CD-EXCALIBUR-FMS0064-M6.2_FiniteElementModelsComplementaryActions2
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ExCALIBUR,Finite,Element,Models,Complementary,Actions,2,M6.2,Version,1.00,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Mile-,stone,M6.2.,It,includes,an,application,of,the,Nektar++,ﬁnite-element,soft-,ware,to,the,classic,problem,of,vertical,natural,convection,,showing,concur-,rence,with,numerical,results,from,a,benchmark,test,case,in,the,literature,and,the,replication,of,some,outputs,from,a,recent,publication.,A,further,section,contains,details,of,a,preliminary,attempt,to,explore,recent,devel-,opments,in,the,use,of,structure-preserving,ﬁnite-element,methods,ie.,dis-,crete,exterior,calculus,,the,vision,being,that,such,methods,will,enable,ﬁnite-,element,outputs,to,integrate,reliably,with,developments,in,particle-based,codes,within,NEPTUNE.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0064,Issue:,Date:,1.00,18,March,2022,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Ed,Threlfall,Signature,N/A,Date,18,March,2022,BD,Reviewed,By:,Wayne,Arter,18,March,2022,Project,Technical,Lead,1,1,Introduction,High-order,methods,,which,offer,the,potential,for,exponential,improvement,in,accuracy,given,poly-,nomial,increase,in,computation,time,,are,seen,as,essential,for,maximizing,the,performance,of,next-generation,software,intended,to,run,on,forthcoming,exascale,machines.,One,speciﬁc,ex-,ample,of,such,an,approach,is,the,spectral,/,hp,ﬁnite,element,method,[1],,of,which,a,modern,,open-source,C++,implementation,is,the,software,Nektar++,[2],co-written,and,maintained,by,one,of,the,NEPTUNE,grantees.,The,Nektar++,framework,forms,the,basis,for,some,of,the,core,NEP-,TUNE,proxyapps,and,is,currently,being,augmented,with,capabilities,to,handle,fusion-relevant,problems,([3],,[4]),and,also,extension,to,allow,optimized,operation,on,a,wider,range,of,computa-,tional,architectures.,This,involvement,justiﬁes,continued,investment,of,UKAEA,time,in,veriﬁcation,and,validation,of,the,framework,,e.g.,by,comparison,to,published,results,,and,,with,a,view,to,fu-,ture,extensions,,in,understanding,its,operation.,In,view,of,its,overarching,focus,on,ﬁnite-element,methods,,this,deliverable,relates,primarily,to,Work,Package,FM-WP1,-,Numerical,Representation.,In,Section,2,,a,comparison,between,Nektar++,outputs,and,a,well-known,benchmark,result,for,convective,heat,transfer,is,performed,,and,also,some,results,from,a,recent,paper,on,convection,modelling,are,replicated.,This,work,extends,ﬂuid,methods,studied,in,the,prequel,[5],in,terms,of,comparison,to,results,from,the,literature.,The,simulation,of,the,plasma,edge,will,require,models,going,beyond,the,purely,ﬂuid-mechanical,treatment,in,2,in,order,to,describe,the,physics,in,regions,where,collisionality,is,low.,This,kinetic,regime,can,be,accessed,by,the,use,of,particle,methods,,which,literally,attempt,an,atomistic,dy-,namical,simulation,(though,each,computational,particle,may,correspond,to,very,many,physical,particles).,The,current,wisdom,associated,to,the,coupling,of,particles,to,continuum,ﬁelds,(e.g.,electromagnetism),harbingers,deeper,changes,to,the,existing,Nektar++,framework.,Speciﬁcally,,numerical,properties,of,the,coupled,particle-ﬁeld,solver,are,known,to,beneﬁt,greatly,,for,example,in,terms,of,stability,guarantees,,from,the,use,of,structure-preserving,methods,,in,which,aspects,of,the,continuum,physics,survive,the,transition,to,a,discrete,representation.,These,techniques,go,also,under,the,name,of,‘discrete,exterior,calculus’,or,‘ﬁnite-element,exterior,calculus’,as,they,preserve,some,of,the,attributes,of,continuum,exterior,calculus.,As,a,prelude,to,these,expected,changes,,a,brief,exposition,of,such,a,translation,is,given,in,Section,3,for,a,simple,toy,problem,using,a,method,taken,from,the,literature;,the,example,is,shown,to,achieve,spectral,convergence,(via,the,use,of,a,small,stand-alone,code).,The,test,problems,treated,herein,are,sufﬁciently,small,so,as,not,to,beneﬁt,much,from,execution,on,HPC,hardware,,and,the,results,were,obtained,using,only,a,modest,desktop,PC.,The,at-scale,capa-,bility,of,the,Nektar++,framework,is,the,subject,of,current,study,and,future,work,by,the,developers,of,that,code,and,is,beyond,the,ambit,of,this,report.,2,2,Vertical,natural,convection,in,Nektar++,2.1,Introduction,This,work,follows-on,from,the,investigation,described,in,the,previous,report,[5],,which,treated,the,problem,of,heat,transport,in,vertical,natural,convection,(for,an,illustration,of,the,relevance,of,such,problems,to,the,fusion,use,case,,see,[6]).,One,new,aspect,of,the,current,work,is,the,replication,of,numerical,results,published,in,the,literature.,In,vertical,natural,convection,an,upright,cavity,has,a,hot,side,and,a,cold,side,and,is,ﬁlled,with,a,ﬂuid,that,experiences,a,vertical,buoyancy,force,when,heated.,The,strength,of,this,force,then,leads,to,different,regimes,of,ﬂow,and,heat,transport.,When,the,force,is,negligible,,heat,is,transferred,diffusively,from,the,hot,surface,to,the,cold,one,,in,rough,analogy,with,classical,transport.,Greater,buoyancy,leads,to,ﬂuid,moving,up,the,hot,side,,across,the,top,of,the,cell,,and,down,the,cold,side,,creating,a,steady,(laminar,,or,‘classical’),circulation,and,transporting,heat,by,advection.,As,the,force,is,increased,,the,vertical,ﬂow,becomes,increasingly,conﬁned,to,a,velocity,boundary,layer,near,the,vertical,walls;,likewise,,the,temperature,gradients,are,increasingly,restricted,to,a,thermal,boundary,layer,near,the,vertical,walls.,Further,increasing,the,force,,one,ﬁnds,that,the,sheared,ﬂow,near,the,vertical,boundaries,leads,to,an,instability,of,Kelvin-Helmholtz,type,that,takes,the,form,of,waves,that,co-move,with,the,convective,ﬂow,,and,which,are,chaotic,in,nature;,these,boundary,effects,also,lead,ultimately,to,complicated,,also-chaotic,,time-dependent,ﬂow,structures,in,the,bulk,-,turbulent,ﬂow.,As,in,the,previous,report,,the,simulations,are,two,dimensional,and,thus,truly,three-dimensional,effects,-,including,expected,instabilities,-,are,excluded,a,priori.,In,dimensionless,form,,the,system,of,equations,describing,vertical,natural,convection,is,,for,ﬂuid,velocity,u,,temperature,T,and,pressure,p,,for,a,system,coordinatized,such,that,the,vertical,,gravity-,opposing,direction,is,ˆy,,1,P,r,(cid:18),∂u,∂t,(cid:18),∂T,∂t,(cid:19),(cid:19),+,u,·,∇u,+,u,·,∇T,=,−∇p,+,Ra,T,ˆy,+,∇2u,,=,∇2T,,∇,·,u,=,0.,(1),(2),(3),The,Boussinesq,approximation,is,assumed,,with,the,density,ﬂuctuation,entering,only,as,a,temperature-,dependent,buoyancy,force,(hence,the,incompressibility,condition),-,for,this,reason,,the,system,is,simulated,using,the,incompressible,Navier-Stokes,solver,of,Nektar++.,Dirichlet,boundary,condi-,tions,were,used,for,the,vertical,wall,temperatures,(T,=,1,on,the,left,and,T,=,0,on,the,right),and,zero,Neumann,conditions,(i.e.,no,heat,ﬂux),on,the,horizontal,walls,(top,and,bottom).,The,ve-,locity,boundary,conditions,were,no-slip,/,no-penetration,and,the,pressure,used,a,zero,Neumann,condition,,for,all,walls.,The,dimensionless,parameters,are,the,Rayleigh,number,,where,β,is,the,thermal,expansion,coef-,ﬁcient,,∆T,the,temperature,difference,across,cavity,width,L,,3,Ra,=,βg∆T,L3,κν,(4),and,the,Prandtl,number:,the,ratio,of,kinematic,viscosity,ν,to,thermal,diffusivity,κ,(a,property,of,the,ﬂuid,ﬁlling,the,cavity;,in,these,simulations,,the,value,0.71,,appropriate,to,air,,was,used),,P,r,=,ν,κ,.,The,other,relevant,parameter,is,the,aspect,ratio,λ,=,L/H,of,the,cavity,(width,L,,height,H).,The,heat,ﬂux,the,spatial-averaged,Nusselt,number,deﬁned,as,N,u,=,(cid:82),H,0,−∂/∂xT,(x0,,y),+,∂/∂xT,(x1,,y)dy,(cid:82),H,0,−∂/∂xT,(x0,,y),+,∂/∂xT,(x1,,y)dy|Ra=0,.,(5),(6),Computational,parameters,enter,as,the,mesh,resolution,(for,simplicity,,meshes,comprising,uniform,squares,are,used,for,this,study,,with,h,denoting,the,size,of,a,single,element),and,the,global,order,p,(a,natural,number),of,the,spectral,elements,,with,the,lowest,allowed,value,(p,=,1),meaning,that,linear,intra-element,basis,functions,are,used,,and,greater,p,corresponding,to,higher-order,Lagrange,polynomials.,For,reference,,a,continuous,Galerkin,formulation,was,employed.,2.2,New,numerical,results,Veriﬁcation,of,the,incompressible,Navier-Stokes,solver,of,Nektar++,was,performed,for,the,case,of,two,convective,ﬂows.,The,ﬁrst,of,these,,which,will,be,the,main,focus,,is,identical,to,the,MIT,benchmark,[7],and,uses,a,Rayleigh,number,Ra,of,3.4,×,105,,giving,a,steady,ﬂow,with,a,time-,periodic,ﬂow,instability.,The,second,differs,only,in,using,a,larger,value,of,Ra,,5.5,×,105,,at,which,the,ﬂow,instability,is,no,longer,periodic,(see,Figs.,1,and,2),,probably,due,to,the,operation,of,multiple,linear,instability,modes,of,incommensurate,period.,The,setup,involves,a,computational,domain,with,aspect,ratio,8,:,1,i.e.,a,tall,narrow,tank,of,ﬂuid.,The,MIT,benchmark,was,established,in,order,to,identify,the,time-dependent,solution,for,the,cavity,described,above,,to,ﬁnd,the,critical,Rayleigh,number,for,the,onset,of,ﬂow,instability,,and,to,identify,computational,methods,capable,of,providing,these,results.,Two,of,the,quantities,of,interest,in,the,study,are,the,time-averaged,,spatially-averaged,Nusselt,number,and,the,time,series,of,the,temperature,at,the,point,(x,,y),=,(0.181,,7.37).,The,former,is,presented,in,a,table,of,results,obtained,by,various,authors,,while,the,latter,is,available,for,download,,both,at,[7].,A,uniform,grid,of,800,p,=,10,spectral,elements,was,used,,unlike,the,MIT,benchmark,which,uses,computational,meshes,that,are,reﬁned,near,the,boundaries,to,better,resolve,the,boundary,layers,(note,that,the,paper,[8],,which,appears,to,use,a,uniform,mesh,,contains,evidence,that,the,reso-,lution,used,herein,is,sufﬁcient,to,resolve,relevant,physics,of,both,ﬂows).,Timestep,sizes,were,set,(by,trial-and-error),to,be,close,to,but,not,in,violation,of,the,stability,limit,on,the,maximum,allowed,timestep.,Regarding,computational,resources,,the,simulations,are,not,particularly,onerous,,with,4,Figure,1:,Temperature,ﬁelds,for,Ra,of,3.4,×,105,(left),and,5.5,×,105,(right).,Note,the,appearance,of,the,‘wall,waves’,in,the,larger-Ra,case,,in,which,the,steady,conﬁguration,is,replaced,by,a,time-,dependent,quasi-steady,state.,5,Figure,2:,Field-at-point,outputs,showing,Nektar++,and,MIT,benchmark,data.,Temperature,ﬁeld,recorded,at,point,‘1’,described,in,[7],i.e.,(x,,y),=,(0.181,,7.37).,Note,absence,of,periodicity,of,the,ﬂow,instability,in,the,higher-Ra,case.,ten,periods,of,the,lower-Ra,example,taking,approx.,30mins,to,run,on,a,single,Core-i7-10700KF,3.8GHz,PC,(the,higher-Ra,case,is,somewhat,slower,due,to,the,use,of,a,smaller,timestep).,The,Nusselt,number,was,computed,as,a,time,series,using,a,customized,Nektar++,ﬁlter,,described,in,[5],and,representing,the,sole,modiﬁcation,of,the,existing,code,base,for,the,work,described,in,this,section.,The,mean,value,of,the,time,series,data,was,evaluated,as,the,average,of,the,maximum,and,minimum,of,the,time,series,(using,a,small,timestep,for,all,simulations,meant,that,this,approach,was,sufﬁciently,accurate).,The,most,accurate,of,the,Nusselt,number,results,reported,in,[7],for,the,Ra,=,3.4×105,ﬂow,appears,to,be,that,obtained,by,le,Qu,´er,´e,,with,a,value,of,4.57946.,This,is,0.002%,different,from,the,converged,value,obtained,from,Nektar++,and,presented,in,the,table,1,(note,these,results,all,used,the,same,timestep,size).,This,level,of,agreement,falls,slightly,short,of,the,exact,(to,6,sf),concurrence,shown,in,[8],-,this,minor,discrepancy,may,be,down,to,details,of,the,solver,implementation,,or,perhaps,the,method,used,for,the,time-averaging.,The,value,of,the,time-averaged,Nusselt,number,obtained,from,Nektar++,can,also,be,seen,,in,the,ﬁgure,4,,to,converge,spectrally,with,the,element,order,p.,Note,also,the,wall-clock,run,times,tabulated,in,Table,1,,which,indicate,that,the,computation,is,not,bound,by,arithmetic;,the,number,of,degrees,of,freedom,scales,as,(p,+,1)2,and,these,execution,times,obviously,do,not,scale,even,nearly,as,strongly,as,p2.,Good,agreement,with,the,point,temperature,was,also,obtained,for,the,lower,Ra-value,case,(Fig.2),,in,which,the,MIT,benchmark,data,is,available,for,download,from,[7],-,amplitude,and,period,can,be,seen,to,match.,The,higher-Ra,value,generates,a,time,series,that,is,chaotic,and,thus,the,compari-,son,is,less,straightforward,since,only,its,statistical,properties,can,meaningfully,be,compared,(not,done,here).,The,lower-Ra,simulation,contains,a,periodic,perturbation,,the,ﬂow,ﬁeld,of,which,can,be,visualized,and,compared,to,Figs.5-7,of,[8].,Excellent,visual,agreement,with,those,ﬁgures,is,obtained,in,Nek-,tar++,outputs,(note,the,amplitude,of,the,perturbation,is,not,meaningful,as,the,phase,is,not,quoted,6,Element,order,p,1,2,3,4,5,6,7,8,9,10,11,time-av.,Nusselt,number,N,u,Execution,time,/,s,(16,logical,cores),3.66566,4.95898,4.62167,4.54045,4.56850,4.57956,4.57977,4.57943,4.57936,4.57935,4.57935,38,97,171,336,601,799,1047,1627,1903,2466,3029,Table,1:,Table,of,time-averaged,Nusselt,number,values,for,Ra,=,3.4×105,obtained,from,Nektar++.,in,the,paper).,The,procedure,for,extracting,these,plots,from,Nektar++,involves,identiﬁcation,of,the,time-dependent,parts,of,the,solution,(see,e.g.,Fig.5,of,[8]),in,a,relatively,large-amplitude,steady-,state,background,-,the,laminar,convective,cell,-,which,needs,to,be,subtracted.,This,is,done,using,the,Nektar++,time-averaged,ﬁeld,ﬁlter,to,compute,the,time-averaged,background,ﬁeld,,and,then,processing,the,outputs,to,subtract,the,background,using,the,FieldConvert,postprocessing,utility,of,Nektar++,(the,command,is,fieldconvert,-m,addfld:fromfld=MIT,Ra,3pt4e5,averaged.fld:scale=-1,MIT,Ra,3pt4e5.xml,MIT,Ra,3pt4e5.fld,MIT,Ra,3pt4e5,fluctuation.fld),(See,Fig.3,for,Nektar++,output).,Note,a,couple,of,points:,the,Nektar++,simulations,use,a,differently-scaled,time,variable:,to,convert,RaP,r,(P,r,is,the,Prandtl,number,,from,that,used,in,Nektar++,to,that,used,in,[8],and,[7],,multiply,by,0.71,for,both,cases).,Also,note,that,[7],uses,a,temperature,of,0.5,on,the,hot,side,and,−0.5,on,the,cold,side,,while,[8],and,the,Nektar++,ﬁles,use,values,of,1,and,0,respectively,-,conversion,trivial.,√,In,summary,,very,good,agreement,between,published,results,and,Nektar++,simulation,results,is,shown,for,averaged,Nusselt,numbers,computed,close,to,but,on,either,side,of,the,critical,Rayleigh,number,,and,linear,perturbations,to,ﬂow-ﬁeld,proﬁles,in,time,and,space,just,beneath,the,critical,value,of,the,Rayleigh,number.,Note,that,these,Rayleigh,number,values,fall,rather,short,of,resulting,in,a,truly,turbulent,overall,ﬂow,of,the,sort,reported,in,[5].,3,Preliminary,study,of,discrete,exterior,calculus,3.1,Introduction,In,In,recent,years,,structure-preserving,methods,have,been,applied,to,ﬁnite-element,methods.,short,,these,techniques,ensure,that,certain,important,properties,of,the,continuum,system,(e.g.,local,conservation,laws),are,preserved,in,the,discretized,version,(i.e.,the,action,of,discretization,commutes,with,certain,operations,,such,as,forming,curls,or,divergences).,There,is,a,growing,literature,on,the,advantages,of,such,methods,as,relevant,to,e.g.,stability,and,charge,conservation,7,Figure,3:,Nektar++,ﬂuctuation,ﬁelds,for,T,,,v,,u,(respectively,temperature,,vertical,velocity,,hor-,izontal,velocity),,corresponding,to,Figs.5-7,of,[8],,computed,by,subtracting,ﬁelds,averaged,over,one,period,of,Ra,=,3.4,×,105,case.,Scales,are,excluded,as,phase,does,not,correspond,to,that,used,in,ﬁgures,from,the,reference.,8,Figure,4:,Demonstration,of,spectral,convergence,for,the,Nusselt,numbers,displayed,in,Table,1.,[9],,often,using,a,discretization,of,Whitney,type,[10,,11,,12].,Complete,understanding,of,the,literature,on,this,subject,is,something,of,a,challenge,for,the,non-,specialist;,however,,a,clear,presentation,can,be,found,in,[13],for,the,case,of,a,globally-spectral,implementation,of,discrete,exterior,calculus.,The,formalism,there,described,is,here,used,to,treat,a,simple,toy,problem,,allowing,a,transparent,,elementary,presentation,of,the,recipe,including,such,relevant,actors,as,the,mesh,,discrete,exterior,derivative,operator,,and,discrete,Hodge,star,-,and,the,duals,of,all,three,of,the,aforementioned.,The,reference,[13],should,be,consulted,for,details,of,various,attractive,properties,that,make,the,formulation,work,as,the,presentation,here,concerns,itself,only,with,the,basics.,3.2,A,simple,problem,solved,using,discrete,exterior,calculus,The,problem,treated,is,a,one-dimensional,ordinary,differential,equation,of,Helmholtz,type,,d2u(x),dx2,+,u(x),π,=,f,(x).,(7),The,domain,of,interest,is,taken,to,be,the,interval,[0,,2π],with,a,periodic,boundary,condition.,It,is,clear,that,the,choice,of,a,periodic,domain,avoids,complications,with,what,to,do,with,the,mesh,and,dual,mesh,at,the,end,points.,In,exterior,calculus,language,,the,equation,is,expressed,as,∗,d,∗,du,+,u,π,=,f,(8),for,0-forms,u,and,f,,,exterior,derivative,d,and,Hodge,star,∗.,The,composite,∗d,∗,d,is,referred,to,as,the,Laplace-Beltrami,operator,in,the,literature,(in,this,simple,case,it,represents,the,Laplacian,operating,on,a,scalar).,9,It,is,clear,that,the,equation,is,self-consistent,in,the,pattern,of,forms,,i.e.,u,is,a,0-form,,du,a,1-form,,∗du,a,0-form,,and,so,forth.,The,basis,functions,taken,from,[13],are,αN,(x),=,1,N,(cid:40),cot,x,cosec,x,2,sin,N,x,2,sin,N,x,2,(N,even),2,(N,odd),βN,(x),=,,,1,2π,−,1,4,cos,N,x,,1,2π,(cid:80),N,−1,2,n=1,(cid:80),N,2,+,1,2,n=1,N,n,cos,nx,sin,nπN,(N,odd),n,cos,nx,sin,nπN,(N,even),(9),(10),Further,deﬁnitions,are,αN,n(x),≡,αN,(x,−,nh),and,βN,n(x),≡,βN,(x,−,nh),,for,h,≡,2πN,.,Note,that,these,functions,have,the,properties,αN,n(xm),=,δmn,and,(cid:82),xm+,h,2,xm−,h,2,βN,n(x)dx,=,δmn.,The,basis,functions,for,the,primal,forms,,deﬁned,on,a,primal,mesh,of,N,evenly-spaced,nodes,in,the,interval,[0,,2π],,are,then,and,φ0,N,n(x),=,αN,n(x),φ1,N,n(x),=,βN,n+,1,2,(x),dx.,(11),(12),The,corresponding,dual,forms,,described,on,the,Voronoi,dual,mesh,which,has,nodes,half-way,between,those,of,the,primal,mesh,,are,(noting,that,a,tilde,is,used,to,denote,a,dual,quantity,in,the,discrete,representation),and,˜φ0,N,n(x),=,αN,n+,1,2,(x),˜φ1,N,n(x),=,βN,n(x),dx.,(13),(14),Examination,of,the,equation,8,makes,it,clear,that,an,expression,for,the,discrete,Hodge,star,acting,on,a,1-form,is,required.,This,is,obtained,from,and,for,the,dual,Hodge,star,operator,H,k,ij,=,(cid:90),˜σn−k,i,∗φk,j,,,˜H,k,ij,=,∗,˜φk,j,,,(cid:90),σn−k,i,10,(15),(16),with,n,=,1,and,k,=,1,in,both,cases,(note,that,in,this,problem,,the,α,functions,are,used,only,to,plot,the,solution,,and,the,β,functions,are,only,used,in,evaluating,the,discrete,Hodge,operators).,The,ﬁnal,pieces,are,the,discrete,exterior,derivative,operators,on,the,primal,and,dual,meshes.,These,have,the,expected,form,though,one,subtlety,is,that,the,primal,mesh,operator,is,a,forward,difference,while,the,dual,is,a,backward,difference;,explicitly,,in,a,small,example,(N,=,4),,,,,,,,,D,=,˜D,=,for,the,primal,and,for,the,dual.,−1,1,0,−1,0,1,0,1,0,−1,0,0,0,1,0,−1,,,,,0,1,−1,1,0,−1,0,0,−1,0,0,0,1,1,0,−1,,,,,(17),(18),Having,assembled,these,matrices,,the,method,can,be,applied,and,tested.,The,matrix,equation,is,˜H,1,˜DH,1Du,+,1,π,u,=,f,.,(19),The,matrix,multiplication,and,inversion,routines,included,in,Intel,Math,Kernel,Library,(MKL),[14],were,used;,the,system,is,small,enough,that,a,direct,solver,,based,on,a,LU,decomposition,,is,effective.,Note,that,the,inclusion,of,the,reaction,term,u,π,in,the,equation,removes,from,the,com-,plementary,function,a,constant,term,and,a,general,solution,with,periodicity,2π,(were,this,not,the,case,,the,matrix,obtained,would,be,expected,to,be,singular);,this,difﬁculty,can,also,be,avoided,by,abandoning,the,periodic,boundary,condition.,Analytic,solutions,of,the,differential,equation,7,are,easily,generated,using,the,method,of,manufactured,solutions,,in,this,case,by,evaluating,the,right-hand-side,(f,(x)),corresponding,to,the,periodic,solution,u,=,e3,sin,2x.,A,graph,of,the,conver-,gence,with,N,of,the,resulting,numerical,problem,is,shown,in,Fig.,5.,Note,the,Linf,error,(assessed,against,the,analytic,solution),was,used,as,the,basis,for,this,assessment,,as,is,the,case,in,[13].,The,exponential,convergence,seen,here,can,be,compared,to,that,obtained,by,more,elementary,approaches,such,as,second-order,ﬁnite-difference,for,which,the,convergence,would,be,polynomial,(error,∝,h2).,In,conclusion,,a,spectral,method,using,the,machinery,of,discrete,exterior,calculus,has,been,de-,scribed,and,used,to,solve,a,simple,problem.,These,methods,have,much,potential,for,solving,more,interesting,systems,,potential,that,is,achieved,at,the,expense,of,more,involved,mathematics,than,that,outlined,here.,It,is,anticipated,that,the,implementation,chosen,for,NEPTUNE,will,be,a,spectral,/,hp,method,rather,than,the,globally-spectral,method,illustrated,here.,11,Figure,5:,Exponential,convergence,of,the,discrete,exterior,calculus,implementation,explained,in,the,main,text,(left).,Solution,curve,for,N,=,40,shown,to,right.,12,4,Summary,This,report,has,provided,further,numerical,results,using,the,Nektar++,framework,,continuing,in-,vestigations,started,in,[5].,Explicit,numerical,agreement,between,Nektar++,simulations,of,vertical,natural,convection,and,corresponding,results,from,the,literature,was,demonstrated.,Additionally,,a,simple,application,of,discrete,exterior,calculus,to,a,toy,problem,,exhibiting,spectral,convergence,,was,shown,using,a,simple,code,written,by,the,author.,It,is,apparent,that,the,exten-,sion,of,the,Nektar++,framework,to,support,less,elementary,implementations,will,entail,a,signiﬁcant,amount,of,effort,in,order,to,understand,the,theory,of,higher-order,discrete,exterior,calculus,and,to,incorporate,this,within,the,framework,in,a,modular,and,sustainable,way.,The,author,would,like,to,thank,the,developers,of,Nektar++,for,useful,discussions.,Acknowledgement,The,support,of,the,UK,Meteorological,Ofﬁce,and,Strategic,Priorities,Fund,is,acknowledged.,References,[1],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.,[2],D.,Moxey,et,al.,Nektar++,website.,https://www.nektar.info,,2020.,Accessed:,June,2020.,[3],Nektar-diffusion,proxyapp.,https://github.com/ExCALIBUR-NEPTUNE/nektar-diffusion,,2021.,Accessed:,September,2021.,[4],Nektar++,solver,for,Hasegawa-Wakatani,equations.,https://github.com/,ExCALIBUR-NEPTUNE/nektar-driftwave,,2021.,Accessed:,September,2021.,[5],E.,Threlfall,and,W.,Arter.,Finite,Element,Models:,I.,Technical,Report,CD/EXCALIBUR-FMS/0051-M6.1,,UKAEA,,2021.,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0051-M6.1.pdf.,Complementary,Activities,https:,[6],F.,Wilczynski,,D.W.,Hughes,,S.,Van,Loo,,W.,Arter,,and,F.,Militello.,Stability,of,scrape-off,layer,plasma:,A,modiﬁed,Rayleigh–B,´enard,problem.,Physics,of,Plasmas,,26(2):022510,,2019.,https://doi.org/10.1063/1.5064765.,[7],MIT,Benchmark.,http://www.mathematik.tu-dortmund.de/~featflow/en/benchmarks/,cfdbenchmarking/mit_benchmark.html.,Accessed:,March,2022.,13,[8],V.,Zucatti,,H.F.S.,Lui,,D.B.,Pitz,,and,W.R.,Wolf.,Assessment,of,reduced-order,modeling,strategies,for,convective,heat,transfer.,Numerical,Heat,Transfer:,Part,A:,Applications,77:7,,pages,702–729,,2020.,[9],S.,O’Connor,,Z.D.,Crawford,,and,O.H.,Ramachandran.,Time-integrator-agnostic,charge-,conserving,ﬁnite-element,PIC.,Phys.,Plasmas,28,,09211;,https:,//,doi.,org/,10.,1063/,5.,0046842,,,2021.,[10],L.,Kettunen,,J.,Lohi,,J.,R,¨abin,¨a,,S.,M,¨onk,¨ol,¨a,,and,T.,Rossi.,Generalized,ﬁnite-difference,schemes,with,higher-order,Whitney,forms.,ESAIM,Mathematical,Modelling,and,Numerical,Analysis,,55,,pages,1439–1460,,2021.,[11],F.,Rapetti.,High-order,edge,elements,on,simplicial,meshes.,ESAIM,Mathematical,Modelling,and,Numerical,Analysis,,41,,pages,1001–1020,,2007.,[12],F.,Rapetti,and,A.,Bossavit.,Whitney,forms,of,higher,degree.,SIAM,Journal,on,Numerical,Analysis,Vol.47,,Iss.3,10.1137/070705489,,2009.,[13],D.S.,Rufat.,Spectral,exterior,calculus,and,its,implementation.,PhD,thesis,,Caltech,,2017.,[14],Intel,oneAPI,Math,Kernel,Library.,https://www.intel.com/content/www/us/en/develop/,documentation/get-started-with-mkl-for-dpcpp/top.html,,2022.,Accessed:,March,2022.,14

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