Wakefield Code ECHO 2(3)D How it works, limitations, pros/cons and new/current developments Igor Zagorodnov ICFA mini-Workshop on “Electromagnetic wake fields and impedances in particle accelerators“ Erice, Sicily 23-28. April 2014

Overview Motivation Numerical Methods Examples Code Status low-dispersive schemes boundary approximation indirect integration algorithm modelling of conductive walls Examples Code Status

Motivation Wake field calculation – estimation of the effect of the geometry variations on the bunch Maxwell‘s Equations First codes in time domain ~ 1980 A. Novokhatski (BINP), T. Weiland (CERN)

Motivation TBCI*, MAFIA*** rotationally symmetric and 3D longitudinal and transverse wakes triangular geometry approximation; arbitrary materials moving window dispersion error NOVO*** only rotationally symmetric only longitudinal wake “staircase” geometry approximation; only PEC moving mesh low dispersion error * T. Weiland, TBCI and URMEL - New Computer Codes for Wake field and cavity mode calculations, IEEE Trans. Nuclear Science, 30 (1983) 2489 ** MAFIA Collaboration, MAFIA Manual, CST GmbH, Darmstadt, 1997. *** A. Novokhatski, M. Timm, T. Weiland, Transition Dynamics of the Wake Fields of Ultra Short Bunches. Proc. of the ICAP 1998, Monterey, California, USA.

Low-dispersive schemes New projects – new needs short bunches; long structures; tapered collimators New methods are required without dispersion error accumulation; “staircase” free; fast 3D calculations on PC Solutions zero dispersion in longitudinal direction; “conformal” meshing; moving mesh and “explicit” or “split” methods

Low-dispersive schemes Dual grid Maxwell Grid Equations* Primary grid * T. Weiland, A discretization method for the solution of Maxwell’s equations for six-component fields, Electronics and Communication (AEÜ) (1977) 31, p. 116.

Low-dispersive schemes transversal plane longitudinal direction

Low-dispersive schemes FDTD Implicit Scheme* (2002) * I. Zagorodnov, R. Schuhmann, T. Weiland, Long-Time Numerical Computation of Electromagnetic Fields in the Vicinity of a Relativistic Source, Journal of Computational Physics, vol. 191, No.2 , pp. 525-541, 2003.

Low-dispersive schemes Implicit Scheme (2002) Our scheme in staircase approximation with =0.5 is reduced to the scheme realized in code NOVO.

E/M and TE/TM* splitting (2004) Low-dispersive schemes E/M and TE/TM* splitting (2004) Subdue the updating procedure to the bunch motion * Zagorodnov I.A, Weiland T., TE/TM Field Solver for Particle Beam Simulations without Numerical Cherenkov Radiation, Phys. Rev.  ST Accel. Beams, vol. 8, 042001 (2005) E/M splitting TE/TM splitting MAFIA ECHO

Low-dispersive schemes transversal plane longitudinal direction

Low-dispersive schemes E/M splitting TE/TM splitting dispersion error dispersion error suppressed for

Low-dispersive schemes Implicit and explicit TE/TM formulations Implicit TE/TM scheme

Low-dispersive schemes Implicit and explicit TE/TM formulations TE/TM-ADI scheme - splitting error Explicit TE/TM scheme - splitting error

Low-dispersive schemes Explicit TE/TM scheme E/M scheme (FDTD)

Low-dispersive schemes Stability, energy and charge conservation The stability condition E/M splitting TE/TM implicit TE/TM explicit* (FDTD scheme) * Dohlus M., Zagorodnov I., Explicit TE/TM Scheme for Particle Beam Simulations// Journal of Computational Physics, vol. 225, No. 8, pp. 2822-2833, 2009.

Low-dispersive schemes Explicit TE/TM vs FDTD Explicit TE/TM scheme E/M scheme (FDTD) Explicit TE/TM requires the same memory 6N as FDTD Explicit TE/TM requires only 18% more computational time than FDTD (the same transverse mesh) Explicit TE/TM allows “magical” time step to avoid the dispersion error

Low-dispersive schemes Dispersion relation in the transverse plane Implicit TE/TM scheme Explicit TE/TM scheme explicit implicit No dispersion in z-direction + no dispersion along XY diagonals

Low-dispersive schemes Transverse Deflecting Structure Gaussian bunch with sigma=300mm explicit implicit

Standard Conformal Scheme* Boundary approximation Standard Conformal Scheme* PEC reduced cell area time step must be reduced * Dey S, Mittra R. A locally conformal finite-difference time-domain (FDTD) algorithm for modeling threedimensional perfectly conducting objects. IEEE Microwave and Guided Wave Letters 1997; 7(9):273–275.. Thoma P. Zur numerischen Lösung der Maxwellschen Gleichungen im Zeitbereich. Dissertation Dl7: TH Darmstadt,1997.

Time step is not reduced! Boundary approximation New Conformal Scheme* (2002) PEC PEC virtual cell Time step is not reduced! * Zagorodnov I., Schuhmann R.,Weiland T., A Uniformly Stable Conformal FDTD-Method on Cartesian Grids, International Journal on Numerical Modeling, vol. 16, No.2, pp. 127-141, 2003.

Square rotate by angle p/8. Boundary approximation 20 40 60 80 100 0.2 0.4 0.6 0.8 1 PFC USC 10 1 2 -4 -3 -2 -1 USC PFC staircase Square rotate by angle p/8. PFC (Partially Filled Cells) ~ Dey-Mittra USC (Uniformly Stable Conformal)

Simple Conformal (SC) Scheme* Boundary approximation Simple Conformal (SC) Scheme* * Zagorodnov I., Schuhmann R., Weiland T., Conformal FDTD-methods to avoid Time Step Reduction with and without Cell Enlargement , Journal of Computational Physics, vol. 225,  No. 2, pp. 1493-1507, 2007.

Error in loss factor for sphere Boundary approximation ? 1 1 staircase staircase () Oh conformal 0.01 0.01 conformal 2 () Oh r r r z 1e 1e - - 4 4 z z h ? 10 10 100 100 Error in loss factor for sphere Gaussian bunch with σ=0.5 cm is passing through a spherical resonator. The sphere has the diameter 1.8 cm. The analytical loss factor L is equal to 0.152444 Vp/C The error for stationary mesh is demonstrated by lines. The results for moving mesh are shown by triangles and circles.

Error in loss factor for a taper Boundary approximation 5 10 15 20 25 30 35 40 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 staircase conformal Error in loss factor for a taper The error δ relative to the extrapolated loss factor L=-7.63777 Vp/C for bunch with σ=1 mm is shown.

Indirect Integration Algorithm 1 C -1 C C r z O. Napoly, Y. Chin, and B. Zotter, Nucl. Instrum. Methods Phys. Res., Sect. A 334, 255 (1993)

Indirect Integration Algorithm 1 C -1 C C r 2 C z I. Zagorodnov, R. Schuhmann, T. Weiland, Journal of Computational Physics, vol. 191, No.2 , pp. 525-541, 2003.

Indirect Integration Algorithm

Indirect Integration Algorithm Zagorodnov I., Indirect Methods for Wake Potential Integration, Phys. Rev.  STAB vol. 9, 102002 (2006) H. Henke and W. Bruns, in Proceedings of EPAC 2006, Edinburgh, Scotland (WEPCH110, 2006)

Indirect Integration Algorithm

Modelling of Conductive Walls

Modelling of Conductive Walls

Modelling of Conductive Walls

3D simulation. Cavity Examples 3 geometry elements Moving mesh 20 TESLA cells structure 3 geometry elements The geometric elements are loaded at the instant when the moving mesh reach them. During the calculation only 2 geometric elements are in memory.

Examples E/M ECHO-3D (TE/TM) Comparison of the wake potentials obtained by different methods for structure consisting of 20 TESLA cells excited by Gaussian bunch E/M splitting TE/TM splitting

3D simulation. Collimator Examples 3D simulation. Collimator

Examples 10 20 30 40 500 1000 1500 2000 2500 3000 250.6% 82.5% 30% 12.2% 5.4% 12.4% 2.8% 0.8% 0.36% ref. E/M TE/TM -5 5 -500 The transverse dipole wake function (left) and loss factor for the collimators with and L=20cm (right). The solid lines show the results for the E/M scheme (Yee’s scheme) and the dashed lines display the results for the TE/TM scheme. The relative errors are given regarding the reference value (marked as ref. on the graphs) calculated by TE/TM method with the finest mesh.

Examples E/M ECHO-3D (TE/TM) Comparison of the wake potentials obtained by different methods for round collimator excited by Gaussian bunch TE/TM method – fast, stable and accurate with coarse mesh

Examples Coupler Kick

Code Status ECHO 2D https://www.desy.de/~zagor/WakefieldCode_ECHOz/

Code Status ECHO 3D