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4.2 General description of the method

The prospects previously described will imply several requirements, one of them being of course the possibility of importing our algorithm in Astra. A stand alone main program has been implemented in order to test our space charge calculation algorithm before its incorporation in astra's program infrastructure. The adaptation of the meshing will require an interface between the absolute coordinates and the rotated grid. Indeed, Astra uses absolute laboratory coordinates whereas the grid is defined in a rotated frame. This interface will be provided by storing the coordinates of the corner of the grid and the components of the three axis defining the rotated grid (see figure 4.3). The other main change, would be the Lorentz transformation of the coordinates and fields, so that they could take into account any momentum direction. The global process of space charge calculation is schematised on figure 4.2 where the steps concerning the lorentz boost transformation and the rotation of the frame are presented. Attention has to be paid on these different steps because from their simple appearence, underlying subtle manipulation of coordinates and components are requiered. For instance, after the resolution of Poisson algorithm, the fields are known on each points of the rotated grid. These fields have components in the rotated frame and are expressed with respect to coordinates also in the rotated frame.

**Figure 4.2:** Global process of space charge calculation. Poisson equation has to be solved in the rest frame, where by hypothesis, there is no motion of particles. In this frame, a frame adapted to the bunch is found by diagonalization of the covariance matrix of the particles.

Emmanuel Branlard
http://emmanuel.branlard.free.fr/