A PSEUDOSPECTRAL METHOD FOR SOLVING THE BLOCH EQUATIONS OF THE POLARIZATION DENSITY IN e − STORAGE RINGS

We consider the numerical evolution of the Bloch equations of Derbenev and Kondratenko (DK) for the polarization density in high-energy electron storage rings, such as the proposed FCC-ee and CEPC. Equilibrium spin polarization is well characterized by the DK formulas for current rings (see [1] and [2]), but deviations may be important at the high energies we have in mind. We believe these Bloch equations, in three degrees of freedom (DOF), derived in [3] give a more complete description at all energies. The equations are a system of three coupled linear partial differential equations for the three components of the polarization density and include the spin flip polarization effect. Following [4] we formulate the equations in action-angle variables and approximate the Fokker-Planck terms. Our goal is to integrate these equations numerically in order to approximate the equilibrium and compare with the DK formulas. For 3 DOF the polarization density has 6+ 1 independent variables. For simplicity, suppose that each of the space-like variables has been discretized on a grid with N grid-points, then the computational cost of each time step will scale no better than O(N6). The presence of parabolic terms in the governing equations necessitates implicit time stepping and thus solutions of linear systems of equations. For a fully coupled problem this will bring the per time step cost to O(N6q), with 1 ≤ q ≤ 3, depending on the algorithms used for the linear solve. However, only algorithms with q ≈ 1 are feasible (for Gaussian elimination q = 3). Fortunately, as we outline below, the structure of the equations allow us to group the space-like variables into two groups resulting in a cost that, to leading order, scales as O(M N3q). In the next section we present the full and the reduced Bloch equations and the underlying physics. Then we discuss the details of the numerical algorithm outlined above and the issue of complexity in higher dimensions. Finally we present two numerical results showing a depolarization calculation in 1DOF and spectral convergence in a 2DOF example.