Accuracy and Efficiency of 2d and 3d Fast Poisson's Solvers for Space Charge Field Calculation of Intense Beam

Design of particle accelerators with intense beams requires careful control of space charge problem. To obtain accurate treatment of the problem, solution of the Poisson's equation for electrostatic potential created by an arbitrary space charge distribution of the beam is required. Numerical routines developed for 2D and 3D space charge field calculation of high current beam are examined. Two numerical techniques are used: (i) finite-difference method, combining Fourier expansion and Gauss elimination and (ii) spectral method, utilizing Fourier expansion of electrostatic potential. Accuracy and time consuming for calculation of test problem are compared. 1 NUMERICAL ERRORS AND CONSERVATION LAW Particle-in-cell code BEAMPATH has been developed for study of wide range of problems with intense beams [1]. Space charge field of the beam at every time step of particle trajectories integration is calculated from the Poisson's equation ∆U = Q, where U is a space charge potential and Q is a space charge density of the beam. Calculation includes distribution of space charge of macroparticles among the grid nodes, solution of Poisson's equation on a grid, and differentiation of grid potential function to find components of electrostatic field of the beam. Important point in numerical simulations is a balance between accuracy and required resources of computer to get an efficient solution of the problem. Good accuracy in space charge problem is obtained, if number of macroparticles per cell is large enough and if a mesh size is much smaller, than the beam size. Meanwhile, even if these conditions are fulfilled, the following errors are unavoidable: (i) errors due to discrete charge representation in particle method, (ii) errors due to substitution of exact derivatives of Poisson's equation by approximation formulas, (iii) errors of differentiation of potential function to obtain values of electric field components, and (iiii) computer round-off errors. To control errors of calculations, the following parameter can be used