  In this paper we describe our work on spin polarization in high-energy
electron storage rings which we base on the Bloch equation for the polarization
density and which aims towards the e-/e+ option of the proposed Future Circular
Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC).
The Bloch equation takes into account non spin-flip and spin-flip effects due
to synchrotron radiation including the spin-diffusion effects and the
Sokolov-Ternov effect with its Baier-Katkov generalization as well as the
kinetic-polarization effect. This mathematical model is an alternative to the
standard mathematical model based on the Derbenev-Kondratenko formulas. For our
numerical and analytical studies of the Bloch equation we develop an
approximation to the latter to obtain an effective Bloch equation. This is
accomplished by finding a third mathematical model based on a system of
stochastic differential equations underlying the Bloch equation and by
approximating that system via the method of averaging from perturbative ODE
theory. We also give an overview of our algorithm for numerically integrating
the effective Bloch equation. This discretizes the phase space using spectral
methods and discretizes time via the additive Runge-Kutta method which is a
high-order semi-implicit method. We also discuss the relevance of the third
mathematical model for spin tracking.

Accelerator Physics