applied math seminar
Title: A pseudospectral method for solving the Bloch equations for the polarization density in electron storage rings
In this talk we present some numerical and analytical results
of our work on the initial value problem
of what we call the reduced Bloch equations (RBEs). The latter are a system of
three uncoupled Fokker-Planck equations plus one coupling term.
Thus they are a generalization of a system of
three uncoupled homogeneous heat equations.
Because of the Fokker-Planck terms,
the RBEs are of first-order in time and second order in phase space
(the coupling term is linear and of zeroth order in phase space).
Our numerical approach uses the pseudospectral method
providing a system of linear first-order ODEs in time which is then solved by
a time stepping scheme.
The presence of parabolic terms in the governing equations necessitates
implicit time stepping and thus solutions of linear systems of equations.
The phase space variables come in $d$ pairs of polar coordinates
where d=1,2,3 is the number of DOF allowing for a d-dimensional
Fourier expansion. Thus the Pseudospectral Method is applied to the radial variables
using a Chebychev grid for each of them.
Dealing with 2d+1 independent variables poses a computational
challenge. However the coupling of the Fourier modes is weak in the sense
that it disappears after the time discretization if
an appropriate time stepping scheme is used.
For the time stepping scheme
we use the second order Adams-Bashforth/Adams-Moulton splitting method
which is explicit in the mode coupling terms making the
coupling of the Fourier modes weak.
The issue of global versus local mode coupling is briefly mentioned as well.
Since all three of our Fokker-Planck operators
are of Ornstein-Uhlenbeck type (if one transforms the polar coordinates
into cartesian variables) we were able to compute their eigenfunctions
and to use these functions to exactly solve
the RBEs for simple models of the coupling term. These exact solutions are used
to check the spectral convergence of our codes
for those models.
We will also say a bit about the application of the
RBEs. They are the most important part of the
so-called full Bloch equations (FBEs) which are a system of
three uncoupled Fokker-Planck equations
plus several coupling terms,
one of them being the coupling term of the RBEs
(we will apply the same numerical methods to the FBEs and to the RBEs).
The solutions of the FBEs are, as for the RBEs,
three-component functions of time and phase space,
called polarization densities. The polarization density in combination
with the phase space density provide a mesoscale description
of the electron beam in a high-energy storage ring.
The phase space density covers those aspects of the
beam which are independent of the electron spins whereas the
polarization density takes care of the electron spins.
In the language of quantum physics,
the phase space density and the
polarization density are the four components of the semiclassical
spin-1/2 Wigner function of the beam taking
into account the Lorentz force on the particle motion and
the spin-orbit coupling effects of
the external electromagnetic field as well as the effects of the
synchrotron radiation emitted by the electrons.
The FBEs were obtained by Derbenev and Kondratenko, around 1975.
For each storage ring the parameters in the FBEs are to be chosen appropriately.
The main focus of our work is on future high-energy
electron storage rings, e.g., the FCC-ee at CERN and the CEPC in China,
for studying the Higgs particle etc.
Contact Name: Pavel Lushnikov