Computational framework for particle and spin simulations based on the stochastic Galerkin method

An implementation of the polynomial chaos expansion is introduced as a fast solver of the equations of beam and spin motion of charged particles in electromagnetic fields. We show that, based on the stochastic Galerkin method, our computational framework substantially reduces the required number of tracking calculations compared to the widely used Monte Carlo method.

Figure 1

With the ($m=5$)-dimensional problem and an expansion order of $p=4$, the number of basis functions is $P=126$ [see Eq. (7) of 11], which results in a $(126\times 126\times 126$) $C_{ijl}$ tensor. The sparsity of the rank-3 tensor $C_{ijl}$ is illustrated by fixing, e.g., the first index to values of $i=14$ [panel (a)], $i=26$ (b), $i=37$ (c), $i=45$ (d), $i=69$ (e), $i=77$ (f), $i=86$ (g), $i=93$ (h), $i=105$ (i), and $i=116$ (j), which yields the 10 sparse matrices shown. The multiplication of the PCE coefficients involving this tensor is very fast, as most of the tensor elements are zero, while computing, storing, and loading of this tensor constitutes a CPU and memory-intensive operation.

Figure 2

Results of tracking ${10}^5$ particles in a uniform electric and magnetic field $[\vec{E}=(1,0,0)\mathrm{V}/\mathrm{m}$, $\vec{H}=(0,1/173,0)\mathrm{A}/\mathrm{m}]$. On the horizontal axis, the time in seconds is shown over which the tracking simulation evolved, and on the vertical axis the resulting solutions for $x\left(t\right)$ in (a), $y\left(t\right)$ (b), $v_x\left(t\right)$ (c), and $v_y\left(t\right)$ (d). The dotted blue lines represent the solution computed using the Monte Carlo simulations (MC), while the dashed red lines visualize the solutions of the stochastic Galerkin method (SGM). In all four cases, positions and velocities are nearly indistinguishable, indicating a very good agreement between MC and SGM (see also Fig. 4).

Figure 3

Result of spin tracking simulations in uniform fields using the MC and the SGM. The horizontal axis shows the time $t$ in seconds, and the vertical axis shows the vertical component $S_y\left(t\right)$ of the spin vector $\vec{S}\left(t\right)$.

Figure 4

(a) The error analysis of the uniform field scenario (Sec. 5a) of the mean value ${\epsilon }_{\mu }\left(t\right)$ using Eq. (32a), and (b) the standard deviation ${\epsilon }_{\sigma }\left(t\right)$ using Eq. (32b) for the quantities $x$, $y$, $v_x$, $v_y$, and $S_y$ in the time interval from $t=0$ to 20 s.

Figure 5

Comparison between the phase-space ellipses of the MC tracking results (blue crosses) and the SGM (red dots) along the beam direction in the waveguide RF Wien filter at $z=1$ (a), 210 (b), 610 (c), and 810 mm (d) [5] (Sec. 5b).

Figure 6

Trajectory simulation of the $xz$ plane in the drift region along the RF Wien filter (Sec. 5b) using the MC and the SGM.

Figure 7

Error analysis involving the mean value ${\epsilon }_{\mu }\left(z\right)$ and the standard deviation ${\epsilon }_{\sigma }\left(z\right)$ of the quantities $x$ and $v_x$ along the beam direction inside the waveguide RF Wien filter (Sec. 5b).

Figure 8

Comparison of the simulation time required for the parallelized MC and the SGM. For particle numbers below about ${10}^5$, the methods are comparable. For larger particle numbers with a constant expansion order of $p=4$, the time required for the SGM stays constant, while the demand for the MC increases exponentially.