Low-Dispersion Wake Field Calculation Tools ∗ Mikko K¨ arkk¨ ainen † , Erion Gjonaj ‡ , Thomas Lau § , Thomas Weiland ¶ Technische Universitaet Darmstadt, Institut fuer Theorie Electromagnetischer Felder (TEMF) Schlossgartenstrasse 8, 64289 Darmstadt, Germany Abstract Wide-band finite-difference time-domain (FDTD) algo- rithms for wake field simulations in accelerator structures are presented. The schemes are based on enlarged sten- cils enabling explicit updating of the variables over time. An elaborated dispersion analysis verifies that the schemes propagate plane waves in major coordinate axis directions without numerical dispersion. The methods are validated by comparing numerical results with results obtained by other methods. The important issue of calculating wake potentials in general 3D structures is addressed. INTRODUCTION Very short bunches (rms length in the sub-millimeter range) will be used in future linear colliders. To be able to propagate these short bunches numerically through a long structure (several kilometers), billions of mesh cells are needed in the longitudinal direction. To avoid accumula- tion of dispersion errors during the numerical simulation, a numerical scheme should ideally be free of numerical dis- persion at least in the longitudinal direction. Several meth- ods with this feature have been published [1, 2, 3, 4, 5]. In large problems, the simulation time becomes very long, and it is desirable to parallelize the codes. Explicit schemes are usually computationally more effective and easier to paral- lelize than implicit schemes. THE PROBLEM FORMULATION The beam is assumed to propagate rigidly, i.e., the wake fields generated by the bunch do not affect the particle dis- tribution within the bunch. The beam is also assumed to propagate with the velocity of light along the z-axis. An ultra-relativistic bunch (v = c) with a linear charge distri- bution ρ(x,y,z,t)= δ(x)δ(y)λ(z − ct) (1) in free space creates a radial electric field distribution of the form E r (x,y,z,t)= ρ(x,y,z,t) 2πǫ 0  x 2 + y 2 . (2) The field dynamics is governed by the Maxwell’s equa- tions: ∗ Work supported by EUROTeV (RIDS-011899), EUROFEL (RIDS- 011935), DFG (1239/22-3) and DESY Hamburg † mikko@temf.tu-darmstadt.de ‡ gjonaj@temf.tu-darmstadt.de § lau@temf.tu-darmstadt.de ¶ thomas.weiland@temf.tu-darmstadt.de ∇× H = ∂ D ∂t + J, ∇× E = − ∂ B ∂t , (3) ∇· D = ρ, ∇· B =0. The scattered field formalism is used in this paper. Hence, the excitation is embedded in boundary conditions and the current density J vanishes inside the computational do- main. Cylindrically symmetric problems are dealt with us- ing a scalar potential as described in the next section. THE NUMERICAL SCHEMES Finite-difference schemes are usually based on solving the Maxwell’s curl equations (4) iteratively in time and space. The divergence equations are not explicitly enforced in FDTD/FIT schemes [6, 7]. While the numerical diver- gence vanishes for some schemes [6, 7], it is non-zero for others [3]. A general operator splitting method is discussed in [8], and has been applied to derive specific schemes in [3]. We will focus on the curl equations in 3D and use a scalar potential in 2D problems. 3D Scheme Consider updating the x-component of the electric field. The basic idea of the scheme is to enlarge the stencil al- lowing a larger time step than with the Yee scheme. The finite-difference operators presented for narrow-band ap- plications in [9, 10] are modified and used here for wide- band calculations. Thus, the operators are used without as- sumptions on the frequency. The update equation for E x in free space reads: E x | n+1 i+1/2,j,k = E x | n i+1/2,j,k − α ∆t ǫ 0 D z,0 H y | n+1/2 i+1/2,j,k − 4β ∆t ǫ 0 D z,1 H y | i+1/2,j,k − 4γ ∆t ǫ 0 D z,2 H y | n+1/2 i+1/2,j,k + (4) α ∆t ǫ 0 D y,0 H z | n+1/2 i+1/2,j,k + 4β ∆t ǫ 0 D y,1 H z | n+1/2 i+1/2,j,k + 4γ ∆t ǫ 0 D y,2 H z | n+1/2 i+1/2,j,k . Here the three spatial derivative operators are all second- order accurate by construction (center differences). The Proceedings of ICAP 2006, Chamonix, France MOM2IS03 Numerical Methods in Field Computation Solver Techniques 35