Start-to-end simulations are needed for sensitivity stud- ies and online analysis of experimental data of the Plasma Wakefield Acceleration experiment COMB at SPARC_LAB facility, Frascati (Italy). Ad hoc tools are needed for the plasma section modeling. Particle in cell codes are the most widely used tools for this purpose, but they suffer from the considerable amount of computational resources they re- quire. We seek for a simple, portable, quick-to-run approach. For this purpose we introduce a time-explicit cylindrical hybrid fluid-kinetic code: Architect. The beam particles are treated with PIC-like kinetic approach, while the plasma wake is treated as a fluid. Since the number of computational particles used by the hybrid model is significantly reduced with respect of full PIC codes with the same number of di- mensions, the time required for a simulation is reduced as well

Chiadroni, E.

Ferrario, M.

MAROCCHINO, ALBERTO

MASSIMO, FRANCESCO

MOSTACCI, Andrea

PALUMBO, Luigi

Rossi, A. R.

Archivio della ricerca- Università di Roma La Sapienza

EFFICIENT PLASMAWAKEFIELD ACCELERATION SIMULATIONSVIA KINETIC-HYDRO CODE ARCHITECTF. Massimo1,2, A. Marocchino1,2, E. Chiadroni3, M. Ferrario3,A. Mostacci1,2, L. Palumbo1,2, A.R. Rossi41Dipartimento SBAI, “Sapienza” University of Rome, Rome, Italy2INFN-ROMA 1, Rome, Italy3INFN, Laboratori Nazionali di Frascati, Frascati, Italy4INFN, Milano, ItalyAbstractStart-to-end simulations are needed for sensitivity stud-ies and online analysis of experimental data of the PlasmaWakefield Acceleration experiment COMB at SPARC_LABfacility, Frascati (Italy). Ad hoc tools are needed for theplasma section modeling. Particle in cell codes are the mostwidely used tools for this purpose, but they suffer from theconsiderable amount of computational resources they re-quire. We seek for a simple, portable, quick-to-run approach.For this purpose we introduce a time-explicit cylindricalhybrid fluid-kinetic code: Architect. The beam particlesare treated with PIC-like kinetic approach, while the plasmawake is treated as a fluid. Since the number of computationalparticles used by the hybrid model is significantly reducedwith respect of full PIC codes with the same number of di-mensions, the time required for a simulation is reduced aswell.INTRODUCTIONThe use of accelerating electromagnetic fields in the wakeof a perturbation of the neutrality of a plasma channel ap-pears as one of the most promising ways to overcome theconventional accelerators’ limits [1]. In particular in thePlasma Wakefield Acceleration (PWFA) scheme high en-ergy driving particle beams created in a conventional beamline are injected into a plasma channel, to excite wakefieldsused to accelerate a witness bunch injected in the properphase of the wake [2]. The most widely used technique tostudy the underlying physics and to guide the experimentalefforts in PWFA is the Particle in Cell method [3]. Theamount of computational resources (i.e. number of coresand simulation time) required for 3D full PIC simulationsof typical experimental setups of plasma acceleration is stilltoo high to use PIC codes for systematic scans. For thisreason much interest has been devoted to reduced models,to perform faster simulations. In particular the codes usingthe quasi-static (QSA) approximation [4–6] proved to signif-icantly reduce the computational time required to simulateplasma acceleration when the characteristic timescales ofthe beam evolution are significantly greater than the charac-teristic timescales of the plasma evolution.An alternative efficient technique is the hybrid kinetic-fluid approach, which treats the relativistic beams kineticallyand the background plasma electrons as a fluid. The reducednumber of computational particles needed in this techniquedrastically reduces the simulation time with respect to a PICcode of the same dimensions. This approach has been provensuccessful for the Laser Wakefield Acceleration scheme (e.g.INF&RNO [7]) and for the PWFA with QSA (LCODE [8]).We present the 2D time-explicit hybrid kinetic-fluid codeArchitect, developed to aid the preliminary parameter skim-ming for the PWFA experiments planned at at SPARC_LABfacility in Frascati, Italy [9]. The time-explicit formulationof the code allows to initialize the bunch electromagneticfields in vacuum and to investigate the transient phase of thebeam entrance in the plasma channel.IMPLEMENTATION OF A HYBRIDMODEL FOR PWFAArchitect simulates PWFA treating the relativistic elec-tron beams as an ensemble of macroparticles as in a PICcode and the background plasma electrons as a relativisticcold fluid. The plasma ions are treated as a uniform immo-bile background. The current densities of the two speciesare projected on the grid, using PIC techniques [3] and com-puting the fluid current on the grid respectively. The sum ofthe current densities acts as a source for the evolution equa-tions of the electromagnetic fields, i.e. Faraday’s Law andAmpere-Maxwell equation. The macroparticle positionsand momenta in the 6D space evolve using the updated elec-tromagnetic fields as source terms. The plasma backgroundfluid density and momentum on the grid evolve using theupdated electromagnetic fields as source terms.For each particle of the kinetic bunch(es) we identifya position, xparticle, a velocity, βparticlec and a relativisticmomentum, pparticle = meβparticlec/√1 − |βparticle |2 (me isthe electron mass).Thus, the the macroparticle positions and momenta evolvefollowing the equations of motion:dtxparticle = βparticlecdtpparticle = q(E + cβparticle × B)(1)where q is the electron charge, E is the electric field, B themagnetic field, c the speed of light.Proceedings of ICAP2015, Shanghai, China THDWC1D-2 Plasma, Laser, Dielectric and Other Acceleration SchemesISBN 978-3-95450-136-6175 Copyright©2015CC-BY-3.0andbytherespectiveauthorsThe background plasma electron density ne and momen-tum pe evolve according to [10]:∂tne + ∇ · (βec ne) = 0∂tpe + cβe · ∇pe = q(E + cβe × B),βe =pemec√1 + |pe/mec|2. (2)The first equation is the mass conservation equation; thesecond equation is the momentum conservation equation.The electromagnetic fields, induced by both the beamparticles and the fluid background, evolve according to Fara-day’s Law and Ampere-Maxwell’s equation:∇ × E + ∂tB = 0∇ × B − c−2∂tE = qµ0c (neβe + nbβb) , (3)where βbc the velocity for the electron bunch and nb thebunch density.The fluid Eqs. (2) and the evolution equations for the elec-tromagnetic fields, i.e. Eqs. (3) are integrated on a cartesianr − z grid assuming cylindrical symmetry, i.e. the partialderivatives along the poloidal direction are all zero. To savememory and reduce the simulation time, the fluid and electro-magnetic quantities are only integrated in a window aroundthe beam (depicted in Fig. 1), which moves with the beamcenter of mass. The direction of propagation for the beamis z. Cylindrically symmetric boundary conditions are as-sumed on the window boundary corresponding to the z axis;free flux conditions are assumed on the other window edges.The code loop for each timestep is composed as follows; theFigure 1: Architect r − z grid. In blue a gaussian electronbunch, which moves in the z direction.beam particles’ position xparticle and momentum pparticle areused to project the beam current Jb = nbβbc by means ofPIC techniques [3]. The fluid current Je = neβec is thencomputed on the grid. The sum of the two currents is thenused as a source term to integrate Eqs. (3) through the FiniteDifference Time Domain (FDTD) method [3]. The updatedfields are then used as source terms to integrate the equationsof motion for every macroparticle, i.e. Eqs. (3), with a sec-ond order leap-frog with Boris rotation around the magneticfield [3]. The fields acting on each macroparticle are extrapo-lated from the grid through PIC techniques [3]. The updatedfields are used also as source terms to integrate the fluidEquations (3). Such integration is performed through theFlux Corrected Transport (FCT) scheme [11,12], a simpli-fied shock-capturing scheme to catch the shock-like featuresof the ion bubble closing-up region. Once the fluid quantitiesand the macroparticle positions and momenta are updatedthe loop of the code timestep, summarized in in Fig. 2, isiterated. The self-consistent simulation of the beam-plasmaIntegrate electromagnetic eld equations  on the grid (FDTD)J         E, BCompute force acting on each particleIntegrate macroparticles equations of motionxparticle , βparticleProject beam current density on the grid xparticle , βparticle        JbSum background and beamcurrent densities on the grid J = Je+ JbCompute force acting on each !uid elementIntegrate !uid equationson the grid (FCT)ne , βeCompute background !uid current density on the grid ne , βe        JeFigure 2: Architect Loop. On the left branch are the stepswhich involve the fluid quantities; on the right branch arethe steps involving the beam macroparticles, identical to thesteps of a standard PIC code.interaction requires the correct initialization of the beamelectromagnetic field in vacuum before it enters the plasmachannel, which is initially at rest. Typical COMB simula-tions have an initial beam energy spread of 0.1%, thus allthe beam particles move initially with a nearly equal velocity.The electromagnetic fields induced by the electron beam canthus be computed solving Poisson’s equation in the beamrest frame and transforming the fields back in the laboratoryframe. Poisson’s equation is solved in Architect throughfinite differences, and using a Successive Over Relaxation(SOR) method to accurately solve the Poisson associatedproblem. Appropriate optimizations reduces the requiredcomputation time down to seconds even with very refinedmeshes.RUNNING TIMETo highlight the speed of the hybrid approach, we re-port the time required for a 1 cm-simulation of an electronbeam injected in a uniform plasma channel of density n0 =106 cm−3. The initial beam parameters, chosen from a real-istic SPARC_LAB scenario, are: charge Q = 113 pC, rms-lengh σz = 50 µm, rms-transverse size σx = σy = 8 µm,THDWC1 Proceedings of ICAP2015, Shanghai, ChinaISBN 978-3-95450-136-6176Copyright©2015CC-BY-3.0andbytherespectiveauthorsD-2 Plasma, Laser, Dielectric and Other Acceleration Schemesenergy E0 = 100MeV, energy spread ∆γ/γ = 0.1%, trans-verse normalized emittance εx = εy = 1mm-mrad. Suchbenchmark is in the weakly nonlinear regime [13,14]. Thesimulation parameters are: integration timestep ∆t = 0.88 fs,mesh cell size ∆z = 2 µm in the z direction and ∆z = 0.8 µmin the r direction. The colormap of the beam and backgroundelectron density after 0.1 cm of propagation are shown inFig. 3. The simulation reported in the figure used 50000particles to model the electron beam.z−ct (µm)x (µm)nb/n0  −200 −100 0 100−40−2002040012048121620ne /n0Figure 3: Beam and background electron density for theconsidered benchmark, after a 0.1 cm beam propagation.The time scaling of the described benchmark, varyingthe number of beam particles is reported in Table 1. Thereported running times are referred to a single cpu, since thecode does not need parallelization. From the table we notethat a preliminary run of 1 cm in the plasma channel, withthe same number of beam particles of Fig. 3, needs less thanone hour on a single cpu.Table 1: Benchmark Simulation Time, Referred to an 1 cm-Run on a Single CPUNumber of beam particles Running time (hours/cm)30000 0.2550000 0.3870000 0.51100000 0.71200000 1.37500000 3.341000000 6.63ACKNOWLEDGMENTThis work has been partially funded by the ItalianMinisterof Research in the framework of FIRB- Fondo per gli Inves-timenti della Ricerca di Base, Project no. RBFR12NK5K.REFERENCES[1] I. Blumenfeld et al., Nature 445 741-744 (2007).[2] M. Litos et al., Nature, 515 92-95 (2014).[3] C. K. Birdsall, A. B. Langdon, Plasma Physics Via ComputerSimulation, CRC Press, Boca Raton (2004).[4] P. Mora, T.M. Antonsen, Phys. Rev. E, 53 R2068 (1996).[5] D.H. Whittum, Phys. Plasmas, 4 1154 (1997).[6] P. Sprangle, E. Esarey, A. Ting, Phys. Rev. A, 41 4463 (1990).[7] C. Benedetti, C.B. Schroeder, E. Esarey, C.G.R. Geddes, W.P.Leemans, AAC 2010 Proc., 56 084012 (2010).[8] K. V. Lotov, Phys. Plasmas, 5 785 (1998).[9] M. Ferrario et al., Nucl. Instr. and Meth. B 309, 183 (2013).[10] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Ox-ford University Press (2013).[11] J.P. Boris, D.L. Book, J Comput Phys, 11 38 (1973)[12] S.T. Zalesak, J Comput Phys, 1-28 (1979)[13] J.B. Rosenzweig, N. Barov, M. Thompson, R. Yoder, Phys.Rev. ST Accel. Beams., 7 061302 (2004).[14] P. Londrillo, C. Gatti, M. Ferrario, Nuclear Inst. and Methodsin Physics Research, A, 740 236-241 (2014).Proceedings of ICAP2015, Shanghai, China THDWC1D-2 Plasma, Laser, Dielectric and Other Acceleration SchemesISBN 978-3-95450-136-6177 Copyright©2015CC-BY-3.0andbytherespectiveauthors

Efficient plasma wakefield acceleration simulations via kinetic-hydro code

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Efficient plasma wakefield acceleration simulations via kinetic-hydro code

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