Boriscorrection:exactandapproximate

K.G.Miller1,P.Q.Elias2,R.A.Fonseca3,B.J.Winjum1,F.S.Tsung1,V.K.Decyk1,W.B.Mori1

1UCLA,2ONERA,3IST

KyleMiller,OSIRISWorkshop2017,2

Outline• OSIRIScurrentdeposits

1. Charge-conservingcurrentdeposit• StandardinOSIRIS,satisfiescontinuityequation

2. Directcurrentdeposit• Useparticlepositions/shapefunctions• Lessnoisy,doesnotsatisfycontinuityequation

• Divergencecleaning(Boriscorrection)1. Iterativemultigridsolver2. Exactspectralsolver3. Exactspectralandfinite-differencesolver

KyleMiller,OSIRISWorkshop2017,3

Particle push

𝑑𝒙#𝑑𝑡 = 𝒗#

𝑑𝒑#𝑑𝑡 = 𝑞 𝑬# + 𝒗#×𝑩#

Charge/current deposition

Calculate 𝜌./ and 𝒋./from the 𝒙# and 𝒑#.

Field solve

𝜕𝑩./𝜕𝑡 = −𝛻×𝑬./

𝜕𝑬./𝜕𝑡 = 𝑐5𝛻×𝑩./ − 𝜇7𝑐5𝒋./

Field interpolation

Interpolate 𝑬./ and 𝑩./ onto the 𝒙#.

Δ𝑡

𝑘: continuous (particles)𝑖𝑗: gridded (fields)

PICmethod

KyleMiller,OSIRISWorkshop2017,4

Gauss’slaw• Gauss’slawnotexplicitlysolvedinOSIRIS• Ifinitial𝜌 and𝑬 satisfyGauss’slaw,thencontinuityequationguaranteesGauss’slawatfuturetimes

𝜕𝜌./𝜕𝑡 + 𝛻 ⋅ 𝒋./ ≠ 0

⇓𝛻 ⋅ 𝑬./ ≠

𝜌./𝜖7

KyleMiller,OSIRISWorkshop2017,5

Currentdeposits• Charge-conservingcurrentdepositinherentlysatisfiescontinuityequation(andthusGauss’slaw)• J.VillasenorandO.Buneman,Comput.Phys.Commun.69,306-316(1992).

• Directdeposit—usesparticleshapes

Default

Averageparticleshapes

KyleMiller,OSIRISWorkshop2017,6

Boriscorrection• SolvePoisson’sequationtocorrectelectricfield

𝛻5𝜓 = 𝛻 ⋅ 𝑬 −𝜌𝜖7≡ 𝜌C

• where𝜌C istheresidualchargeerror.Then

𝑬C = 𝑬 − 𝛻𝜓

𝛻 ⋅ 𝑬C =𝜌𝜖7

KyleMiller,OSIRISWorkshop2017,7

Iterativemultigridsolver:V-cycle

• Solvethematrixequation𝐴𝒙 = 𝒃 i. e. , 𝛻5𝜓 = 𝜌C ,

• Startwithaguessof𝜓 = 0 𝒙 = 0• Computeresidual 𝒓 = 𝐴𝒙 − 𝒃• Restrict ontocoarsergrid 𝒓 → 𝒓M• Smooth solutionusingpoint-Jacobi 𝐴M𝒆M = 𝒓M• Direct matrixsolveofsmallproblem 𝐴O𝒆O = 𝒓O• Interpolate ontofinergrid 𝒆O → 𝒆OPM∗

• Correct thesolution 𝒆OPM = 𝒆OPM − 𝒆OPM∗ , 𝒙 = 𝒙 − 𝒆7∗

• Iterate

KyleMiller,OSIRISWorkshop2017,8

Iterativemultigridsolver

KyleMiller,OSIRISWorkshop2017,9

Exactspectralsolver• AssumingFourierbehavior,ODEà algebraicequation

𝛻5𝜓 = 𝜌C → −𝑘5𝜓 = 𝜌C

• However,finite-differencederivativeisapproximate:

ℱ𝜓 𝑥 + ℎ − 2𝜓 𝑥 + 𝜓 𝑥 − ℎ

ℎ5 = −4ℎ5 sin

5 𝜋𝑘𝑁

• Onedimension(𝑥)mustbeentirelylocal

KyleMiller,OSIRISWorkshop2017,10

Hybridspectral/finite-difference

• 3Dsolver• Spectralin𝑥 and𝑦• Finite-differencein𝑧

𝜓 𝑧 + ℎ − 2 + 𝑘]5 + 𝑘^5 𝜓 𝑧 + 𝜓 𝑧 − ℎℎ5 = 𝜌 𝑧

• Shouldbemoreconsistentthanfullyspectral

KyleMiller,OSIRISWorkshop2017,11

Speedup• Comparedtopoint-Jacobi,multigridismuchfaster• Spectralmethodsshouldbeevenfaster

KyleMiller,OSIRISWorkshop2017,12

Test:stimulatedRamanscattering

KyleMiller,OSIRISWorkshop2017,13

Test:stimulatedRamanscattering

KyleMiller,OSIRISWorkshop2017,14

Usedirectcurrentdeposit• #defineDIRECT_DEPOSIT

• inos-config.h• #defineAVG_SHAPES

• inos-config.h

KyleMiller,OSIRISWorkshop2017,15

UseBoriscorrection• if_boris =.true.,• boris_solver =’multigrid’,‘fft’,‘hybrid’,or’point-Jacobi’,

• inel_mag_fld ofinputfile

• Otheroptions:• boris_tol =1.e-14,• boris_n =4,

KyleMiller,OSIRISWorkshop2017,16

Importantfiles• Multigrid:

• pmg/*(esp m_parallel_multigrid_solver.f90)• os-emf-es-solver-pmg.f03• os-emf-marder.f90• os-emf-math.f03• os-vdf-math.f90

• ExactBoris• boris/*

• Branchparallel-multigridonGithub

KyleMiller,OSIRISWorkshop2017,17

Conclusions• Implementations:

• Directcurrentdeposit• Usingaveragepositionoraverageparticleshapes

• Boriscorrectionforusewithdirectdeposit• Usingiterativemultigrid,exactspectral,orpoint-Jacobi

• Uses• Lessnoisycurrentdeposit• Solvermaybeusedforotherinitializationpurposes

• Futurework• Finishimplementation,runmorephysicstestcases