physics,such,as,diffusion,of,heat.,The,strength,of,the,forcing,is,scaled,to,a,parameter,akin,to,the,Rayleigh,Number,Ra,,so,that,the,functional,form,is,(cid:18),(cid:18),(cid:19)(cid:19),F,=,Ra,sin9,2π,y,−,1,2,;,(4),and,the,kinematic,viscosity,was,set,to,a,value,of,0.1.,Three,examples,were,obtained,using,different,values,of,Ra,,see,Figs.,5,,6,and,7.,It,is,seen,that,the,resulting,Ra,=,10,case,is,a,steady,flow,,Ra,=,102,is,an,oscillatory,flow,and,Ra,=,103,is,a,7,Figure,5:,Horizontal,velocity,component,(red,is,right-moving,and,blue,is,left-moving),(left),,and,streamlines,coloured,according,to,vorticity,(right),for,Ra,=,101.,The,flow,is,steady,and,the,vortex,field,is,largely,featureless.,The,scales,are,such,that,the,largest,values,are,approximately,±1.4,for,the,velocity,component,and,up,to,7.4,for,the,vorticity,in,the,units,used,by,the,code.,Figure,6:,Horizontal,velocity,component,(red,is,right-moving,and,blue,is,left-moving),(left),and,streamlines,coloured,according,to,vorticity,(right),for,Ra,=,102.,The,flow,is,oscillatory,as,there,is,not,quite,enough,energy,present,for,a,stable,three-vortex,pattern,to,form.,The,scales,are,such,that,the,largest,values,are,approximately,±14,for,the,velocity,component,and,up,to,140,for,the,vorticity,in,the,units,used,by,the,code.,steady,vortex,flow.,It,is,expected,that,larger,Ra,values,would,lead,to,turbulent,flows,,possibly,via,steady,states,with,higher,numbers,of,vortices.,3,Coupling,Finite,Element,Codes,Thanks,to,the,present,limitations,of,NEKTAR++,it,is,necessary,to,treat,problems,with,one,or,more,regions,of,solid,with,different,thermal,conductivities,using,code,coupling.,This,situation,is,mocked,up,by,coupling,two,models,with,different,thermal,conductivities,using,FIREDRAKE,[9].,This,section,contains,some,simple,examples,of,coupling,NEPTUNE,-relevant,codes,,which,could,be,used,with,coupling,frameworks,such,as,CWIPI,and,MUI.,Note,that,in,this,section,and,the,following,one,,diffusion,and,heat,conduction,are,treated,as,being,interchangeable.,8,Figure,7:,Horizontal,velocity,component,(red,is,right-moving,and,blue,is,left-moving),(left),and,streamlines,coloured,according,to,vorticity,(right),for,Ra,=,103.,The,three,vortex,pattern,is,stable,,though,subject,to,minor,wobbles.,The,scales,are,such,that,the,largest,values,are,approximately,±66,for,the,velocity,component,and,up,to,750,for,the,vorticity,in,the,units,used,by,the,code.,3.1,Time-independent,problems,3.1.1,Prelude:,one-dimensional,diffusion,or,heat,conduction,The,model,described,here,involves,two,regions,of,thermally,conducting,material,,both,unit,square.,Let,1.,the,temperature,of,the,left-hand,side,of,the,composite,be,unity,and,that,of,the,right,side,zero.,2.,the,top,and,bottom,of,both,squares,be,insulated,,and,boundary,conditions,be,chosen,so,that,the,heat,flow,is,one-dimensional.,3.,the,parameters,of,the,left-hand,region,be,labelled,with,index,1,and,those,of,the,right-hand,with,index,2;,thus,the,thermal,conductivities,are,κ1,for,the,leftmost,region,and,κ2,for,the,other.,4.,the,ratio,of,thermal,conductivities,be,β,≡,κ1,κ2,.,It,is,useful,to,know,that,for,a,one-dimensional,conduction,problem,the,mid-boundary,temperature,is,given,by,Tmid,=,β,1+β,and,also,that,β,is,the,ratio,of,the,temperature,gradient,in,the,second,region,to,that,in,the,first.,These,results,are,trivially,obtained,by,matching,the,temperature,and,heat,flux,at,the,internal,boundary.,The,natural,approach,is,to,couple,the,two,regions,(left,and,right),at,their,common,boundary,,sepa-,rately,updating,the,boundary,conditions,in,a,series,of,iterations,,leading,to,the,following,algorithm:,•,Select,initial,guess,T0,for,the,internal,boundary,temperature.,•,Solve,the,left-hand-side,(LHS),conduction,problem.,•,Use,the,value,of,the,left-hand-side,temperature,gradient,to,give,a,Neumann,boundary,con-,dition,for,the,right-hand-side,(RHS),conduction,problem,,by,matching,heat,flux.,9,Figure,8:,One-dimensional,conduction.,Figure,9:,Sketch,of,coupling,algorithm,in,1-D.,10,•,Solve,the,RHS,conduction,problem.,•,Use,a,weighted,average,of,the,internal,boundary,temperatures,of,the,LHS,and,RHS,region,to,overwrite,the,initial,guess,for,the,internal,Dirichlet,boundary,condition,for,the,LHS,region,,and,go,to,the,second,step.,•,Iterate,all,but,the,first,step,until,convergence.,The,above,algorithm,successfully,gives,a,converged,answer,for,a,range,of,values,of,β.,The,algorithm,is,sufficiently,simple,that,its,working,can,be,reproduced,analytically:,let,the,midpoint,temperature,after,the,nth,iteration,be,labelled,by,Tn,,then,there,follows,the,recurrence,relation,(letting,the,LHS,temperature,be,weighted,by,w−1,w,in,the,Dirichlet,boundary,condition),w,and,the,RHS,one,1,which,has,solution,Tn+1,=,1,w,((w,−,1)Tn,+,β(1,−,Tn))),Tn,=,(cid:18),w,−,1,−,β,w,(cid:19)n,(cid:18),T0,−,(cid:19),β,1,+,β,+,β,1,+,β,.,(5),(6),This,has,the,obvious,form,of,the,desired,answer,β,1+β,plus,an,error,term,proportional,to,the,distance,between,the,initial,guess,T0,and,the,desired,answer.,It,is,not,difficult,to,see,that,if,the,recurrence,relation,converges,then,it,converges,as,a,nested,product,to,β,1,+,β,=,1,w,(cid:18),β,+,(w,−,1,−,β),(cid:18),β,+,(w,−,1,−,β),1,w,1,w,(β,+,(w,−,1,−,β),...,),.,(7),(cid:19)(cid:19),If,expanded,this,is,seen,to,be,equivalent,to,the,Taylor,series,for,just,β,1+β,around,β,=,w,−,1,which,is,β,1,+,β,=,w,−,1,w,+,(−1)n+1,wn+1,(β,−,w,+,1)n,,(8),∞,(cid:88),n=1,that,obviously,has,radius,of,convergence,rc,=,w,and,for,real,β,converges,for,−1,<,β,<,2w,−,1.,Another,approach,is,to,use,the,analytic,solution,in,one,dimension,to,choose,the,weighting,co-,β,1+β,and,efficients.,This,leads,to,the,conclusion,that,the,LHS,temperature,should,have,weight,the,RHS,1,1+β,(obviously,these,sum,to,unity).,In,1-D,this,value,yields,the,exact,solution,after,one,iteration,because,β,1,+,β,≡,β,1,+,β,T0,+,1,1,+,β,β(1,−,T0).,(9),(The,Taylor,series,terminates,after,the,constant,term.),This,choice,of,weighting,will,be,referred,to,as,the,‘optimal’,value.,For,more,complex,cases,,notably,where,one,of,the,regions,contains,moving,fluid,,it,is,possible,to,draw,on,the,theory,of,domain,decomposition,/,Schwarz,method,,eg.,[10].,11,Figure,10:,Solid,conducting,regions,temperature,profile,for,β,=,0.5,(top),and,β,=,2.0,(bottom).,3.1.2,Two-dimensional,diffusion,,fluid,heat,transfer,,and,anisotropic,diffusion,The,method,outlined,above,has,been,found,to,be,effective,for,less-trivial,two-dimensional,prob-,lems,,ie.,a,PDE,rather,than,just,an,ODE.,In,this,case,,the,1-D,boundary,profile,of,the,temperature,and,its,x-derivative,are,used,as,boundary,conditions.,The,problem,is,made,two-dimensional,by,having,a,non-constant,temperature,profile,on,the,left-hand,boundary,of,the,composite,domain.,The,function,T,=,y2(1,−,y),was,used,(chosen,not,to,have,any,particular,symmetry).,The,tempera-,ture,and,the,heat,flux,output,from,the,code,match,on,either,side,of,the,internal,boundary,,see,Fig.,10.,Note,that,these,results,used,w,=,2,in,the,above,formulae,rather,than,the,superior,‘optimal’,value.,Further,,since,only,the,‘zero’,transverse,Fourier,mode,ever,transmits,non-zero,heat,flux,,in,important,aspects,higher-dimensional,cases,resemble,the,one-dimensional,problem.,It,can,be,shown,that,the,same,approach,works,when,the,first,conductor,is,replaced,with,a,lam-,inar,convecting,fluid,cell.,The,fluid,problem,is,incompressible,Navier-Stokes,with,Boussinesq-,approximation,buoyancy,term,and,is,parametrized,by,the,usual,Rayleigh,(Ra),and,Prandtl,(Pr),numbers.,The,temperature,at,the,internal,boundary,is,allowed,to,vary,,giving,what,is,described,by,Nield,[11],as,the,‘Rayleigh-Jeffreys,problem’.,Parameter,β,now,represents,the,ratio,of,the,thermal,conductivity,of,the,fluid,to,that,of,the,solid.,An,example,output,temperature,profile,is,shown,in,Fig.11,,which,has,β,=,0.5,,Ra,=,105,,Pr,=,1.0.,The,temperature,on,the,left-most,boundary,is,set,to,be,unity.,Example,streamlines,in,the,fluid,12,Figure,11:,Fluid,(left),and,solid,conducting,regions,temperature,profile,for,Ra,=,105,,Pr,=,1.0,,β,=,0.5.,Figure,12:,Fluid,region,flow,streamlines,coloured,by,temperature,profile,for,Ra,=,105,,Pr,=,1.0,,β,=,0.75.,region,are,shown,in,Fig.12,(for,β,=,0.75),where,it,can,be,seen,that,the,coupled,solution,does,not,have,the,same,symmetry,as,would,be,found,in,a,typical,slot,problem,with,constant,temperatures,on,the,left,and,right,uprights.,The,method,works,also,for,problems,with,anisotropic,diffusion,ie.,the,diffusivity,in,either,domain,can,be,a,symmetric,tensor.,The,temperature,and,the,normal,component,of,the,heat,flux,are,made,continuous,across,the,boundary,,and,it,is,found,that,the,method,works,correctly,provided,the,expression,for,the,flux,component,is,correct,-,this,is,easily,obtained,using,the,usual,rules,for,tensor,transformation,,giving,the,x-component,of,the,LHS,flux,needed,for,the,Neumann,data,as,κ∥,κ2,(cid:18),cos2,θ,∂T,∂x,+,sin,θ,cos,θ,(cid:19),∂T,∂y,+,κ⊥,κ2,(cid:18),sin2,θ,∂T,∂x,−,sin,θ,cos,θ,(cid:19),.,∂T,∂y,(10),This,has,been,tested,for,cases,with,κ∥,=,10,and,κ⊥,=,0,and,1,and,non-zero,angles.,13,Figure,13:,Anisotropic,diffusion,,tightly-coupled,,κ⊥,=,0,in,the,left-hand-side,region,and,β,=,10.,At,top:,all,one,FIREDRAKE,solve,,using,a,position-dependent,diffusivity,tensor.,Below:,the,same,problem,done,using,two,solves,coupled,as,discussed,in,the,main,text,,showing,very,good,agreement,which,was,further,checked,by,inspection,of,1-D,sections.,The,outputs,from,coupling,of,this,type,can,be,checked,against,a,tight-coupled,FIREDRAKE,im-,plementation,(ie.,the,whole,problem,is,one,big,domain,with,a,diffusivity,tensor,that,depends,on,position),-,see,Fig.13.,Note,that,this,two-way,coupled,implementation,extends,a,simple,one-way,coupled,model,used,earlier,in,the,project,and,supplied,to,the,holder,of,the,grant,T/AW085/22,concerning,uncertainty,quantification,and,reduced-order,models.,3.2,Time-dependent,problems,It,is,necessary,to,study,time-dependent,problems,in,order,to,treat,cases,with,turbulence,or,with,particle,noise,,both,of,which,preclude,a,strictly,time-independent,solution.,One,approach,is,to,treat,each,timestep,as,an,elliptic,problem,and,use,the,‘optimal’,method,outlined,above.,This,has,been,tested,for,a,coupled,time-dependent,diffusion,problem,with,scalar,diffusivity,D,=,1,in,both,coupling,regions,and,using,a,simple,finite-difference,code,(a,finite-element,code,could,easily,be,substituted).,The,method,to,solve,the,elliptic,problem,is,to,call,the,time-step,algorithm,but,not,update,the,initial,data,(in,the,code,this,is,achieved,by,copying,back,the,original,14,Figure,14:,One,frame,at,t,=,0.5,from,a,time-dependent,simulation,of,diffusion,with,coefficient,D,=,1,,coupling,in,the,middle,of,the,domain.,The,initial,data,was,T,=,0,everywhere,,with,boundary,Dirichlet,values,of,1,and,0.,The,horizontal,axis,is,x,and,the,vertical,axis,is,temperature,T,.,data,after,the,timestep).,For,the,backward,Euler,(implicit),scheme,the,elliptic,problem,is,easily,formulated,from,T,n+1,−,T,n,D∆t,=,∇2,T,n+1,where,the,solution,Tn,is,assumed,known,at,the,start,of,the,timestep,,as,−,d2T,n+1,dx2,+,(D∆t)−1,T,n+1,=,(D∆t)−1,T,n.,(11),(12),The,initial,guess,is,the,Dirichlet,data,at,step,n,on,the,right-hand-side,of,the,left-hand-side,re-,gion.,The,Neumann,boundary,condition,used,to,solve,the,RHS,region,comes,from,a,simple,finite,difference,at,the,RHS,end,of,the,LHS,region.,The,Crank-Nicolson,time-stepping,method,used,in,the,LHS,region,needed,to,be,downgraded,to,backward,Euler,since,oscillations,inherent,to,the,former,method,resulted,in,non-convergence.,These,oscillations,are,a,well-known,feature,of,Crank-,Nicolson,which,is,absent,in,backward,Euler,,appearing,when,the,initial,data,is,T,=,0,and,the,LHS,boundary,of,the,region,is,at,T,=,1.,The,problem,is,to,calculate,diffusion,on,[0,,2],with,unit,Dirichlet,on,the,LHS,and,homogeneous,Dirichlet,boundary,conditions,on,the,RHS,,for,which,the,analytic,solution,is,T,=,1,−,x,2,−,∞,(cid:88),n=1,2,πn,e−,Dn2π2t,4,sin,nπx,2,;,(13),Outputs,from,the,coupled,problem,,see,Fig.14,agree,with,the,analytic,solution.,15,4,Estimating,Accuracy,in,Advection-Diffusion,Problems,5,Model,Problem,This,section,develops,ref,[12,,§,2.2],,concerning,discretisations,of,the,time,dependent,advection-,diffusion,equation,for,a,scalar,field,f,(x,,t),in,a,flow,u(x,,t),with,diffusion,κ,and,source,term,S(x,,t),∂f,∂t,+,∇,·,(uf,),=,∇,·,(κ∇f,),+,S,(14),Suppose,that,f,only,depends,on,a,single,spatial,coordinate,x,,u,has,a,single,component,u,,and,also,that,the,diffusion,κ,=,const.,Neglecting,gradients,of,u,,the,analytic,model,equation,may,be,taken,∂f,∂t,+,u,=,κ,∂f,∂x,∂2f,∂x2,+,S,(15),Solution,is,made,in,the,finite,domain,0,≤,x,≤,L,over,a,time,interval,[0,,T,].,Note,that,the,assump-,tions,made,do,not,exclude,the,nonlinear,case,f,=,u,,when,Equation,(15),is,known,as,Burgers,equation,and,κ,becomes,the,viscous,diffusion.,It,will,be,important,for,subsequent,work,to,distinguish,different,approaches,to,a,dimensionless,treatment,of,Equation,(15),and,thence,Equation,(14),,see,Section,5.1.,The,following,Section,5.2,shows,how,discretisation,of,Equation,(15),introduces,new,dimensionless,quantities,based,on,mesh,spacing,,and,then,Section,5.3,goes,onto,discuss,the,implications,of,these,parameters,for,numerical,stability,and,accuracy.,Section,5.2,also,includes,treatment,of,the,hyper-diffusion,term,which,may,be,represented,as,an,addition,to,the,source,in,Equation,(15),of,form,where,κ4,is,the,hyper-diffusion,,aka,hyper-viscosity,in,the,nonlinear,case,that,f,=,u.,S4,=,−κ4,∂4f,∂x4,(16),5.1,Dimensionless,Parameters,Naturally,because,of,the,finite,domain,,distances,are,scaled,by,L,,when,by,far,the,common-,est,approach,,particularly,when,S,=,0,is,to,make,time,t,dimensionless,in,terms,of,the,diffusion,timescale,L2/κ.,It,will,be,recalled,that,this,leads,to,a,model,in,dimensionless,variables,of,the,form,where,the,Peclet,number,∂f,∂t,+,P,e,·,u,∂f,∂x,=,∂2f,∂x2,P,e,=,U0L,κ,(17),(18),with,U0,=,||u||,as,a,representative,absolute,value,of,u,,usually,the,maximum,flow-speed.,Making,t,dimensionless,in,terms,of,the,turnover,timescale,L/U0,,similarly,leads,to,the,appearance,of,(reciprocal,of),P,e,in,the,coefficient,of,the,diffusion,term.,The,point,is,that,in,the,linear,problem,,not,only,is,κ,fixed,at,least,in,order,of,magnitude,,but,also,so,is,u/U0,similarly,of,order,unity,throughout.,16,When,u,is,varying,appreciably,,eg.,in,time,as,a,result,of,instability,in,a,coupled,momentum,equation,(or,say,due,to,driving,by,S,when,f,=,u),,it,is,often,customary,to,omit,the,Peclet,number,,and,work,with,a,model,in,dimensionless,variables,of,the,form,∂f,∂t,+,u,∂f,∂x,=,∂2f,∂x2,(19),where,the,dimensionless,u,may,vary,over,orders,of,magnitude,,and,of,course,even,when,u,is,fixed,,this,formulation,is,workable.,The,fixed,flow,case,using,the,model,Equation,(19),is,then,apparently,parameter-free,,although,it,should,be,evident,that,the,scale,of,the,initial,values,of,u,serves,to,set,an,implicit,or,‘hidden’,Peclet,number.,(Similar,remarks,apply,in,respect,of,hyperdiffusion,upon,introducing,P,e4,=,||u||h3/κ4.),5.2,Discrete,Dimensionless,Parameters,Suppose,h,is,the,spatial,mesh,separation,implicitly,assumed,to,be,uniform,,ie.,h,=,L/N,,,where,N,+,1,with,N,≫,1,is,the,number,of,mesh-points,in,0,≤,x,≤,L.,Let,∆t,be,the,timestep,,then,the,discrete,time,advance,is,of,general,form,∆tf,∆t,+,u,h,∆xf,=,κ,∆2,xf,h2,(20),where,∆x,and,∆2,x,are,discrete,difference,operators,in,the,x-coordinate,,eg.,∆xf,=,fi+1,−,fi,,operating,on,the,discrete,values,of,fi,at,the,mesh-points,i.,(∆t,is,defined,analogously.),If,κ,=,0,and,there,is,only,hyper-diffusion,∆tf,∆t,+,u,h,∆xf,=,−κ4,∆2,xf,h4,(21),Evidently,the,discrete,Equation,(20),may,be,made,dimensionless,using,the,mesh-scale,h,,intro-,ducing,the,mesh,Peclet,number,parameter,P,eh,=,U0h,κ,(22),There,are,questions,concerning,the,role,of,P,eh,inherited,from,a,good,deal,of,controversy,about,the,role,of,the,mesh,Reynolds,number,Reh,,viz.,P,eh,for,the,nonlinear,case.,Nonetheless,,at,least,heuristically,it,would,seem,that,when,P,eh,=,O(1),,there,is,an,approximate,balance,between,the,discrete,advective,and,diffusive,terms.,For,smaller,P,eh,the,latter,dominates,,which,is,numerically,significant,in,that,the,discrete,Laplacian,operator,,being,symmetric,and,definite,,is,much,easier,to,treat,numerically,than,the,discrete,advection,operator.,(Similar,remarks,apply,to,Equation,(21),upon,introducing,P,e4h,=,),(23),U0h3,κ4,Other,important,well-known,dimensionless,groups,for,Equations(20),and,(21),are,the,Courant,or,CFL,number,and,the,diffusion,limit,parameters,,respectively,ch,=,U0∆t,h,,,dh,=,κ∆t,h2,,,and,d4,=,κ4∆t,h4,(24),17,When,these,are,separately,unity,,the,corresponding,timestep,has,a,simple,physical,interpreta-,tion,as,respectively,the,timescale,for,the,turnover,,diffusion,and,hyper-diffusion,of,one,cell,of,the,discretisation.,Note,also,the,identities,P,eh,=,ch,dh,and,P,e4h,=,ch,d4,(25),Note,finally,that,,particularly,in,the,case,of,nonlinear,problems,,it,is,unclear,how,long,the,simulated,duration,T,needs,to,be,,and,indeed,what,the,smallest,physically,significant,lengthscale,will,be,,thus,it,is,inadequate,to,consider,the,simple,dimensionless,ratios,∆t/T,and,∆x/L.,5.3,Stability,Parameters,and,Accuracy,If,the,temporal,discretisation,of,Equation,(20),is,explicit,,it,is,plausible,(as,is,confirmed,by,rigorous,analysis,up,to,factors,of,order,unity),that,for,stability,the,timestep,must,be,less,than,or,equal,to,the,turnover,and,diffusion,times,of,one,cell,,viz.,ch,≤,1,,dh,≤,1,,and/or,d4,≤,1,(26),Since,maximal,size,of,timestep,is,to,be,preferred,for,computational,efficiency,,then,to,within,O(1),factors,,ch,=,dh,=,1,will,be,taken,and,so,P,eh,=,1,at,the,stability,boundary.,It,is,important,to,consider,accuracy,,again,this,is,posed,in,heuristic,terms,by,assuming,that,the,smallest,lengthscale,ℓ,important,in,the,dynamics,of,f,is,capable,of,numerical,estimation,,or,known,It,follows,that,for,minimally,accurate,results,h,<,ℓ.,For,say,on,the,basis,of,analytic,results.,advection-diffusion,problems,,whether,or,not,they,are,linear,,provided,S,=,0,,then,typically,ℓ,=,LP,e−α,,or,ℓ4,=,LP,e−β,4,(27),for,positive,α,≤,1,and,β,≤,1.,Simple,balancing,of,coefficients,gives,α,=,β,=,1,,whereas,smaller,values,follow,in,higher,dimensions,when,the,flow,is,incompressible,(typically,α,=,1/2),or,from,estimates,of,turbulent,microscales,in,the,nonlinear,case,,giving,α,=,3/4,[13].,Suppose,further,that,hyper-diffusion,is,added,to,an,advection-diffusion,problem,for,additional,nu-,merical,smoothing,,then,it,is,desirable,that,the,minimum,scale,h4,attained,by,the,latter,satisfies,ℓ4,<,ℓ,,equivalently,P,e−β,4,<,P,e−α,,which,when,α,=,β,implies,κ4,<,κL2,,and,hence,P,e4,>,P,eh/N,2.,Remembering,that,L,=,N,h,and,P,e,=,U0L/κ,,then,h,<,LP,e−α,gives,h,<,N,1−α,α,κ,U0,,,ie.,P,eh,<,N,−1+,1,α,,,implying,ch,<,dhN,−1+,1,α,Similarly,for,hyper-diffusion,h3,<,N,1−3β,β,κ4,U0,,,ie.,P,e4h,<,N,−3+,1,β,,,implying,ch,<,d4N,−3+,1,β,(28),(29),The,results,Equations(28),and,(29),are,key.,From,Equation,(28),it,follows,that,stability,of,the,explicit,scheme,implies,accurate,advection-diffusion,for,all,reasonable,values,of,α.,Moreover,there,18,is,benefit,from,use,of,implicit,schemes,,even,on,uniform,meshes,if,α,<,1,,since,then,accuracy,is,possible,even,when,ch,≫,1.,Whether,accuracy,is,achieved,in,a,time,dependent,calculation,,depends,on,the,smallest,important,timescale,τ,.,The,work,of,Hunt,et,al.,[13],implies,τ,=,ℓ2/κ,,ie.,then,∆t,<,τ,implies,dh,<,1,since,h,<,ℓ,has,been,assumed,,so,that,temporal,accuracy,may,also,be,consistent,with,ch,≫,1.,(For,hyperdiffusion,,d4,<,κ4/(κh2),,so,if,κ4,<,κL2,,d4,<,N,2.),However,if,τ,=,ℓ/U0,,which,could,occur,if,there,were,say,large,initial,transients,,then,for,minimal,accuracy,in,the,time,advance,,∆t,<,τ,implies,ch,<,1.,By,contrast,,in,an,approach,to,a,steady-state,solution,,the,timescale,for,change,of,field,variables,could,become,very,long,,meaning,that,an,implicit,scheme,would,benefit,from,its,ability,to,take,very,large,timesteps,whilst,maintaining,accuracy.,6,Numerical,issues,with,anisotropic,diffusion,problems,6.1,Anisotropic,diffusion,-,simple,FIREDRAKE,treatment,6.1.1,Straight,magnetic,fieldlines,case:,null,space,problem,This,section,presents,a,case,similar,to,that,in,the,paper,by,Deluzet,and,Narski,[14,,§,3.2].,From,their,p.12,,they,have,an,anisotropy,corresponding,to,ϵ,=,10−15,below.,Consider,the,diffusion,problem,∂2u,∂x2,+,ϵ,∂2u,∂y2,=,−ϵ.,(30),The,corresponding,magnetic,field,lines,in,this,case,are,straight,and,horizontal,,ie.,the,more-,diffusive,direction,lies,parallel,to,the,x-axis.,The,domain,is,a,unit,square,and,the,boundary,con-,ditions,are,homogeneous,Dirichlet,at,y,=,0,,1,and,homogeneous,Neumann,at,x,=,0,,1,,where,‘homogeneous’,implies,zero,values,for,function,and,normal,gradient,respectively.,The,solution,is,u,=,1,2,y(1,−,y),and,the,problem,is,singular,as,ϵ,→,0,because,then,any,function,of,y,that,satisfies,the,Dirichlet,conditions,is,a,solution,,ie.,there,is,a,null,space,in,this,limit.,This,problem,is,implemented,in,FIREDRAKE.,Using,a,unit,square,domain,discretized,into,20,×,20,first-order,Classical,Galerkin,elements,,the,solver,finds,the,correct,solution,until,ϵ,is,approx.,10−14,,below,that,the,quality,of,the,solution,degrades,,see,Fig.15.,The,nature,of,the,failure,is,not,that,the,solution,takes,excessive,time,to,obtain,but,rather,than,the,numerics,appear,to,converge,to,the,It,is,found,that,the,problem,is,worse,at,higher-order,,with,order-2,elements,only,wrong,answer.,remaining,good,down,until,about,ϵ,=,10−13,and,order-6,down,to,about,ϵ,=,10−11.,The,problem,seems,to,level,off,for,sufficiently,high,order,,since,order-22,gives,correct,results,at,ϵ,=,10−10,but,not,at,10−11.,All,these,cases,run,in,seconds,on,the,20,×,20,mesh.,The,singular,limit,can,be,removed,by,replacing,the,homogeneous,Neumann,conditions,with,a,Dirichlet,condition,corresponding,to,the,analytic,solution.,In,this,case,there,is,not,a,null,space,for,ϵ,→,0,,and,it,is,found,that,this,case,works,fine,even,for,ϵ,=,10−20,(and,indeed,,only,one,of,the,constant-x,boundaries,needs,to,be,Dirichlet).,19,Figure,15:,Anisotropic,diffusion,solution,as,a,function,of,y,(analytic,solution,is,u,=,1,is,ϵ,=,1,,10−14,,10−15;,straight,magnetic,fieldlines,case.,2,y(1,−,y)),,L-R,Exploring,the,parameter,space,by,taking,ϵ,=,0,gave,the,result,that,the,solver,returned,a,everywhere-,zero,solution,without,throwing,any,errors.,Again,,the,replacement,of,one,or,more,of,the,Neumann,boundary,conditions,with,Dirichlet,meant,that,the,solver,gave,the,correct,solution.,6.1.2,Curved,magnetic,fieldlines,case:,locking,Now,consider,the,case,with,non-straight,magnetic,field,lines.,The,anisotropy,field,is,given,by,calculating,the,flux,as,,with,k∥,=,1,and,k⊥,=,ϵ,,F,=,k∥,ˆb,ˆb,·,∇u,+,k⊥,(cid:16),(cid:17),∇u,−,ˆb,ˆb,·,∇u,using,the,unit,vector,ˆb,=,1,(cid:112)1,+,λ2x2y2(1,−,x)2(1,−,y)2,(1,,λxy(1,−,x)(1,−,y)),(31),(32),to,define,the,direction,of,the,magnetic,field,,ie.,the,more-diffusive,direction.,The,fieldlines,for,this,configuration,are,exhibited,in,Fig.16.,With,the,above,definitions,,the,diffusion,equation,is,∇,·,F,=,−ϵ.,Apart,from,the,curved,fieldlines,,the,equation,is,as,it,was,in,the,preceding,section,,and,is,identical,if,the,fieldline-bending,parameter,λ,=,0.,The,interaction,with,the,boundary,is,as,before,,ie.,the,field,lines,are,parallel,to,the,y,=,0,,1,(Dirichlet),boundaries,and,normal,to,the,x,=,0,,1,(Neumann),ones.,Selecting,λ,=,10,and,ϵ,=,10−6,an,accurate,solution,is,obtained,by,running,the,solver,with,order-5,elements,,see,Fig.16,,where,the,results,presented,compare,well,with,order-6,elements.,The,paper,[14],indicates,that,there,is,an,problem,additional,to,the,null-space,issue,,if,the,finite-,element,function,spaces,do,not,contain,functions,that,are,constant,in,the,direction,of,the,fieldlines,20,Figure,16:,Anisotropic,diffusion,,ϵ,=,10−6,,in,case,with,curved,magnetic,fieldlines,as,shown,at,top.,Order-5,elements,were,used,to,produce,the,left-hand,picture,,order-1,to,produce,the,right-hand,contours,,which,although,they,have,the,correct,general,shape,correspond,to,much,too,small,an,amplitude.,This,example,used,the,same,triangular,mesh,as,in,Fig.17,,but,the,over-plotting,of,the,mesh,has,been,suppressed,in,order,to,bring,out,small,differences,between,the,solutions.,in,the,anisotropic,case,,and,also,that,this,problem,manifests,itself,for,less,extreme,ϵ,values,than,those,needed,to,reveal,the,null-space,problem,described,in,the,preceding,subsection,(which,is,,of,course,,still,present,for,the,curved,magnetic,fieldlines,case).,Deluzet,and,Narski,refer,to,this,as,‘locking’,,which,is,apparently,a,known,problem,for,low-order,finite-element,expansions.,The,paper,[14],indicates,that,there,is,an,problem,additional,to,the,null-space,issue,,if,the,finite-,element,function,spaces,do,not,contain,functions,that,are,constant,in,the,direction,of,the,fieldlines,in,the,anisotropic,case,,and,also,that,this,problem,manifests,itself,for,less,extreme,ϵ,values,than,those,needed,to,reveal,the,null-space,problem,described,in,the,preceding,subsection,(which,is,,of,course,,still,present,for,the,curved,magnetic,fieldlines,case).,Deluzet,and,Narski,refer,to,this,as,‘locking’,,which,is,apparently,a,known,problem,for,low-order,finite-element,expansions.,‘Locking’,is,is,again,a,case,of,numerics,converging,to,the,wrong,answer;,as,stated,in,[14],,for,low-order,elements,the,solution,amplitude,is,much,smaller,than,it,ought,to,be,,see,the,right-hand,plot,in,In,this,case,,increasing,the,element,order,quickly,fixes,the,problem,,as,expected,from,Fig.16.,ref,[15],,and,it,does,not,currently,seem,necessary,to,examine,how,rapidly,the,situation,is,remedied,by,using,meshes,with,a,larger,number,of,low-order,elements.,Note,that,changing,the,Neumann,boundary,condition,for,a,Dirichlet,condition,(,1,2,y(1,−,y),was,used,for,this,test),,at,both,or,even,at,one,of,the,two,boundaries,somewhat,unexpectedly,makes,the,problem,go,away.,This,indicates,that,locking,might,be,avoided,by,careful,choice,of,boundary,conditions.,21,Figure,17:,Outputs,of,script,for,ϵ,=,10−14,(left,,correct),and,ϵ,=,10−15,(right,,incorrect).,Note,that,smaller,values,of,ϵ,result,in,the,solution,becoming,increasingly,negative.,6.2,A,Dirac,equation-like,formulation,of,anisotropic,diffusion,Consider,once,more,the,anisotropic,diffusion,problem.,∂2u,∂x2,+,ϵ,∂2u,∂y2,=,−ϵ.,(33),Since,u,is,a,scalar,,this,will,be,referred,to,as,the,‘scalar,case’,in,spite,of,the,fact,that,the,diffusivity,is,described,by,an,anisotropic,tensor.,The,domain,is,[−0.5,,0.5]2,,a,different,unit,square,displaced,relative,to,that,used,before.,The,magnetic,field,lines,are,straight,,ie.,the,more-diffusive,direction,lies,parallel,to,the,x-axis.,The,boundary,conditions,are,homogeneous,Dirichlet,at,y,=,−0.5,,0.5,and,homogeneous,Neumann,at,x,=,−0.5,,0.5.,The,solution,is,u,=,1,2,y2,and,the,problem,is,singular,as,ϵ,→,0,because,then,any,function,of,y,that,satisfies,the,Dirichlet,conditions,is,a,solution,,ie.,there,is,a,null,space,in,this,limit.,8,−,1,A,fairly,crude,triangle,mesh,is,used,as,shown,in,Fig.17,,with,second-order,Classical,Galerkin,elements.,The,numerics,are,seen,to,give,the,correct,answer,until,ϵ,falls,below,a,critical,value,in,the,range,[10−14,,10−15],,with,the,answer,clearly,wrong,for,smaller,ϵ.,Now,consider,the,same,problem,reformulated,as,a,spin-half,,Dirac-like,equation,[16,,§,22],,noting,in,particular,the,appearance,of,a,square,root:,With,the,choice,and,(cid:18),Γx,∂,∂x,+,√,ϵ,Γy,∂,∂y,(cid:19),ψ,=,ψ0.,Γx,=,(cid:19),(cid:18)1,0,0,−1,Γy,=,(cid:19),(cid:18)0,1,1,0,.,(34),(35),(36),the,two,matrices,obey,the,relation,ΓaΓb,+,ΓbΓa,=,2gabI,,which,is,the,Clifford,algebra,for,2-D,rotations,,ie.,they,anticommute,when,gab,=,0,,a,̸=,b,and,separately,square,to,unity.,The,two-component,spinors,are,ψ,=,(cid:19),(cid:18)u,v,22,(37),Figure,18:,Outputs,of,the,spin-half,equation,for,a,mesh,of,triangular,first-order,(left),and,second-,order,(right),elements;,ϵ,=,1.,Figure,19:,Outputs,of,the,spin-half,equation,for,a,mesh,of,triangular,second-order,elements,ϵ,=,10−20,(left),and,10−22,(right).,The,latter,is,beginning,to,diverge,from,the,correct,solution.,(Do,not,be,misled,by,the,labelling,of,the,coordinate,axis,as,[0,,1],instead,of,[−0.5,,0.5],caused,by,a,PARAVIEW,quirk.),and,ψ0,=,(cid:19),.,(cid:18)f,g,Spinor,theory,then,implies,f,=,g,=,−,ϵ,y,,and,the,spin-half,Dirac,form,follows,as,√,√,(cid:32),∂,∂x,√,ϵ,∂,ϵ,∂,∂y,∂y,−,∂,∂x,(cid:33),(cid:18)u,(cid:19),v,√,=,−,ϵ,y,(cid:19),(cid:18)1,1,.,(38),(39),The,advantages,of,the,above,formulation,are,that,now,the,system,is,first-order,and,the,ϵ,in,the,original,equation,is,replaced,by,ϵ,so,the,anisotropy,is,expected,to,be,less,of,a,problem.,√,The,above,system,Equation,(39),is,implemented,as,a,simple,FIREDRAKE,script.,Promising,outputs,are,obtained,for,the,triangle,mesh,as,used,in,the,scalar,case,described,above,,see,Fig.18.,As,the,anisotropy,is,increased,in,the,spin-half,equation,,it,is,found,that,for,ϵ,=,10−20,the,numerical,solution,is,still,good,,but,that,it,is,becoming,unreliable,at,ϵ,=,10−22;,see,Fig.19.,If,we,take,ϵ,=,10−20,as,the,limit,of,effective,operation,of,the,solver,,it,is,possible,to,claim,a,factor,of,greater,than,105,improvement,in,the,accessible,range,of,ϵ,over,the,scalar,solver,in,this,case.,Unfortunately,the,Dirac,form,appears,to,lead,to,incorrect,results,if,square,Classical,Galerkin,el-,ements,are,used,,even,for,ϵ,=,1,,see,Fig.20.,This,failure,is,the,most,concerning,of,a,number,of,issues,concerning,the,spinor,approach,,which,extend,to,unexpected,sensitivity,to,the,order,of,23,Figure,20:,Output,of,the,spin-half,equation,for,a,mesh,of,first-order,elements,,20,×,20,uniform,squares,,ϵ,=,1,,indicating,a,problem,for,the,approach.,element,used,,and,to,the,need,for,a,formulation,to,treat,curved,magnetic,fieldlines.,More,detailed,investigations,might,yet,lead,to,understanding,and,overcoming,of,these,difficulties,,but,they,could,be,paused,,since,for,the,project,to,proceed,,it,is,only,strictly,necessary,to,estimate,quantitatively,the,level,of,numerical,diffusion.,Should,it,become,necessary,to,proceed,with,the,approach,,scripts,and,documentation,for,this,work,have,been,placed,in,the,repository,[17].,7,Summary,This,report,has,presented,work,towards,3-D,plasma,turbulence,simulations,using,NEKTAR++.,Results,of,an,on-going,,incremental,development,strategy,have,been,presented,,showing,progress,towards,the,development,of,a,solver,capable,of,addressing,problems,of,current,research,interest.,Its,outputs,will,be,coupled,to,the,UKAEA’s,particle,modelling,capability,in,order,to,include,kinetic,effects,from,,in,particular,,the,neutral,particle,simulation,capability,,to,be,described,in,subsequent,NEPTUNE,reports.,The,current,report,has,also,outlined,work,towards,understanding,the,coupling,between,different,fi-,nite,element,solvers,and,mitigating,the,numerical,difficulties,associated,with,demanding,advection-,diffusion,problems,and,highly,anisotropic,diffusion,problems.,Lines,of,enquiry,which,were,begun,,but,then,subsequently,paused,in,the,light,of,ongoing,discoveries,have,been,recorded.,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,,and,also,from,the,holders,of,Grants,T/AW085/22,(particularly,David,Moxey,and,Chris,Cantwell),,T/AW087/22,(particularly,Hussam,al-Daas),and,T/AW088,(particularly,Mike,Kryjak).,24,A,Higher-dimensional,harmonics,This,work,builds,on,the,work,on,harmonic,functions,in,the,report,[18],,particularly,§,2.4,,and,also,,for,an,introduction,to,their,relationship,with,fluid,vortices,,see,Appendix,B.,The,motivation,for,Section,A.1,and,Section,A.2,is,to,gain,greater,analytic,understanding,of,certain,types,of,fluid,flow,that,could,produce,additional,test,cases.,To,improve,understanding,,it,is,important,to,achieve,informative,visual,representations,,see,Section,A.3.,A.1,Two-dimensional,harmonics,and,complex,numbers,The,understanding,of,functions,of,a,complex,variable,is,aided,by,effective,means,of,visualizing,them.,This,is,complicated,by,the,fact,that,a,complex,function,w(z),=,u(x,,y),+,iv(x,,y),is,a,two-,dimensional,function,of,a,two-dimensional,variable;,separate,contour,or,3-D,plots,of,u,and,v,contain,in,principle,all,the,information,about,the,function,but,do,not,give,a,feel,for,how,it,all,fits,together.,The,utility,of,plots,of,the,complex,phase,as,a,function,of,(x,,y),is,advocated,in,eg.,[19].,Another,appealing,way,of,visualizing,a,complex,function,is,to,note,that,any,function,of,the,complex,conjugate,,w(¯z),(aka,anti-holomorphic,function),,is,a,2-D,harmonic,in,the,Hodge,sense,in,the,(x,,y)-plane,,ie.,it,represents,a,divergence-,and,curl-free,two,dimensional,fluid,flow,,referred,to,as,the,P,`olya,vector,field.,This,gives,the,simple,picture,that,the,magnitude,of,w,is,the,local,flow,speed,and,its,phase,gives,the,local,direction,of,the,flow.,Zeros,in,the,complex,plane,correspond,to,stagnation,points,of,the,flow,,and,the,relationship,has,many,other,important,consequences,,such,as,the,picture,of,dipoles,as,a,merger,of,two,simple,poles,[18,,App.,B],,and,the,fact,that,flows,corresponding,to,the,neighbourhood,of,an,essential,singularity,exhibit,all,possible,velocity,values,infinitely,many,times,,by,the,Great,Picard,Theorem,,see,Henrici,[20,,Theorem,4.4g,,Ex.,5].,An,illustration,of,the,P,`olya,vector,field,is,the,simple,vortex,given,by,w,=,i,¯z,.,Observe,that,the,multiplication,of,this,function,by,complex,i,switches,between,vortex,solutions,and,draining,bathtub,solutions,-,the,90o,phase,of,i,causes,the,local,direction,of,the,flow,field,to,rotate,through,a,right,angle,,hence,the,vortex,flow,and,the,draining,bathtub,flow,are,given,by,the,one-forms:,u0,=,u1,=,−ydx,+,xdy,x2,+,y2,−xdx,−,ydy,x2,+,y2,.,(40),Intermediate,values,of,the,phase,give,a,draining,or,filling,bathtub,solution,where,the,flow,lines,have,a,certain,amount,of,‘twist’,around,the,drain,/,source.,Another,example,is,the,function:,w(¯z),=,¯z,1,+,¯z,(41),for,which,the,corresponding,flow,field,is,shown,in,Fig.21,along,with,a,plot,showing,the,magnitude,of,the,velocity.,The,flowfield,for,the,Taylor,series,,truncated,to,10,terms,,and,expanded,about,¯z,=,0,is,shown,also,,along,with,the,corresponding,velocity,magnitude.,These,plots,show,the,expected,agreement,in,the,flowfields,within,the,unit,circle,of,convergence.,The,zeros,of,the,Taylor,polynomial,can,be,seen,to,be,clustering,about,the,circle,of,convergence,,in,accordance,with,Jentzsch’s,theorem.,25,Figure,21:,Streamlines,at,left,and,flow,speed,at,right,corresponding,to,at,top,,w(¯z),=,¯z,1+¯z,and,at,bottom,,the,tenth-order,Taylor,expansion,of,w(¯z),about,¯z,=,0.,The,flow,speed,scales,are,common.,Note,the,yellow,area,in,the,top,RHS,plot,is,a,plotting,artifact,due,to,the,singularity.,Noting,the,identity,∇(u2),≡,2,(u,·,∇u,+,u,×,(∇,×,u)),and,using,the,fact,that,the,P,´olya,vector,field,is,curl-free,,there,follows,Bernoulli’s,principle:,(cid:18),1,2,u2,+,p,=,0,∇,(cid:19),Applied,to,the,simple,vortex,,for,which,u,=,(cid:18),−y,x2,+,y2,,,x,x2,+,y2,(cid:19),.,the,pressure,follows,as,1,x2,+,y2,.,which,is,lower,near,the,centre,of,the,vortex,as,expected.,p,=,−,1,2,(42),(43),(44),(45),Another,interesting,relation,for,time-stationary,harmonic,flows,follows,from,their,satisfaction,in,two,dimensions,of,the,inertial,law,Writing,out,in,Cartesians,u,=,(u,,v),,using,u,·,∇u,=,−∇p.,∂u,∂x,∂u,∂y,=,−,∂v,∂y,=,∂v,∂x,26,(46),(47),and,taking,the,divergence,,gives,the,eikonal,equation,(cid:19)2,(cid:18),∂u,∂x,+,(cid:18),∂u,∂y,(cid:19)2,=,−,∇2p.,1,2,(48),This,relation,implies,a,method,of,finding,solutions,to,the,eikonal,equation,in,the,case,where,the,field,is,known,to,be,harmonic.,When,the,pressure,is,specified,the,squared,modulus,of,the,holomorphic,velocity,field,is,known,which,means,the,vector,field,is,defined,up,to,a,local,rotation,,ie.,to,within,an,arbitrary,factor,of,complex,phase,expiφ.,However,,the,argument,φ,of,the,phase,function,must,be,purely,real,must,also,be,holomorphic,or,the,harmonic,property,is,lost,,implying,that,the,rotation,is,global,,since,the,only,pure,real,holomorphic,functions,are,constants.,An,example,is,the,vortex,/,bathtub,solution,family,introduced,in,ref,[18,,App.,B].,It,is,worth,noting,here,that,as,well,as,the,anti-holomorphic,functions,,ie.,functions,of,¯z,considered,above,,holomorphic,functions,,ie.,functions,of,z,,also,play,a,role,in,hydrodynamics.,The,simplest,example,is,w(z),=,−iz,,which,corresponds,to,the,linear,field,in,the,terminology,of,Arter,[21],,viz.,u,=,(y,,−x),in,the,(x,,y)-plane,,which,brings,non-zero,vorticity,into,play.,Its,streamlines,correspond,to,the,phase-space,representation,of,an,harmonic,oscillator.,A.2,Four-dimensional,harmonics,and,quaternions,The,equations,for,harmonics,are,[18,,§,2.4],du,=,0,,δu,=,0.,where,the,harmonic,u,is,a,one-form.,In,the,two-dimensional,Euclidean,space,previously,consid-,ered,,du,represents,the,curl,and,δu,the,divergence,of,a,2-vector,u,,but,henceforth,the,space,is,four-dimensional,Euclidean,space.,Evaluating,du,where,u,=,ϕ,dw,+,ψ,·,dx,gives,the,two-form,du,=,−,(cid:18),∂ϕ,∂x,dw,∧,dx,+,∂ϕ,∂y,dw,∧,dy,+,dw,∧,dz,(cid:19),(cid:18),∂ψ2,∂x,+,−,∂ψ1,∂y,(cid:19),dx,∧,dy,+,(cid:18),∂ψ3,∂y,−,∂ψ2,∂z,(cid:19),dy,∧,dz,+,(cid:18),∂ψ1,∂z,−,(cid:19),∂ϕ,∂z,∂ψ3,∂x,∂ψ3,∂w,dw,∧,dy,+,so,that,du,=,0,is,equivalent,to,+,∂ψ1,∂w,dw,∧,dx,+,∂ψ2,∂w,∇ϕ,=,∂ψ,∂w,+,∇,×,ψ.,27,dz,∧,dx,dw,∧,dz.,(49),Evaluating,δu,≡,∗d,∗,u,,taking,∗dw,=,−dx,∧,dy,∧,dz,,∗dx,=,dy,∧,dz,∧,dw,,∗dy,=,−dz,∧,dw,∧,dx,,∗dz,=,dw,∧,dx,∧,dy,,gives,the,equation,δu,=,0,as,(cid:18),−,∂ϕ,∂w,−,∂ψ1,∂x,−,∂ψ2,∂y,−,∂ψ3,∂z,(cid:19),dw,∧,dx,∧,dy,∧,dz,=,0.,which,is,the,equation,of,a,vanishing,four-divergence,∂ϕ,∂w,=,−∇,·,ψ.,(50),(51),Now,,it,is,known,that,a,left-regular,(right-regular,in,+,sign,case),quaternion,function,satisfies,(see,eg.,[22]),∂ϕ,∂w,=,∇,·,ψ,,(52),∂ψ,∂w,Thus,,the,differential,form,equations,Equations(49),and,(52),have,solutions,corresponding,to,the,conjugate,ϕ,−,iψ,of,a,left-regular,function,of,a,quaternion,variable,q,≡,w,+,ix,+,jy,+,kz.,∇ϕ,=,−,∓,∇,×,ψ.,(53),A,simple,quaternion,function,(left-,and,right-regular),is,r(q),=,1,q,≡,w,−,(xi,+,yj,+,zk),(w2,+,x2,+,y2,+,z2)2,.,which,corresponds,to,the,harmonic,u,=,wdw,+,xdx,+,ydy,+,zdz,(w2,+,x2,+,y2,+,z2)2,.,Post-multiplications,by,i,,j,,k,respectively,generate,the,harmonics,(with,the,progenitor,first):,u0,=,u1,=,u2,=,u3,=,wdw,+,xdx,+,ydy,+,zdz,(w2,+,x2,+,y2,+,z2)2,xdw,−,wdx,+,zdy,−,ydz,(w2,+,x2,+,y2,+,z2)2,ydw,−,zdx,−,wdy,+,xdz,(w2,+,x2,+,y2,+,z2)2,zdw,+,ydx,−,xdy,−,wdz,(w2,+,x2,+,y2,+,z2)2,,,,,,,.,(54),(55),(56),This,is,the,quaternionic,analogue,of,obtaining,the,draining,bathtub,by,multiplying,by,i,to,get,the,vortex.,There,are,thus,three,quaternion,analogues,of,the,vortex,solution,,separately,corresponding,to,a,rotation,about,one,of,the,three,axes,in,a,Cartesian,three-space.,(Pre-multiplication,does,not,generate,harmonics.),28,A.3,Visualization,of,4-D,harmonics,Visualization,of,quaternionic,functions,is,obviously,even,more,of,a,challenge,than,that,of,complex,functions.,A,projection,may,be,attempted,from,the,four-dimensional,vector,field,into,a,three-,dimensional,subspace.,Here,a,stereographic,projection,,equivalent,to,a,conformal,representation,from,an,embedded,three-sphere,onto,the,w,=,0,subspace,,is,performed.,First,is,derived,the,coordinate,transform,from,the,3-sphere,w2,+,x2,+,y2,+,z2,=,a2,to,the,w,=,0,subspace:,The,mapped,point,is,where,the,line,connecting,the,‘North,Pole’,(a,,0,,0,,0),and,the,point,with,coordinates,(w,,x,,y,,z),intersects,the,w,=,0,subspace,(0,,x′,,y′,,z′).,The,formula,for,the,new,coordinates,is,derived,from,(a,,0,,0,,0),+,λ(w,−,a,,x,,y,,z),=,(0,,x′,,y′,,z′),(57),ie.,x′,=,y′,=,z′,=,ax,a,−,(cid:112)a2,−,x2,−,y2,−,z2,ay,a,−,(cid:112)a2,−,x2,−,y2,−,z2,az,a,−,(cid:112)a2,−,x2,−,y2,−,z2,,,,,.,(58),(Do,not,get,confused,and,assume,w,=,0,-,the,correct,version,is,w′,=,0.),The,covector,field,is,harmonic,1,and,its,scaling,factor,may,be,neglected,as,this,is,constant,on,the,4-sphere,,thus,The,transformation,law,for,this,type,of,object,is,the,covariant,form,u,=,(x,,−w,,z,,−y).,ua′,=,∂xa,∂xa′,ua,,(59),(60),where,the,indices,range,over,1,,2,,3,since,the,mapping,is,from,the,3-sphere,to,the,equatorial,3-,space.,The,derivatives,of,the,pre-transformation,coordinates,with,respect,to,the,post-transformation,ones,are,required.,They,may,be,obtained,more,easily,by,squaring,the,coordinate,relations,and,summing,to,form,r′2,=,(a,−,a2r2,√,a2,−,r2)2,.,where,r,and,r′,are,defined,by,the,expected,Pythagorean,formulae.,Writing,∆,≡,a,−,algebra,simplifies,to,give,∆,=,2a3,r′2+a2,and,the,relations,w,=,a,x,=,y,=,z,=,r′2,−,a2,r′2,+,a2,,,2a2x′,r′2,+,a2,,,2a2y′,r′2,+,a2,,,2a2z′,r′2,+,a2,.,29,(61),√,a2,−,r2,the,(62),Figure,22:,Streamlines,for,the,4-D,field,projected,into,3-D,as,discussed,in,the,text.,The,plot,was,performed,on,a,cubical,volume,,hence,streamlines,intersecting,the,boundary,of,the,cube,appear,to,terminate.,whence,it,now,straightforward,to,perform,the,coordinate,transformations,on,the,components,of,the,1-form.,The,vector,field,in,the,(x′,,y′,,z′),3-subspace,resulting,from,the,above,transformations,can,be,vi-,sualized,as,a,streamline,plot,using,PARAVIEW.,Thus,for,example,,the,harmonic,u1,in,Eq.,(56),,transformed,from,(x,,y,,z),to,(x′,,y′,,z′),using,the,covariant,transformation,law,,gives,the,following,components:,ux′,=,(−x′2,+,y′2,+,z′2,+,a2)(a2,−,x′2,−,y′2,−,z′2),uy′,=,2x′y′(x′2,+,y′2,+,z′2,−,a2),+,2az′(x′2,+,y′2,+,z′2,+,a2)),uz′,=,2x′z′(x′2,+,y′2,+,z′2,−,a2),−,2ay′(x′2,+,y′2,+,z′2,+,a2)),Streamlines,of,the,above,vector,field,are,plotted,in,Fig.22,on,a,cube,of,side,2a.,B,Advanced,visualization:,Nvidia,IndeX,PARAVIEW,plugin,Rendering,of,volumetric,fields,in,3-D,involves,the,processing,of,large,amounts,of,data.,Nvidia,offers,Nvidia,IndeX,for,visualizing,volumetric,data,,see,[23],,which,includes,an,example,showing,the,visualization,of,150TB,of,data.,This,software,is,,at,the,time,of,writing,,offered,free,of,charge,for,a,single,workstation,or,for,academic,use.,There,is,also,a,plugin,for,PARAVIEW,(v5.11,and,later).,A,workflow,for,a,simple,demonstration,of,the,plugin,is:,•,Open,PARAVIEW.,30,Figure,23:,Output,of,the,NVIDIA,IndeX,test,outlined,in,the,text.,•,Go,to,tools/manage,plugins;,load,pvNVIDIAIndeX.,•,Open,.vtu,file.,Click,Apply.,•,Add,a,calculator,to,input,a,test,function,in,3-D.,One,possible,example,is,cos(coordsX,+,coordsY,+,coordsX*coordsZ).,•,Visualize,as,Volume,to,render,volumetric,data,using,the,default,renderer.,•,Switch,visualization,to,NVIDIA,IndeX,to,render,using,Nvidia,IndeX.,The,performance,using,the,plugin,was,found,to,be,significantly,faster,than,the,default,renderer,in,the,case,of,the,cube-shaped,domain,used,for,Fig.23,and,using,the,plugin,allowed,the,visualization,to,be,rotated,and,zoomed,in,real,time,(the,default,renderer,shows,a,defeatured,visualization,during,rotate,/,zoom).,The,plugin,also,produced,slightly,higher-quality,renderings,than,the,default.,It,was,,however,,found,that,the,plugin,appeared,slower,than,the,default,renderer,in,one,case,that,involved,an,elongated,domain.,C,A,plotting,bug,in,PARAVIEW,A,simple,FIREDRAKE,script,(Fig.24),was,used,to,integrate,the,trivial,ODE,u′(x),=,−λu(x),,u(0),=,1.,It,was,found,that,the,PARAVIEW,Plot,Over,Line,filter,worked,as,expected,for,first-order,CG,elements,but,produced,strange,artifacts,when,higher-order,CG,elements,were,used.,The,artifacts,are,not,evident,if,the,same,data,is,output,to,a,.csv,file,and,plotted,by,other,methods,,see,Fig.25.,31,Figure,24:,FIREDRAKE,script,for,solving,the,ODE,in,the,text.,Figure,25:,FIREDRAKE,outputs,for,the,numerical,solution,of,the,ODE,in,the,text,using,second-order,elements.,The,left-hand,plot,was,generated,by,exporting,the,data,to,.csv,and,plotting,in,Excel;,the,right-hand,plot,used,the,Plot,Over,Line,filter,of,PARAVIEW,and,is,clearly,incorrect.,The,concerning,aspect,of,the,bug,is,that,it,causes,the,appearance,of,features,that,indicate,a,particular,numerical,method,is,not,working,correctly.,In,short,,caution,must,be,applied,when,using,higher-order,plots,generated,by,Plot,Over,Line,when,working,with,one-dimensional,data.,This,bug,was,observed,in,version,5.10.1,of,PARAVIEW,on,Windows.,Since,an,earlier,version,(5.8.0),was,seen,to,crash,every,time,when,attempting,to,run,Plot,Over,Line,for,orders,higher,than,one,,there,is,the,suggestion,that,the,relevant,part,of,the,software,is,still,actively,being,devel-,oped,,so,that,reporting,the,problem,has,been,deferred,as,unlikely,to,speed,a,fix.,32,References,[1],Nektar-Driftwave.,https://github.com/ExCALIBUR-NEPTUNE/nektar-driftwave.,Ac-,cessed:,March,2023.,[2],Saunders,W.,,E.,Threlfall,,and,W.,Arter.,Finite,Element,Models:,Performance.,Technical,Report,CD/EXCALIBUR-FMS/0047,,UKAEA,Project,Neptune,,2021.,[3],Nektar-Driftplane.,https://github.com/ExCALIBUR-NEPTUNE/nektar-driftplane.,Ac-,cessed:,March,2023.,[4],Hermes-3,2D.,https://hermes3.readthedocs.io,,2021.,Accessed:,June,2021.,[5],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.,[6],LArge,Plasma,Device,(LAPD).,https://plasma.physics.ucla.edu/,large-plasma-device.html.,Accessed:,September,2023.,[7],W.,Arter,et,al.,Equations,for,EXCALIBUR/NEPTUNE,Proxyapps.,cal,Report,CD/EXCALIBUR-FMS/0021-1.31-M1.2.1,,UKAEA,,10,2023.,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0021-1.30-M1.2.1.pdf.,Techni-,https:,[8],E.,Threlfall,,Powell,S.,,and,W.,Arter.,Complementary,Actions:,Uncertainty,Quantifica-,tion,Code,Integration,,Acceptance,and,Operation,1.,Technical,Report,CD/EXCALIBUR-,FMS/0073,,UKAEA,Project,Neptune,,2023.,[9],F.,Rathgeber,,D.,A.,Ham,,L.,Mitchell,,M.,Lange,,F.,Luporini,,A.T.T.,McRae,,G.-T.,Bercea,,G.R.,Markall,,and,P.H.J.,Kelly.,Firedrake:,automating,the,finite,element,method,by,composing,abstractions.,ACM,Transactions,on,Mathematical,Software,(TOMS),,43(3):1–27,,2016.,[10],A.,Toselli,and,O.,Widlund.,Domain,Decomposition,Methods:,Algor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