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• W2560159770 abstract "Numerical noise in PIC codes produces artifacts, which affects long term beam simulations needed for future heavyion synchrotrons as the SIS100. We present here a detailed study on the effect of numerical noise occurring in multiparticle tracking codes using a PIC scheme. The influence of the granularity of particle distributions and the fineness of the meshes of Poisson solvers on the particle dynamics is studied. These results are used to discuss the effect of the numerical noise in long term space charge benchmarking studies. PARTICLE TRACKINGWITH SPACE CHARGE Multi-particle tracking codes for circular machines make use of so-called transfer mapsM to track a particle from its initial 4-dimensional phase-space position ~xi (s0) to a final position ~x f (s), i.e. ~x f =   x f xf y f yf   =M   xi xi yi y′ i   =M ~ xi , (1) where s (or resp. s0) is the (initial) longitudinal position of the particle. Space charge forces can be implemented in tracking codes by using non-linear kicks   x f xf y f yf   =   xi xi + ∆sFSC,x (xi , yi , s0) yi y′ i + ∆sFSC,y (xi , yi , s0)   , where (FSC,x ,FSC,y ) is the transverse space charge force and ∆s is the integration length. The transverse self-field of the beam (ESC,x ,ESC,y ) (x, y, z), that determines the selfconsistent space charge forces, is computed for long-term simulation most effectively by using a 2.5-dimensional PIC scheme. This refers to a longitudinal slicing of the beam. In each slice the particles are projected onto a transverse plane, where the self-field is calculated by a 2D Poisson solver. Then, the fields in between planes are calculated by a linear interpolation. An approximation of the 3-dimensional field is thus retrieved by a set of Poisson equations in the transverse plane. To achieve a higher computational efficiency, the Poisson equation is only solved on a finite set of N transverse grid points and the particle distribution is approximated by M macroparticles. Since the number of grid points N is finite and the number of macro-particles M is much smaller than the number of physical particles, numerical noise is induced. A theory to understand the effects of numerical noise was first proposed in 2000 by J. Struckmeier [1], and extended recently by I. Hofmann and O.Boine-Frankenheim ( [2] and this proceedings). Their approach is to use a Fokker-Planck description to predict the evolution of the particle distribution in the presence of numerical noise. A complementary approach is presented in this proceeding, in which instead the effect of numerical noise on the single particle dynamics is studied to estimate emittance growth rates. FLUCTUATIONS IN FIELD CALCULATIONS We consider as a transverse particle distribution f (x, y) the Gaussian distribution, i.e. f (x, y) = 1 √ 2πσ e− x2+y2 2σ2 , (2) where σ gives the standard deviation in x and y for circular beams. The associated space charge field is given by ESC,r (x, y) = E0 √ x2 + y2 ( 1 − e− x2+y2 2σ2 ) , (3) where E0 is proportional to the field gradient in the vicinity of x = y = 0. To proceed in the study, we adopt the random start technique (opposite to the quiet start). For the random start, the positions of M macro-particles are randomly initialized according to f (x, y). The resulting self-field is not only determined by f (x, y), but varies according to the random initialization of macro-particles. A first study is done to estimate the effect of a granular approximation of the particle distribution and the fineness of the grid. To this purpose, a random start initialization and a calculation of the resulting electric field is repeated several times to study possible fluctuation for different setups of N and M at different spacial positions (x, y). With this technique, the standard deviation for the random start initialization for the electric field is found to be δESC (x, y) = δ0 ξ (x, y) √√ N M e− x2+y2 4σ , (4) where the constant δ0 can be retrieved numerically and depends on the perveance of the beam. The factor ξ (x, y) describes the effect due to the bilinear interpolation, that is Proceedings of IPAC2014, Dresden, Germany THPRO058 05 Beam Dynamics and Electromagnetic Fields D02 Non-linear Dynamics Resonances, Tracking, Higher Order ISBN 978-3-95450-132-8 3005 C op yr ig ht © 20 14 C C -B Y3. 0 an d by th e re sp ec tiv e au th or s x position [a.u.] y p o s it io n [ a .u .] δ E S C (x ,y ) [a .u .] Figure 1: Standard deviation δESC (x, y) of the transverse electric field in arbitrary units. An exponential dependence of the standard deviation on the distance to the center is found. It is superimposed by the PIC grid texture (due to the bilinear interpolation in between grid points). used to compute the electric field in between grid points. For illustration, a sample snapshot is given in Fig. 1. The scaling law Eq. (4) is derived for circular beams for random start initializations. It is valid, as long as the beam shape is reasonably resolved, which is the case for N ≥ 16 grid points within [−2σ,2σ] of the particle distribution f (x, y). The field fluctuations in the longitudinal plane are derived for arbitrary transverse fluctuations. Assume, we know the transverse fluctuations of two slices situated at s = si and s = si+1, given by δESC (x, y, si ) and δESC (x, y, si+1). Then, due to the Gaussian law of error propagation, the field fluctuation at the positon s is given by δESC (x, y, s) = √ dsδESC (x, y, si ) + ds+1δESC (x, y, si+1) (5) where ds = (si+1− s)/(si+1− si ) and ds+1 = (s− si )/(si+1− si ) for si < s < si+1. This scaling is confirmed by sytematic simulation studies, carried out in the same manner as for the transverse plane. The results for a sample bunched beam are presented for x = y = 0 in Fig. 2. As expected, in between two transverse planes the standard deviation declines due to the linear interpolation. TRACKING OF BEAMS AFFECTED BY NOISE For the study of the effect of numerical noise on the single particle beam dynamics, it is most important to study the correlations of fluctuations of the electric field while tracking. Therefore a spectral analysis of the electric field for a choosen setup (beam and machine parameters) has to be done. A flat spectrum refers to random (white) numerical noise, while peaks in the spectrum (f.e. due to resonances) refer to directed (coloured) noise. If the fluctuations are solely random, the effect of noise can be understood in the s S ta n d a rd d e v ia ti o n σ E" @default.
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• W2560159770 date "2014-07-01" @default.
• W2560159770 modified "2023-09-23" @default.
• W2560159770 title "Study of the “Particle-in-Cell” Induced Noise on High Intensity Beams" @default.
• W2560159770 doi "https://doi.org/10.18429/jacow-ipac2014-thpro058" @default.
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