Orthogonal Basis Function Approximation of Particle Distribution in Numerical Simulations of Beams

Numerical simulations of charged particle beams require an approximation to the particle distribution being simulated. We present a mathematical formalism for approximating two-dimensional (2D) particle distribution using a basis composed of scaled and translated Gauss-Hermite (STGH) functions. It is computationally efficient, because it only requires the values of the particle distribution at (N +1)× (M +1) nodes, where N andM are the highest basis function retained in the expansion in each coordinate. After outlining the mathematical formalism for the expansion, we compare it to the cosine expansion which is currently used in a code simulating coherent synchrotron radiation. The advantages of the STGH approximation over the cosine expansion are demonstrated by comparing the computational costs and execution times, as well as manifesting that unphysical fluctuations in the tail of the approximation which plague cosine expansion are not a factor in the new method. All these features make the STGH approximation valuable for N -body codes simulating the dynamics of multiparticle systems.