DELPHI and Vlasov solvers used at CERN D.Amorim, S.Antipov, N.Biancacci, E.Métral, N.Mounet, B.Salvant ABP-Computing Working Group 16 March 2017 2017-03-16 ABP-CWG

DELPHI Discrete Expansion over Laguerre Polynomials and HeadtaIl modes Code written by N.Mounet Semi-analytic Vlasov solver: computes the complex coherent frequency shifts caused by a beam coupling impedance and/or a damper Vlasov equation > Perturbation formation > Sacherer Integral Sacherer Integral + Laguerre Polynomials > Eigensystem Eigenvalues give the modes frequency shifts and the associated growth rates Eigenvectors allow to reconstruct the signal which could be observed at the pick-ups Convergence of the eigenvalues is obtained by automatically increasing the number of azimuthal and radial modes computed, thus increasing the impedance/damper matrix size Can treat impedance, damper, chromaticity, single or multi-bunch Functions to account for Landau damping are present and currently under review High impact on CERN studies: assess the stability thresholds for different machine configuration, input for the calculation of the octupole current threshold Impact on CERN studies: next step after impedance model, Assess stability in the machines and impact of different scenarios Compute octupole current needed to stabilize Complementary to tracking simulations (PyHEADTAIL) 2017-03-16 ABP-CWG

DELPHI Code initially implemented in C/C++ Compilation provides a stand-alone executable and a functions library The stand-alone executable is now depreciated Python functions are routinely used. They use the C++ functions library compiled beforehand The code runs on Linux platforms. The C++ core requires three libraries (already installed on LXPLUS): LAPACK (http://www.netlib.org/lapack) BLAS (http://www.netlib.org/blas) GSL (http://www.gnu.org/software/gsl/) No parallelization strategy implemented Simulations can be run on LSF Possibility to perform scans in chromaticity, bunch intensity, number of bunches, damper gain… Each simulation can be run in parallel on LSF Used with SPS/LHC/HL-LHC/FCC-hh/FCC-ee impedance models An individual simulation can last from a few seconds to several minutes, depending on the convergence criterion and the number of points in the impedance file Number of users: in the order of 10 people 2017-03-16 ABP-CWG

DELPHI No performance limit reached so far: current hardware infrastructure matches our needs Open-source code, maintained on CERN IRIS repository, hosted on CERN’s GitLab. A mirrored repository is also available for external users without a CERN account https://gitlab.cern.ch/IRIS/DELPHI https://gitlab.com/IRIS_mirror/DELPHI_mirror Documentation on the code usage is available in the repository For the theoretical development, see N.Mounet presentation https://espace.cern.ch/be-dep/ABP/HSC/Meetings/DELPHI-expanded_Part2.pdf No further developments on the C++ core Foreseen future evolutions for the Python code: Check the functions associated to Landau damping Include new parameters such as Q’’, linear coupling, detuning, space-charge (will need some work on the theory side) Implement job submission to HTCondor to address LSF degrading performance Rewrite some features to be more Object Oriented Code available in GitLab + Mirror Add some plots (benchmarks) to visualize the output ? Future: migration to HTCondor ? Documentation available inside repository License 2017-03-16 ABP-CWG

DELPHI N.Biancacci L.Carver et al. Octupole current threshold with the LHC impedance model: Single bunch, fixed intensity Scan in chromaticity and damper gain The eigenvalues given by DELPHI have been postprocessed to give the current threshold TMCI threshold with the LHC impedance model: Single bunch, zero chromaticity, no damper Scan in bunch intensity Real part of the eigenvalues on the upper plot Imaginary part on the lower plot TMCI threshold Octupole current vs chroma Effect of damper 2017-03-16 ABP-CWG

MOSES MOde-coupling Single bunch instability in an Electron Storage ring Code written by Y.H.Chin Semi-analytic Vlasov solver: computes the complex coherent frequency shifts caused by a beam coupling impedance Output the eigenvalues, giving the modes frequency shifts and the associated growth rates The user inputs the number of azimuthal and radial modes computed: there is no convergence check on the eigenvalues Can only treat a resonator impedance, with a Gaussian longitudinal distribution Include chromaticity, single bunch only Used for studies with simple impedance models Possible benchmark 2017-03-16 ABP-CWG

MOSES Code written in Fortran A Windows executable is provided, as well as a software to plot the results Source code is also provided The code runs on Windows (tested with Windows 7) No parallelization strategy implemented Possible to perform a scan in bunch intensity. If only a few modes are computed, an individual simulation is almost instantaneous A scan in bunch intensity can last up to several minutes depending on the number of steps No performance limitation Code freely available on Y.H. Chin page, no information on licensing http://abci.kek.jp/moses.htm Documentation on the code usage is provided with the sources The documentation includes some theoretical development 2017-03-16 ABP-CWG

MOSES B.Salvant Scan in bunch intensity performed with MOSES (red) and HEADTAIL (white), for the SPS impedance model (broad-band resonator, 𝑓 𝑟 =1𝐺𝐻𝑧, 𝑅 𝑠 =10𝑀Ω/𝑚, 𝑄=1) 2017-03-16 ABP-CWG

Nested Head-Tail Vlasov Solver Author: A.Burov What’s included: Single-bunch or single bunch + coupled bunch Damper: flat or arbitrary frequency response Beam-beam: several IP’s Landau damping: analytic estimate, small betatron tune spread, no synchrotron tune spread Out of scope: Intra-beam scattering Synchrotron radiation damping Space charge Mathematica notebook Runs on any laptop or desktop, supporting Mathematica v. 10 Reference: https://arxiv.org/ftp/arxiv/papers/1309/1309.0044.pdf 2017-03-16 ABP-CWG

NHT: Physics Solving for eigenvalues and eigenfunctions of transverse modes α and R(r). Air-bag approximation: Unperturbed solution Nested air-bags: equal population A. Chao, Physics of collective beam instabilities, 6.6 Transverse modes 2017-03-16 ABP-CWG

NHT: Simulation procedure Generate nested air-bags Compute the impedance matrix of Z = Z(no modes, chromaticities) Most time-consuming, overnight on a laptop Can be saved to be reused in future studies Solve for the eigenvalues One study scenario takes ~ 1 sec on a laptop Can make a scan through gains and chromaticities Growth rates of individual modes Most unstable mode vs gain and chromaticity 2017-03-16 ABP-CWG

Backup slides 2017-03-16 ABP-CWG

Vlasov equation Phase space coordinates Phase space distribution Transverse Longitudinal Phase space distribution Vlasov equation: Phase space conservation From A.W. Chao 2017-03-16 ABP-CWG

Perturbation formalism Perturbed phase space distribution Unperturbed distribution Perturbation term Transverse distribution Longitudinal distribution Assume that a mode is developing At complex frequency While staying close to the unperturbed distribution Re: mode frequency shift Im: mode growth rate 2017-03-16 ABP-CWG

N.Mounet 2017-03-16 ABP-CWG

Decomposition in Laguerre polynomials Decompose the longitudinal functions Unperturbed longitudinal distribution Perturbed longitudinal distribution Orthogonal polynomials 2017-03-16 ABP-CWG

Eigenvalues problem Eigenvalues problem Vlasov equation: how the particle distribution evolves + Perturbation formalism: how the disturbance is treated + Laguerre decomposition : how to treat the problem Eigenvalues problem Eigenvalues Impedance and damper matrix Eigenvectors l,n: azimuthal and radial mode numbers time time From A.W. Chao 2017-03-16 ABP-CWG