Paper  Title  Page 

THPAK144  A Pseudospectral Method for Solving the Bloch Equations of the Polarization Density in e^{} Storage Rings  3589 
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Funding: Work supported by DOE under DESC0018008 We consider the numerical evolution of Bloch equations for the polarization density in highenergy electron storage rings. Equilibrium polarization is well characterized by the DK formulas for current rings, but deviations may be important at the high energies we have in mind. We believe the Bloch equations derived in* give a more accurate description at all energies. These form a system of three coupled linear partial differential equations for the three components of the polarization density. Following** we formulate the equations in actionangle variables and approximate the FokkerPlanck terms. We aim to integrate these equations numerically in order to approximate the equilibrium and compare with the DK formulas. The smoothness and simple geometry of the problem makes it amenable to pseudospectral discretization using Fourier modes in the angles and Chebyshev polynomials in the actions, leading to a large ODE system. We will explore time stepping algorithms for the needed long time integration. Here, we present results for simple models checking the accuracy of the numerical method but note that our ultimate goal is to simulate polarization in the FCC and CEPC rings. * Ya.S.Derbenev, A.M.Kondratenko, Sov. Phys. Dokl., 19, p.438 (1975). ** D.P.Barber, K.Heinemann, H.Mais, G.Ripken, A FokkerPlanck treatment of stochastic particle motion…, DESY91146, 1991. 

DOI •  reference for this paper ※ https://doi.org/10.18429/JACoWIPAC2018THPAK144  
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