CD-EXCALIBUR-FMS0052-M7.1_LiteratureCallTAw08621MathematicalSupportSoftwareImplementation
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ExCALIBUR,Literature,review,for,Call,T/AW086/21:,“Mathematical,Support,for,Software,Implementation”,M7.1,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Milestone,7.1.,This,is,a,lit-,erature,review,performed,to,support,the,Call,T/AW086/21:,“Mathematical,Support,for,Software,Implementation”.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0052,Issue:,Date:,1.00,September,16,,2021,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Joseph,Parker,Wayne,Arter,Signature,N/A,N/A,Date,September,16,,2021,September,16,,2021,BD,Reviewed,By:,Rob,Akers,September,16,,2021,Advanced,Dept.,Manager,Computing,Approved,By:,Rob,Akers,September,16,,2021,Advanced,Dept.,Manager,Computing,2,1,Introduction,Call,T/AW086/21,“Mathematical,Support,for,Software,Implementation”,is,the,second,in,a,series,of,calls,for,mathematical,support,,following,Call,T/NA084/20,“Investigate,matrix-preconditioning,techniques”.,This,task,focuses,upon,the,suitability,of,available,numerical,algorithms,(or,the,devel-,opment,of,new,algorithms),for,Exascale-targeted,plasma,modelling.,The,ideal,numerical,algorithms,for,future,Exascale,edge,plasma,codes,would,have,preferably,at,least,the,following,properties:,P1,Accurate,solution,of,hyperbolic,problems.,P2,Ability,to,deliver,efficient,and,accurate,solutions,of,corresponding,elliptic,problems.,P3,Accurate,modelling,of,highly,anisotropic,dynamics.,P5,Accurate,representation,of,velocity,(phase),space.,P6,Preservation,of,conservation,properties,of,the,underlying,equations.,P7,Scalability,to,likely,Exascale,architectures:,a,interaction,between,models,of,different,dimensionality,,b,interaction,between,particle,and,fluid,models,,c,dynamic,construction,of,surrogates.,P8,Performance,portability,to,allow,rapid,deployment,upon,emerging,hardware.,It,is,unlikely,that,any,algorithm,will,have,all,the,above,,and,part,of,the,call,exercise,will,be,to,rank,the,importance,of,these,properties.,Work,under,the,previous,NEPTUNE,calls,has,provided,1.,a,survey,of,preconditioning,methods,,identifying,suitable,methods,for,Project,NEPTUNE,software,,see,ref,[1,,2],2.,strong,indication,that,time,advance,methods,could,benefit,from,use,of,larger,timesteps,with-,out,significant,loss,of,accuracy,3.,as,yet,,no,identification,of,a,satisfactory,manner,to,treat,extremely,anisotropic,transport,of,plasma,4.,a,study,of,particle,codes,showing,the,importance,of,using,a,control-variates,approach,[2],,ie.,of,carefully,handling,of,statistics,,but,5.,as,yet,,no,identification,of,a,satisfactory,manner,to,coupling,continuum,(fluid),and,particle,models,of,plasma,together,in,the,different,ways,required,by,NEPTUNE,[3].,3,It,has,also,become,evident,that,many,arguments,for,Exascale,suitability,of,a,particular,approach,would,be,strengthened,by,mathematical,analysis,of,the,proposed,algorithm.,Indeed,,many,tech-,niques,used,in,modelling,systems,of,high,dimensionality,modelling,(dimension,five,or,larger),involve,random,sampling,techniques,where,a,knowledge,of,stochastics,is,likely,invaluable.,In,addition,,certain,cutting-edge,algorithms,such,as,“asymptotic-preserving”,methods,and,Variable,Stepsize-Variable,Order,timestepping,require,advanced,mathematical,skills,to,understand,and,thus,to,adapt,for,implementation,in,NEPTUNE,software.,This,report,provides,a,literature,review,designed,to,inform,the,production,of,Call,T/AW086/21,It,contains,two,main,sections,concerning,“Mathematical,Support,for,Software,Implementation”.,recent,advances,in,algorithm,development.,Section,2.1,discusses,methods,used,for,the,time,advance,of,hyperbolic,equations,,while,section,2.2,discusses,the,solution,of,elliptic,equations.,The,work,is,summarized,in,Section,3.,2,Algorithms,for,NEPTUNE-relevant,equation,systems,Modelling,the,physical,systems,relevant,to,NEPTUNE,will,entail,solving,a,partial,differential,equa-,tion,system,where,hyperbolic,equations,are,to,be,evolved,in,time.,Evolving,hyperbolic,equations,implicitly,often,requires,the,solution,of,a,corresponding,elliptic,equation.,Moreover,,in,relevant,physical,systems,hyperbolic,equations,are,often,constrained,by,an,elliptic,equation,which,must,be,solved,at,every,timestep,(for,example,,in,the,Vlasov–Poisson,system,,the,Poisson,equation,must,be,solved,at,every,timestep,to,determine,the,electrostatic,potential,needed,to,advance,the,Vlasov,equation).,In,this,section,,we,discuss,recent,advances,in,algorithms,for,solving,hyperbolic,systems,(in,section,2.1),and,elliptic,systems,(in,section,2.2).,2.1,Hyperbolic,systems,In,this,subsection,,we,discuss,recent,work,on,algorithms,for,the,time-advance,of,hyperbolic,sys-,In,particular,we,discuss,variants,of,Implicit-Explicit,(IMEX),time,advance,schemes,which,tems.,have,additional,favourable,properties,–,here,,schemes,which,have,an,increased,order-of-accuracy,from,incorporating,deferred,corrections,,and,variants,which,are,asymptotic,preserving.,We,also,discuss,Variable,Stepsize,,Variable,Order,(VSVO),schemes.,2.1.1,IMEX,Schemes,Multi-physics,systems,typically,exhibit,multiple,timescales.,In,algorithms,,the,timescales,for,multi-,physics,systems,are,often,separated,into,fast,and,slow,timescales,,where,“slow”,may,refer,to,either,a,slow,physical,timescale,or,to,a,large,computational,cost,of,calculating,the,effect,on,a,parallel,machine.,The,canonical,example,of,this,approach,is,an,Implicit-Explicit,(IMEX),scheme.,Such,schemes,seek,to,integrate,the,equation,du,dt,=,R(x,,u,,t),=,F,(x,,u,,t),+,G(x,,u,,t),,(1),4,in,time,by,decomposing,the,right-hand,side,R,into,terms,F,and,G,which,are,responsible,for,fast,and,slow,timescales,respectively.,Denoting,the,approximation,to,u,at,time,t,as,un,=,u(tn),and,approximating,du/dt,=,(un+1,−,un)/∆t,,equation,(1),may,be,written,as,un+1,=,un,+,∆t,F,(x,,un+1,,tn+1),+,∆t,G(x,,un,,tn),,and,then,integrated,in,two,steps,u∗,=,un,+,∆t,G(x,,un,,tn),,un+1,−,∆t,F,(x,,un+1,,tn+1),=,u∗,,(2),(3a),(3b),where,(3a),is,an,explicit,method,,and,(3b),may,be,solved,by,an,implicit,scheme.,Other,choices,of,approximation,to,du/dt,lead,to,different,schemes,,with,IMEX,schemes,generally,being,classified,as,IMEX,Runge–Kutta,or,IMEX,multistep,methods.,There,is,also,work,on,IMEX,schemes,for,General,Linear,Methods,(which,are,generalizations,of,Runge–Kutta,and,multistep,methods),[4].,This,approach,allows,large,time,steps,to,be,taken,in,the,implicit,integration,of,F,(associated,with,the,fast,time,scale),,rather,than,forcing,∆t,to,limited,by,the,typically-smaller,stability,limit,of,an,explicit,scheme.,This,approach,is,particularly,fruitful,when,G,is,nonlinear,(and,thus,difficult,to,integrate,implicitly),or,expensive,to,compute,(so,that,the,repeated,evaluations,necessary,for,iterative,methods,are,undesirable).,Variations,on,this,fundamental,IMEX,idea,is,an,area,of,ongoing,research,,with,a,typical,approach,being,to,incorporate,desirable,properties,into,IMEX,schemes.,In,the,following,sections,we,dis-,cuss,some,examples,of,this:,using,deferred,corrections,to,increase,the,order,of,schemes,,en-,suring,schemes,are,asymptotic,preserving,,and,combining,IMEX,with,high-order,multiderivative,schemes.,Deferred,Correction,Methods,One,line,of,work,combines,IMEX,schemes,with,Spectral,De-,ferred,Correction,(SDC),[5],and,Integral,Deferred,Correction,(InDC),schemes,[6].,Both,SDC,and,InDC,were,originally,developed,in,the,context,of,solving,ODEs,[7,,8],and,are,based,on,the,older,Deferred,Correction,(DC),method,[9,,10].,In,Deferred,Correction,,one,integrates,the,ODE,for,N,time,steps,with,a,kth-order,method,,and,then,interpolates,the,numerical,solution,with,a,(unique),N,th,order,polynomial.,Substituting,this,polynomial,back,into,the,original,equation,,one,may,derive,an,“error,equation”,,an,ODE,for,the,difference,between,the,true,solution,and,the,polynomial,ap-,proximation.,Solving,this,equation,with,the,same,kth,order,method,yields,corrections,to,add,to,the,original,numerical,approximation,to,make,it,closer,to,the,true,solution.,It,may,be,shown,that,the,corrected,solution,is,(2k)th,order,accurate,[9,,10],,and,therefore,a,higher-order,method,has,been,constructed,automatically,from,a,low-order,method.,SDC,and,InDC,follow,this,idea,,except,in,these,approaches,the,deferred,correction,is,applied,to,the,error,equation,formulated,as,a,Picard,integral,equation,rather,than,an,ODE.,This,approach,has,been,shown,to,have,better,stability,and,accuracy,properties,[7,,8].,The,difference,between,SDC,and,InDC,is,in,the,choice,of,quadrature,grid,for,solving,the,error,equation:,SDC,uses,Gauss,,Lobatto,or,Radau,grids,as,this,improves,stability,and,accuracy,properties,,while,InDC,uses,a,uniform,grid,as,this,ensures,that,increasing,the,order,of,the,low-order,method,will,propagate,through,to,increase,the,order,of,the,corrected,scheme.,This,allows,very,high-order,schemes,to,be,constructed,systematically,using,only,low-order,schemes,[11].,5,Figure,1:,Illustration,of,the,relationship,between,a,microscopic,model,F,ε,,its,macroscopic,limit,F,0,,δ,,,F,0,and,their,discretizations,F,ε,δ,,,where,ε,and,δ,are,parameters,characterising,small,physical,and,numerical,scales,respectively.,The,discretization,scheme,F,ε,δ,is,said,to,be,asymptotic,preserving,if,F,0,δ,is,a,consistent,and,stable,approximation,to,F,0.,Figure,taken,from,[12].,Asymptotic,Preserving,Schemes,In,some,models,of,multiscale,physical,systems,,the,scales,which,the,model,describes,may,be,determined,by,a,parameter.,For,example,the,Vlasov,equation,is,a,model,for,a,plasma,whose,behaviour,depends,on,the,collision,time,(the,typical,time,between,particle,interactions).,When,the,collision,time,is,large,,the,Vlasov,equation,is,a,kinetic,equation,for,the,plasma,,encompassing,microscopic,scales,in,velocity,space.,However,,when,the,collision,time,is,small,,the,particle,distribution,function,does,not,deviate,significantly,from,a,Maxwellian,,and,the,solution,only,contains,macroscopic,velocity,scales.,Discretizations,of,multiscale,systems,are,not,guaranteed,to,correctly,capture,the,behaviour,in,the,macroscopic,limit,,but,numerical,schemes,may,be,derived,to,do,so.,Such,schemes,are,called,asymptotic,preserving,,and,may,be,understood,schematically,as,in,Figure,1,(taken,from,[12]).,Let,F,ε,be,the,multiscale,model,,with,small,scales,characterised,by,the,parameter,ε,,such,that,in,the,δ,be,a,discretization,of,F,ε,,where,δ,charac-,limit,ε,→,0,,we,have,the,macroscopic,model,F,0.,Let,F,ε,terises,small,numerical,scales,,typically,mesh,spacing,or,timestep.,Then,in,the,macroscopic,limit,ε,→,0,,we,will,also,have,a,discretization,of,the,macroscopic,model,,F,0,δ,is,a,consistent,and,stable,approximation,to,F,0,,then,the,discretization,scheme,F,ε,δ,is,said,to,be,asymptotic,preserving.,δ,.,If,F,0,The,value,of,asymptotic,preserving,schemes,is,that,they,allow,the,same,model,to,be,used,to,study,both,microscopic,and,macroscopic,regimes,simply,by,varying,the,parameter,ε.,Without,asymptotic,preserving,schemes,,one,must,either,derive,a,separate,model,for,the,macroscopic,scales,,or,obtain,the,macroscopic,behaviour,by,solving,the,(typically,much,more,expensive),microscopic,system,directly,with,large,ε.1,However,,specific,asymptotic,preserving,schemes,must,be,derived,for,individual,physical,systems.,Moreover,,asymptotic,preserving,schemes,are,typically,stiff,due,to,the,smallness,of,ε,[13],,and,therefore,an,efficient,implementation,of,implicit,terms,is,vital.,1For,the,example,of,the,Vlasov,equation,this,choice,can,be,expressed,as,follows.,One,might,derive,a,separate,model,in,the,collisional,limit,by,taking,velocity,moments,of,the,particle,distribution,function,and,then,providing,a,closure,condition.,This,produces,a,set,of,equations,for,fluid,quantities,,dependent,on,space,but,not,velocity,space.,Alternatively,,one,might,resolve,to,solve,the,Vlasov,equation,directly,,even,though,doing,so,entails,solving,a,six-dimensional,kinetic,system,,rather,than,three-dimensional,fluid,system.,6,Asymptotic,preserving,schemes,were,introduced,in,the,1990s,for,calculations,of,neutron,trans-,port,[14,,15],,and,since,schemes,for,kinetic,systems,relevant,to,Project,NEPTUNE,have,been,developed,(see,[12],for,a,review).,Moreover,,recent,authors,[16,,17],have,developed,high-order,asymptotic,preserving,IMEX,schemes.,These,schemes,take,a,multiderivative,approach,,that,is,,unlike,methods,like,Runge–Kutta,or,multistep,,the,methods,also,use,higher,time,derivatives,of,the,unknowns.,While,this,makes,the,scheme,more,complicated,,it,also,means,the,scheme,is,more,local,,incorporating,more,information,from,each,timestep,,and,therefore,has,the,potential,to,reduce,the,memory,overhead,on,HPC,systems,[16].,2.1.2,Variable,Stepsize,,Variable,Order,Variable,Stepsize,,Variable,Order,(VSVO),schemes,are,a,family,of,schemes,motivated,by,the,idea,of,applying,a,“time,filter”,to,the,approximate,solution,as,a,means,of,increasing,the,order-,of-accuracy,of,the,original,scheme.,Time,filters,are,themselves,motivated,by,an,ad,hoc,tech-,nique,used,in,geophysical,fluid,dynamics,to,reduce,oscillations,in,leapfrog,time,integration.,In,the,Robert–Asselin,filter,[18,,19],,the,approximate,solution,produced,by,the,leapfrog,scheme,,u∗,i,,,is,“filtered”,as,a,post-processing,step,to,produce,the,final,approximation,ui,=,u∗,i,+,ν,2,(cid:0)u∗,i+1,−,2ui,+,ui−1,(cid:1),,,(4),where,ν,∼,0.1,is,a,free,parameter.,Guzel,&,Layton,[20],observed,that,,when,using,this,filter,as,an,on-the-fly,corrector,to,a,low,order,scheme,,judicious,choice,of,ν,could,increase,the,order-of-accuracy,of,the,overall,scheme.,For,example,,first,order,implicit,Euler,,i+1,=,ui,+,∆t,R,(cid:0)u∗,u∗,i+1,,t(cid:1),,,becomes,second-order,accurate,when,filtered,with,ui+1,=,u∗,i+1,−,ν,2,(cid:0)u∗,i+1,−,2ui,+,ui−1,(cid:1),,,(5),(6),with,the,specific,value,ν,=,2/3,(note,also,the,change,in,sign,of,ν,and,shift,in,index,relative,to,(4)).,This,particular,method,is,in,fact,second-order,backwards,differentiation,(BDF2),,though,in,general,the,procedure,does,not,produce,BDF,schemes,[21].,Guzel,&,Layton,[20],also,generalized,this,approach,for,non-uniform,timestep,,while,DeCaria,[21],extended,it,to,produce,VSVO,methods.,For,example,,VSVO-12,adaptively,switches,between,using,the,first,order,approximation,ui+1,and,the,second,order,approximation,u∗,i+1,based,on,error,estimates,,and,if,neither,meets,a,given,tolerance,adapts,the,timestep.,These,methods,are,attractive,because,they,offer,a,means,to,increase,the,order-of-accuracy,of,existing,implementations,with,very,little,coding,overhead.,For,example,,given,an,implementation,of,implicit,Euler,(5),,a,second-order,scheme,could,be,obtained,simply,by,implementing,the,time,filter,(6),and,by,storing,one,additional,time,slice,ui−1.,However,,the,filter,required,is,specific,to,the,integration,method,used,,so,that,each,different,method,requires,careful,derivation,of,a,different,filter.,7,2.2,Elliptic,systems,NEPTUNE-relevant,systems,will,require,the,solution,of,elliptic,equations,in,three,dimensions,and,in,complex,geometries,that,exhibit,strong,anisotropy,in,the,direction,parallel,to,the,magnetic,field,line.,While,multigrid’s,scalability,makes,it,typically,the,favoured,method,for,solving,such,problems,on,distributed,systems,,special,care,is,needed,when,treating,multiscale,systems,or,strongly,irregular,meshes.,Moreover,,iterative,methods,(such,as,the,smoothing,step,in,multigrid),may,require,the,underlying,equations,to,be,modified,in,order,to,deal,with,strong,anisotropy.,In,this,section,we,discuss,the,solution,of,elliptic,systems,particularly,in,regard,to,two,approaches:,asymptotic,preserving,schemes,that,address,the,strong,anisotropy,,and,nested,solvers,that,ad-,dress,the,complex,geometries.,2.2.1,Asymptotic,Preserving,schemes,Asymptotic,preserving,schemes,are,also,relevant,in,the,spatial,component,of,NEPTUNE-relevant,problems,where,the,strong,magnetic,field,means,that,the,system,to,be,inverted,is,highly,anisotropic.,A,typical,system,like,that,studied,by,[22],may,be,written,as,−,1,ε,∆(cid:107)uε,−,∆⊥uε,=,f,uε,=,0,on,Ω,,on,∂Ω,,(7),(8),where,∆(cid:107),and,∆⊥,respectively,denote,the,parts,of,the,Laplacian,parallel,and,perpendicular,to,the,strong,magnetic,field.,The,limiting,system,as,ε,→,0,is,−∆(cid:107)u0,=,0,u0,=,0,on,Ω,,on,∂Ω,,(9),(10),which,is,degenerate,,admitting,an,infinite,number,of,solutions.,Usual,discretization,methods,are,ill-conditioned,for,small,ε,,limiting,anisotropies,that,can,be,studied.,Instead,asymptotic,preserving,discretizations,must,be,derived;,this,is,the,subject,of,ongoing,work,[22,,23,,24,,25].,The,two-field,approach,proposed,in,[22],eliminates,the,stiff,terms,by,introducing,the,auxiliary,field,q,,defined,by,ε∆(cid:107)q,=,∆(cid:107)u.,With,this,,the,strongly,anisotropic,system,(7),may,be,split,into,two,mildly,anisotropic,systems,−(∆(cid:107),+,ε0∆⊥)u,=,ε0f,+,(ε0,−,ε)∆(cid:107)q,u,=,0,−(∆(cid:107),+,ε0∆⊥)q,=,f,+,∆⊥(u,−,ε0q),q,=,0,on,Ω,on,∂Ω,on,Ω,on,∂Ω,(11),(12),where,ε0,is,a,free,parameter,which,should,be,chosen,to,satisfy,ε,(cid:28),ε0,(cid:28),1.,This,system,is,amenable,to,solution,using,an,iterative,method.,Thus,strongly,anisotropic,problems,become,tractable,,but,at,the,cost,of,solving,for,an,additional,field.,Still,,this,extra,cost,is,ameliorated,by,the,fact,that,the,operator,to,be,inverted,is,the,same,in,(11),and,(12).,Moreover,,the,system,is,8,reasonably,well-conditioned,,with,a,condition,number,that,scales,as,1/(ε0h2),,improving,on,the,1/h4,scaling,of,earlier,methods.,The,freedom,to,choose,ε0,also,alleviates,the,problem,of,“locking”,,the,phenomenon,where,the,accuracy,of,a,numerical,approximation,deteriorates,as,a,parameter,approaches,a,limiting,value,(here,ε,→,0),[26].,Finally,,the,method,does,not,place,any,constraint,on,the,discretization,of,the,spatial,operators.,This,means,it,can,be,used,in,arbitrary,magnetic,fields,,and,also,permits,the,method’s,use,with,closed,magnetic,field,lines.,This,is,in,contrast,to,approaches,based,on,field,line,integration,[24,,25],which,typically,require,analytic,expressions,for,the,magnetic,field.,2.2.2,Nested,solvers,Some,mathematical,descriptions,of,physical,systems,possess,a,natural,hierarchy,or,structure,that,may,be,exploited,by,the,numerical,scheme,or,its,implementation,on,HPC,systems.,For,example,,finite,element,methods,contain,matrix,systems,to,be,solved,both,within,and,between,the,basis,elements.,The,system,within,an,element,is,typically,dense,,while,that,coupling,elements,is,typically,sparse.,Such,a,situation,invites,the,use,of,different,,nested,numerical,methods,for,the,different,systems.,Moreover,,modern,HPC,systems,are,typically,hierarchical,in,their,architecture,,with,multiple,processors,grouped,together,in,nodes,or,NUMA,regions.,This,suggests,that,the,numerical,schemes,should,be,implemented,respecting,the,structure,of,the,HPC,architecture.,For,example,,the,communication,cost,of,a,finite,element,method,could,be,minimized,by,implementing,the,dense,inner,problem,locally,to,a,core,(or,,with,a,shared,memory,paradigm,,locally,to,a,NUMA,region),,while,the,sparser,outer,problem,would,be,allowed,to,span,cores,(or,NUMA,regions).,Such,an,approach,could,also,be,applied,problems,where,unstructured,meshes,lead,to,different,spatial,regions,with,different,stiffness.,One,path,to,avoiding,unfavourable,mesh,structures,with,multigrid,would,be,to,confine,difficult,mesh,regions,to,a,single,inner,problem,and,use,a,non-,multigrid,solver,for,that,subdomain.,While,there,is,a,developed,literature,on,nested,solvers,(e.g.,[27,,28,,29]),,we,are,not,aware,of,algorithms,which,subdivide,the,domain,between,solvers,in,this,fashion.,3,Summary,This,report,describes,recent,advances,in,algorithms,for,solving,hyperbolic,and,elliptic,systems.,Of,particular,interest,are,Implicit-Explicit,(IMEX),schemes,,which,allow,for,the,simultaneous,advance,of,fast,and,slow,timescales,typically,found,in,multiphysics,systems,,and,asymptotic,preserving,schemes,,which,can,efficiently,treat,the,extreme,anisotropy,caused,by,the,tokamak’s,strong,mag-,netic,field.,9,Acknowledgement,The,support,of,the,UK,Meteorological,Office,and,Strategic,Priorities,Fund,is,acknowledged.,References,[1],S.,Thorne.,Priority,Equations,and,Test,Cases.,Technical,Report,204735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