CD-EXCALIBUR-FMS0066-M4.1_SupportHighDimensionalProcurement
===========================================================

.. meta::
   :description: technical note

   :keywords: ExCALIBUR,Support,High-dimensional,Procurement,M4.1,Version,1.00,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Mile-,stone,M4.1.,It,collates,technical,material,to,prepare,call,for,high-dimensional,procurement.,In,particular,,it,contains,a,description,of,a,simple,1d1v,kinetic,solver,implemented,within,the,Nektar++,spectral,/,hp,element,framework,and,the,application,of,this,solver,to,the,textbook,problem,of,the,two-stream,instability,,work,which,is,intended,to,prepare,the,ground,for,anticipated,up-,grades,to,Nektar++,and,also,to,deepen,UKAEA,knowledge,of,the,workings,of,the,Nektar++,code.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0066,Issue:,Date:,1.00,18,March,2022,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Ed,Threlfall,Will,Saunders,Signature,N/A,N/A,Date,18,March,2022,18,March,2022,BD,Reviewed,By:,Wayne,Arter,18,March,2022,Project,Technical,Lead,1,1,Introduction,Future,extensions,of,the,spectral,/,hp,method,used,in,the,Nektar++,framework,for,the,fusion,use,case,will,need,to,target,kinetic,models,ie.,solutions,for,matter,that,is,not,accurately,treatable,by,ﬂuid,dynamics,as,in,eg.,[1]).,The,extension,of,the,framework,to,handle,such,models,(physically,,Boltzmann-type,equations),in,some,generality,is,the,subject,of,NEPTUNE,grant,T/AW084/21.,In,anticipation,of,the,latter,,and,having,collateral,beneﬁts,in,terms,of,learning,Nektar++,,a,simple,proxyapp,to,handle,a,1d1v,case,has,been,prototyped,,as,described,herein.,This,is,applied,to,a,classic,kinetic,problem,in,plasma,physics:,the,two-stream,instability,in,a,system,of,collisionless,charged,matter.,Analytic,results,for,instability,linear,growth,rates,are,seen,to,be,replicated,by,this,code,(giving,a,non-trivial,benchmark,that,can,be,applied,immediately,to,the,anticipated,outputs,of,the,aforementioned,grant).,In,addition,,the,nonlinear,evolution,of,the,system,is,demonstrated.,By,virtue,of,the,implementation,of,a,model,incorporating,velocity-space,effects,,this,report,is,relevant,to,Work,Package,FM-WP2,-,Plasma,Multiphysics,Model.,2,2,A,spectral/hp,Vlasov-Poisson,solver,in,Nektar++,2.1,Introduction,This,work,anticipates,the,outputs,of,grant,T/AW084/21,,which,are,explicitly,to,provide,a,framework,for,solving,continuum,kinetic,problems,within,Nektar++,and,also,a,proxyapp,capable,of,solving,1d1v,and,1D-3V,systems.,Brieﬂy,,the,plan,involves,adding,a,new,library,module,TensorRegions,which,will,involve,the,inclusion,of,a,separate,computational,mesh,for,handling,the,velocity,compo-,nents,of,phase,space.,The,results,of,this,section,,which,are,obtained,with,a,simple,modiﬁcation,to,the,existing,Nektar++,code,,will,provide,a,ready-made,benchmark,for,the,expected,proxyapp,outputs,of,the,aforementioned,grant.,This,work,is,,to,the,authors’,knowledge,,the,ﬁrst,application,of,the,Nektar++,framework,to,a,kinetic,problem.,2.2,Theoretical,background,The,kinetic,equation,treated,is,the,1d1v,Vlasov-Poisson,system,,in,the,phase-space,coordinatized,by,(x,,v),,∂f,∂t,+,v,∂f,∂x,+,E,∂f,∂v,=,0.,(1),Here,E,=,−,∂φ,found,from,the,Poisson,equation,∂x,is,the,electric,ﬁeld,,depending,only,on,x,(not,v),,with,the,electrostatic,potential,φ,(cid:18)(cid:90),∂2φ,∂x2,=,ω2,P,(cid:19),−,1,,,f,dv,v0,(2),where,the,plasma,frequency,ωP,is,the,only,physical,parameter,in,the,problem.,The,subtracted,term,adds,a,uniform,(in,x,and,v),neutralizing,background;,v0,is,chosen,so,as,to,give,global,neutrality.,The,system,may,be,interpreted,as,electrons,moving,in,a,background,of,much,heavier,,i.e.,effectively,stationary,,ions.,More,broadly,,this,model,is,a,statistical,,continuum,representation,of,charged,matter,in,the,non-relativistic,limit,-,the,particle,dynamics,are,Newtonian,and,electrodynamics,reduces,to,electrostatics.,It,is,possible,to,include,additional,physics,by,insert-,ing,collision,terms,on,the,right-hand-side,but,that,is,left,for,future,work,,as,classic,and,nontrivial,results,can,be,obtained,in,the,collisionless,limit.,The,deﬁnitions,are,as,in,[2];,note,the,electric,ﬁeld,in,the,equations,is,rescaled,by,a,factor,of,e,m,relative,to,the,physical,value.,It,is,worth,noting,that,the,matter,density,function,comes,with,the,interpretation,of,something,that,is,positive;,f,(x,,v),≥,0.,2.3,Nektar++,implementation,The,implementation,is,made,possible,within,the,existing,Nektar++,framework,by,realizing,that,the,system,1,has,the,form,of,a,two-dimensional,advection,problem.,3,The,solver,derives,from,the,class,AdvectionSystem,and,the,advection,velocity,is,simply,vadv,=,(v,,E).,In,fact,the,new,solver,was,based,on,nektar-driftwave,provided,to,the,NEPTUNE,com-,munity,by,the,developers,of,Nektar,and,available,at,[3].,Most,of,the,work,involves,writing,the,method,ExplicitTimeInt,(which,in,the,implementation,discussed,here,includes,also,the,solu-,tion,of,the,Poisson,problem).,The,advection,part,of,the,problem,is,solved,using,a,discontinuous,Galerkin,formulation,with,an,upwind,numerical,ﬂux.,The,Poisson,problem,is,solved,using,the,API,function,HelmSolve,(with,zero,Helmholtz,constant);,note,that,a,continuous,ﬁeld,is,always,used,for,the,electric,potential,,as,is,the,case,in,nektar-driftwave.,During,a,prototyping,phase,,the,Poisson,problem,was,solved,using,a,simple,ﬁnite,difference,solver,before,being,upgraded,to,the,full,spectral/hp,method.,Note,that,having,all,the,ﬁelds,in,the,proper,spectral/hp,form,enables,use,of,Nektar++,API,functionality,to,provide,diagnostics,of,e.g.,elec-,trostatic,ﬁeld,energy,and,matter,kinetic,energy,using,the,Integral,method,to,perform,numerical,quadrature,accurate,to,the,spectral,order,of,the,code.,The,only,non-API,functionality,that,was,necessary,is,for,the,code,to,integrate,f,over,one,dimension,(v),in,order,to,obtain,the,spatial,charge,density,(Nektar++,does,not,currently,provide,this,capability,-,[4],-,it,can,integrate,along,element,boundaries,but,not,along,a,line,cutting,through,an,element).,This,was,done,using,Gauss-Legendre,quadrature,(see,,for,example,,[5]),and,involved,use,of,the,Nektar++,API,functions,to,obtain,ﬁeld,values,at,the,quadrature,points.,Given,nG,such,points,over,a,(one-dimensional),element,,polynomials,up,to,order,2nG,−1,are,integrated,exactly;,it,is,thus,simple,to,ensure,that,the,integrals,are,exact,up,to,a,given,ﬁnite,element,order.,The,Gaussian,integrals,are,done,on,a,per-element,basis,so,there,is,no,question,of,inaccuracies,due,to,attempting,to,integrate,over,the,ﬁeld,discontinuities,of,the,discontinuous,Galerkin,method;,also,,since,the,Gauss,quadrature,points,lie,strictly,within,the,elements,,there,is,no,issue,with,the,ambiguity,associated,to,non-unique,ﬁeld,values,at,element,boundaries.,Collectively,,the,integrals,form,a,one-dimensional,array,indexed,by,the,node,positions,along,the,x-direction;,these,values,are,converted,to,a,two-,dimensional,array,over,the,entire,(x,,v),domain,in,order,to,provide,an,appropriate,right-hand-side,for,the,HelmSolve,function.,The,spatial,dimension,is,taken,as,periodic,in,the,range,[−1,,1],with,a,periodic,boundary,condi-,tion.,The,velocity,space,is,taken,to,be,in,the,range,[−4,,4],with,homogeneous,Dirichlet,boundary,conditions,on,f,and,homogeneous,Neumann,conditions,on,φ.,Note,initial,tests,used,a,periodic,boundary,condition,in,v,,which,worked,due,to,the,density,vanishing,at,that,boundary,,but,which,caused,a,crash,when,used,with,MPI,(possibly,a,bug).,The,solution,was,made,compatible,with,the,MPI,support,in,Nektar++,by,using,the,shared,array,class,Nektar::Array.,The,possibility,of,more,than,one,thread,contributing,to,the,v-integral,at,a,given,x-point,was,countered,by,performing,a,‘minimum’,reduction,operation,after,the,evaluation,of,the,set,of,v-integrals.,In,addition,,vector,arithmetic,API,functions,for,operating,on,the,shared,arrays,were,utilized,,e.g.,Vmath::Vvtvp,which,is,‘vector,times,vector,plus,vector’.,Note,that,the,API,contains,MPI,reduction,commands,for,summation,,maximum,,and,minimum,(e.g.,the,latter,is,AllReduce(Array,,LibUtilities::ReduceMax));,this,knowledge,means,that,the,temperature-,maximum,ﬁlter,in,[1],can,now,be,made,MPI-compatible.,There,are,a,number,of,properties,about,this,implementation,that,are,non-optimal,from,the,point,of,view,of,efﬁciency:,the,Poisson,problem,is,solved,in,two,dimensions,when,it,only,depends,on,one,of,those,dimensions;,the,electric,ﬁeld,energy,integral,is,done,over,two,dimensions,when,again,the,4,electric,ﬁeld,only,varies,in,one,dimension.,Note,that,these,(and,other),performance,concerns,in,Nektar++,will,be,addressed,by,the,outputs,of,Call,T/AW084/21.,The,(entirely,C++),code,was,written,and,debugged,using,the,Microsoft,Visual,Studio,2019,Com-,munity,Edition,IDE,[6],running,under,Microsoft,Windows,(note,that,the,author’s,experience,with,the,Nektar++,community,suggests,that,Windows,users,are,in,something,of,a,minority).,It,is,intended,that,the,new,solver,will,be,made,available,to,the,NEPTUNE,community,soon,,though,there,are,areas,of,incompleteness,e.g.,dependencies,on,the,particular,mesh,(the,problems,treated,in,this,section,all,used,the,same,regular,computational,mesh,,which,comprised,1600,squares,of,side,0.1).,2.4,Two-stream,instability,A,classic,test,case,for,the,system,1,is,the,case,of,two,counter-propagating,beams,of,like-charged,particles,(in,a,static,uniform,neutralizing,background),interacting,via,electrostatic,forces,only.,The,initial,condition,is,spatially-homogeneous,and,has,zero,electrostatic,ﬁeld,-,such,a,conﬁguration,is,trivially,seen,to,be,a,time-independent,solution,of,Eq.1,-,and,the,question,is,the,stability,and,subsequent,time-evolution,of,the,system.,This,problem,has,a,long,history,of,study.,The,paper,[2],is,an,early,(1967),numerical,investigation,using,the,contour,dynamics,technique,to,track,the,time-evolution,of,boundaries,between,f,=,1,and,f,=,0,regions,of,phase,space,,thereby,reducing,the,dimensionality,of,the,problem,by,one,and,making,it,tractable,on,’60s,computers;,the,limitation,is,that,only,values,of,1,,0,are,allowed,for,f,.,The,latter,is,rather,complementary,to,the,spectral/hp,method,,which,,without,special,modiﬁcations,,is,not,expected,to,work,well,for,non-smooth,functions,(this,is,a,general,feature,of,spectral,methods;,basically,,the,efﬁcacy,of,the,method,results,from,the,presupposition,that,the,solution,is,smooth,,and,if,this,is,not,the,case,then,numerical,problems,arise,via,Gibbs,/,Runge,phenomena).,A,basic,synopsis,of,the,results,in,[2],is,that,the,initial,condition,is,not,stable,and,the,system,evolves,in,phase,space,to,a,distinctive,form,,for,which,a,subjective,justiﬁcation,is,now,given.,In,view,of,the,spatial,periodicity,of,the,system,,it,seems,reasonable,to,expand,the,electric,ﬁeld,in,a,Fourier,series.,In,fact,,some,insight,into,the,steady-state,solution,can,be,found,by,keeping,only,the,leading,term,i.e.,E(x),=,E0,cos(πx).,Changing,variables,,one,ﬁnds,that,steady,solutions,2,v2.,Plotting,contours,are,given,by,functions,of,the,local,particle,energy,density,,E,≡,E0,of,constant,E,(Fig.1),is,certainly,suggestive,of,the,numerical,solutions,exhibited,in,the,sequel.,Note,the,exact,evolution,of,f,is,determined,by,the,initial,data,and,the,value,of,ω2,P,-,some,setups,give,more,than,two,of,the,equilbrium,points,that,are,now,described.,Note,the,two-equilibrium,cosine,electric,ﬁeld,presented,here,gives,a,mathematical,problem,equivalent,to,a,rigid,pendulum,in,a,uniform,gravitational,ﬁeld,with,the,solution,given,in,terms,of,an,elliptic,integral;,the,stable,equilibrium,point,corresponds,to,the,pendulum,bob,at,the,minimum,height,and,the,unstable,point,has,the,bob,at,the,maximum.,The,two,equilibrium,points,-,zeros,of,the,electric,ﬁeld,-,correspond,to,a,phase-space,attractor,and,a,repeller,(near,these,respective,points,,the,governing,equation,is,locally,∂f,∂t,=,0,,with,the,negative,sign,case,representing,simple,harmonic,motion).,∂x,∓,x,∂f,∂t,+,v,∂f,π,+,1,sin(πx),It,will,be,seen,that,these,features,are,evident,in,the,numerical,solutions.,Initial,data:,the,system,was,initialized,with,two,spatially-uniform,counter-propagating,beams,that,5,Figure,1:,Contours,of,constant,energy,assuming,E(x),∝,cos(πx),,for,one,period,of,solution.,Figure,generated,by,[7].,In,terms,of,the,phase-space,attractor,/,repeller,discussed,in,the,main,text,,there,is,an,attractive,point,on,the,x-axis,at,x,=,−0.5,and,a,repeller,at,x,=,0.5.,6,were,both,Gaussian,in,velocity,space:,the,function,used,was,(cid:18),e−,(v−v0)2,2σ2,+,e−,(v+v0)2,2σ2,(cid:19),,,(3),√,1,8πσ2,with,σ,=,0.2,chosen,so,the,initial,data,is,fairly,smooth,(recall,it,is,expected,that,delta,or,step,function,initial,data,will,not,work,well,with,the,spectral,/,hp,element,method),and,v0,=,1.,This,function,is,normalized,to,unity,when,integrated,over,v,at,any,given,x,(by,coincidence,,it,also,has,maxima,which,are,very,close,,but,not,equal,to,unity).,Physically,,σ2,may,be,taken,as,the,temperature,for,each,of,the,non-interacting,beams,(and,note,the,total,energy,of,the,two-beam,conﬁguration,is,given,by,v2,0,+,σ2).,A,small,x-dependent,initial,perturbation,was,applied,to,the,initial,data,Eq.3,in,the,form,of,the,multiplicative,factor,1,+,0.01,sin,πx,in,order,to,seed,the,instability.,Note,that,the,periodic,boundary,conditions,mean,that,the,Poisson,problem,is,not,well-posed,unless,the,total,charge,within,one,period,is,zero,(equivalently,,the,mean,electrostatic,ﬁeld,is,zero).,For,a,modern,study,of,this,system,see,[8].,2.5,Numerical,results,for,two-stream,instability,The,numerical,performance,of,the,solver,was,tested,by,running,simulations,on,a,20,×,80,grid,of,square,elements,with,an,element,size,of,0.1.,The,level,of,computational,accuracy,was,varied,by,using,different,global,values,for,the,number,of,modes,(per,dimension),per,element,-,varied,from,2,(equivalent,to,using,linear,polynomials,within,each,element),to,7.,A,standard,four-stage,fourth-order,Runge-Kutta,(ClassicalRungeKutta4,in,Nektar++),was,used,to,perform,explicit,time-,stepping.,It,is,seen,that,the,solution,represented,by,the,initial,data,Eq.3,is,not,stable,,and,that,the,time-,evolution,leads,to,the,formation,of,a,distinctive,phase-space,pattern,(Fig.5),similar,to,that,seen,in,[2].,Note,it,is,indicated,in,that,paper,that,the,solution,depends,on,the,externally-applied,periodic,boundary,condition.,The,precise,form,of,the,solution,is,dependent,on,the,value,of,ωP,and,the,form,of,the,initial,data,-,an,exposition,of,this,is,given,in,the,following,subsection,,for,the,linear,regime.,Conservation,of,particle,number,-,the,amount,of,matter,is,quantiﬁed,by,(cid:82),f,(x,,v)dx,dv.,It,was,found,that,,for,all,p,>,1,,the,simulation,conserved,particle,number,to,2,×,10−13,(relative,to,unity),over,a,period,of,40,time,units.,Positivity,of,f,(x,,v),-,since,this,function,is,essentially,a,probability,density,,negative,values,are,unphysical,(as,well,as,liable,to,give,difﬁculty,when,evaluating,entropy,,which,involves,forming,ln,f,).,Improvements,can,potentially,be,made,using,the,techniques,outlined,in,[9].,The,table,1,gives,the,magnitude,of,the,most,negative,value,at,the,end,of,a,40,time,unit,simulation,-,note,that,these,do,not,seem,to,reduce,as,the,element,order,p,is,increased.,The,ﬁgure,4,shows,,however,,that,the,negative,values,are,conﬁned,to,relatively,small,regions,of,phase,space.,Energy,conservation,-,energy,conservation,is,exhibited,,to,a,degree,,but,there,is,currently,a,growth,in,the,total,energy,with,time,(Fig.2).,The,expected,scenario,was,a,slow,decay,of,the,total,energy,(because,upwind,DG,schemes,are,known,to,be,slightly,dissipative,-,indeed,,the,lower-p,sim-,ulations,shown,slightly,suppressed,energy,growth,,presumably,for,this,reason).,This,is,being,7,Figure,2:,Time-evolution,of,the,total,energy,change,for,5,,6,,7-mode,solutions,(based,on,a,total,energy,of,approx.,unity).,Figure,3:,Change,in,electric,and,kinetic,energy,from,start,of,simulation;,data,shown,for,6-mode,and,7-mode,solutions.,8,Element,order,p,Magnitude,of,most,negative,value,f,(x,,v),1,2,3,4,5,7,1.1,×,10−1,2.0,×,10−3,2.4,×,10−3,4.1,×,10−3,5.2,×,10−3,8.1,×,10−3,Table,1:,Table,of,maximum,negative,values,of,f,(x,,v),from,Nektar++,Vlasov-Poisson,solver.,investigated,at,the,time,of,writing.,Note,that,the,energy,conservation,problem,seems,to,arise,from,an,imbalance,between,the,kinetic,energy,lost,and,the,electric,ﬁeld,energy,gained,during,the,time,stepping:,the,latter,is,consistently,larger,than,the,former.,2.6,Dispersion,relations,and,linear,instability,growth,rates,Linear,dispersion,relations,can,be,calculated,for,the,growth,rates,of,harmonic,instabilities,over,initial,data,that,is,uniform,in,position,space,and,has,a,speciﬁed,distribution,in,velocity,space.,This,is,based,on,a,linearized,theory,that,can,be,found,in,e.g.,[10].,Note,all,initial,data,here,are,normalized,to,a,total,of,unity,(i.e.,each,beam,is,normalized,to,one,half).,The,linear,dispersion,relation,for,a,single,species,can,be,expressed,as,1,=,ω2,P,(cid:90),∞,−∞,f,(v),dv,(ω,−,kv)2,.,(4),The,simplest,initial,data,case,is,two,delta,functions,in,v,,for,which,the,dispersion,relation,is,x2,=,1,2,1,+,8u2(cid:17),1,+,2u2,±,for,x,≡,ω,ωP,,,u,≡,kv0,ωP,√,(cid:16),.,A,second,example,echoes,[2],by,using,a,background,of,two,counterpropagating,Heaviside,func-,tions,in,velocity,space,,normalized,to,unity,in,total,,centred,on,v0,and,extending,to,±σ.,The,dispersion,relation,for,this,is,x2,=,1,,,with,the,delta,function,2,case,recovered,if,s,=,0.,16s2u2,+,8u2,+,1,1,+,2u2,+,2s2,±,√,(cid:17),(cid:16),The,above,simple,cases,are,not,directly,useful,as,the,spectral,/,hp,implementation,is,not,expected,to,perform,well,with,non-smooth,initial,data.,It,is,,however,,possible,to,obtain,an,analytic,dispersion,relation,for,the,initial,data,in,Eq.3,,as,follows.,The,NRL,Plasma,Formulary,[11],gives,1,√,π,(cid:90),∞,−∞,dt,e−t2,t,−,x,=,e−x2,(cid:18),√,i,π,−,2,(cid:90),x,dt,e−t2(cid:19),.,0,(5),An,expedient,(non-rigorous,,though,it,will,be,seen,to,give,the,correct,answer),path,to,a,closed-form,solution,involves,differentiating,Eq.5,,giving,9,Figure,4:,Negative,values,of,f,(x,,v),in,the,7-mode,solution.,10,Figure,5:,Time-evolution,of,the,bi-Gaussian,initial,data,into,a,state,showing,an,phase-space,at-,tractor,and,a,repeller,(here,ω2,P,=,10).,The,simulation,time,interval,between,successive,plots,is,8.0,units.,Compare,Fig.1.,Note,the,vertical,axis,is,v,and,the,horizontal,x.,Figure,6:,Time-evolution,of,the,electric,potential,plotted,as,a,function,of,x,from,the,simulations,in,Fig.5,-,plots,correspond,to,the,same,intervals.,Note,the,closeness,of,the,solution,to,a,simple,harmonic,wave.,11,d,dx,1,√,π,(cid:90),∞,−∞,dt,e−t2,t,−,x,=,1,√,π,(cid:90),∞,−∞,dt,e−t2,(t,−,x)2,=,−2,(cid:16),√,πxe−x2,1,+,(i,−,erﬁ(x)),(cid:17),(6),wh.,erﬁ(x),≡,2√,π,(cid:82),x,−∞,dt,et2,.,Inserting,Eq.3,into,Eq.4,leads,to,quadratures,of,the,above,form,and,straightforwardly,to,the,dis-,persion,relation,1,=,−,ω2,P,2σ2k2,√,(cid:16),2,−,πAe−A2,(erﬁ(A),−,i),−,√,πA(cid:48)e−A(cid:48)2,(cid:17),(erﬁ(A(cid:48)),−,i),(7),with,A,≡,ω−kv0√,2σk,,,A(cid:48),≡,ω+kv0√,2σk,.,Note,that,the,delta-function,dispersion,relation,can,be,obtained,from,this,in,the,limit,σ,→,0,and,using,erﬁ(x),∼,ex2,(cid:1),−,i,(keep,the,two,leading,terms,and,neglect,the,i,as,the,exponential,dominates,it).,(cid:0),1,x,+,1,2x3,√,π,This,form,of,the,dispersion,relation,does,not,seem,invertible,by,analytic,means.,In,practice,,roots,are,straightforwardly,obtained,by,Newton’s,method;,it,is,found,in,the,cases,examined,that,the,roots,lie,on,the,imaginary,axis,i.e.,the,complex,frequencies,have,zero,real,part.,The,open-source,library,Fadeeva,[12],was,used,to,compute,these,functions,(limited,tests,of,this,library,were,done,e.g.,comparing,outputs,with,those,calculated,via,[7]).,Note,that,it,is,computationally,better,to,use,the,Dawson,function,D(z),≡,2,e−z2erﬁ(z),than,directly,use,the,imaginary,error,function,erﬁ(z).,√,π,Numerical,results,to,compare,with,this,theory,were,obtained,using,a,small,initial,perturbation,amplitude,,1,+,10−7,sin,nπx,(n,is,the,wavenumber,of,the,perturbation),,chosen,in,order,to,prolong,the,duration,of,the,linear,regime.,See,Fig.8,for,an,illustration.,Simulations,were,run,long,enough,that,a,linear,growth,region,was,seen,to,be,established,beyond,the,initial,transient,and,the,growth,rates,were,obtained,by,simple,gradient,measurement.,The,growth,rates,were,assessed,from,the,integral,of,the,electric,ﬁeld,energy,,for,which,the,growth,rate,is,twice,the,linear,mode,growth,rate,on,account,of,bilinearity.,These,simulations,were,found,to,be,sufﬁciently,accurate,using,only,four,modes,and,the,agreement,between,theory,and,numerics,is,good,(Fig.7),,with,the,instability,type,concurring,with,the,wavenumber,used,in,the,initial,data,perturbation.,12,Figure,7:,Theoretical,predictions,for,the,growth,rates,of,modes,n,=,1,−,4,(spatial,wavenumber,k,=,nπ),as,a,function,of,the,plasma,frequency,ωP,.,Overlain,are,numerical,results,from,the,Nektar++,implementation,,showing,good,agreement.,13,Figure,8:,Linear,growth,of,the,electric,ﬁeld,energy,for,ω2,Eq.3.,The,linear,growth,saturates,at,later,times,when,the,nonlinear,dynamics,dominates,it.,P,=,10,and,the,bi-Gaussian,initial,condition,14,3,Summary,This,report,has,provided,further,numerical,results,using,the,Nektar++,framework,,in,which,an,initial,attempt,at,a,1d1v,kinetic,solver,has,been,implemented,and,tested.,Qualitative,agreement,with,the,nonlinear,dynamics,of,the,two-stream,instability,in,the,literature,was,shown.,More,quantitative,results,were,given,for,the,growth,rates,of,linear,wave,instabilities,for,the,same,system,,which,exhibited,good,agreement,with,theory.,The,study,justiﬁes,the,altering,of,emphasis,from,particles,to,ﬁnite,elements,in,the,high-dimensional,procurement.,The,author,would,like,to,thank,the,developers,of,Nektar++,for,useful,discussions,and,Dr,James,Cook,and,Dr,Joseph,Parker,of,UKAEA,for,helpful,discussions,regarding,kinetic,methods.,Acknowledgement,The,support,of,the,UK,Meteorological,Ofﬁce,and,Strategic,Priorities,Fund,is,acknowledged.,References,[1],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:,[2],K.V.,Roberts,and,H.L.,Berk.,Nonlinear,evolution,of,a,two-stream,instability.,Phys.,Rev.,Lett.,Vol.,16,No.,9,,August,1967,,pages,297–300,,1967.,[3],Nektar++,solver,for,Hasegawa-Wakatani,equations.,https://github.com/,ExCALIBUR-NEPTUNE/nektar-driftwave,,2021.,Accessed:,September,2021.,[4],D.,Moxey.,Private,conversation,,2021.,[5],Gaussian,quadrature,wiki.,https://en.wikipedia.org/wiki/Gaussian_quadrature.html.,Accessed:,March,2022.,[6],Microsoft,Visual,Studio.,https://visualstudio.microsoft.com,,2022.,Accessed:,March,2022.,[7],WolframAlpha.,www.wolframalpha.com.,Accessed:,March,2022.,[8],J.T.,Parker,and,P.J.,Dellar.,Poisson,system,in,the,weakly,collisional,doi:10.1017/S0022377814001287.,Fourier-Hermite,spectral,representation,for,the,Vlasov-,J.,Plasma,Physics,Vol.,81,,305810203;,limit.,[9],V.,Zala,,R.M.,Kirby,,and,Narayan,A.,Structure-preserving,nonlinear,ﬁltering,for,continuous,and,discontinuous,Galerkin,spectral/hp,element,methods.,SIAM,Journal,on,Scientiﬁc,Computing,43,(6),,A3713-A3732,,2021.,15,[10],T.H.,Stix.,Waves,in,Plasmas.,American,Inst.,Phys.,,1992.,[11],J.D.,Huba.,NRL,Plasma,Formulary.,US,Naval,Research,Laboratory,,2013.,[12],Fadeeva,library.,http://ab-initio.mit.edu/wiki/index.php/Fadeeva_Package.,Ac-,cessed:,March,2022.,16

:pdfembed:`src:_static/CD-EXCALIBUR-FMS0066-M4.1_SupportHighDimensionalProcurement.pdf, height:1600, width:1100, align:middle`