Spectral Galerkin solver for intense beam Vlasov equilibria in nonlinear constant focusing channels

A numerical method is described for producing stationary solutions of the Vlasov-Poisson system describing a relativistic charged-particle beam in a constant focusing accelerator channel, confined transversely by a general (linear or nonlinear) focusing potential. The method utilizes a variant of the spectral Galerkin algorithm to solve a nonlinear partial differential equation (PDE) in two degrees of freedom for the beam space charge potential in equilibrium. Numerical convergence with an increasing number of computed spectral modes is investigated for several benchmark problems. Preservation of the stationary phase space density is verified using a strongly nonlinear focusing channel based on the Integrable Optics Test Accelerator at Fermi National Accelerator Laboratory.

Figure 1

Two measures of global relative numerical error (34) are shown for computed solutions of the exactly soluble test problems associated with potentials $V_0^{\left(I\right)}$ (a) and $V_0^{\left(II\right)}$ (b) of Appendix pp2. Solid lines denote the corresponding error estimates given in Eq. (37), which are valid for large $l_{\max }$.

Figure 2

(a) Contours of the spatial projection $P_{xy}$ of a stationary waterbag phase space density $f_{\mathrm{eq}}$ obtained for a 2.5 MeV proton beam with current $I=60.7$ mA in a constant focusing channel described by Eq. (38) with $k=0.785{\mathrm{m}}^{-1}$ and $t=32.8{\mathrm{m}}^{-3}$. The domain boundary is located at $a=b=1.69$ cm. (b) Decay of global relative numerical error for the problem (38). The solid line is a fit showing exponential decay with increasing $l_{\max }$.

Figure 3

(a) The potential (41), shown for $\tau =-0.4$. (b) Contours of the spatial projection $P_{xy}$ of the thermal equilibrium phase space density $f_{\mathrm{eq}}$ obtained for a 2.5 MeV proton beam with current $I=60.7$ mA in a constant focusing channel described by Eq. (39b).

Figure 4

Residual in the transverse plane for the numerical solution of Eq. (15) with the applied potential (39b) obtained using $15\times 15$ modes. The result is given relative to the local maximum of ${\nabla }^2{\mathrm{\Phi }}_n$ located at the origin $(x,y)=(0,0)$.

Figure 5

Projected density profiles of a stationary 2.5 MeV proton beam with current $I=60.7$ mA in a constant focusing channel described by Eq. (39b), shown before and after tracking a distance $L=2000/k$. The curves obtained before and after tracking coincide. (a) Spatial projection. (b) Momentum projection. In (b), the two curves corresponding to $p_x$, initial and $p_y$, initial also coincide.

Figure 6

Projected density profiles of a 2.5 MeV proton beam with the same initial phase space density as in Fig. 5, given before and after tracking a distance $L=2000/k$ in the constant focusing channel (39b) at the unmatched current $I=0$. Evolution of the density is evident. (a) Spatial projection. (b) Momentum projection. In (b), the two curves corresponding to $p_x$, initial and $p_y$, initial coincide.