Accelerated three-dimensional quasistatic particle-in-cell code

We introduce a quasistatic particle-in-cell (PIC) code—wand-pic—which does not suffer from some of the common limitations of many quasistatic PICs, such as the need for a predictor-corrector method in solving electromagnetic fields. We derive the field equations under quasistatic (QS) approximation and find the explicit form of the “time” derivative of the transverse plasma current. After that, equations for the magnetic fields can be solved exactly without using the predictor-corrector method. Algorithm design and code structure are thus greatly simplified. With the help of explicit quasistatic equations and our adaptive step size, plasma bubbles driven by the large beam charges can be simulated efficiently without suffering from the numerical instabilities associated with the predictor-corrector method. In addition, wand-pic is able to simulate the sophisticated interactions between high-frequency laser fields and beam particles through the method of subcycling. Comparisons between the wand-pic and a first-principle full PIC code (vlpl) are presented. wand-pic is open-source, fully three-dimensional, and parallelized with the in-house multigrid solver. Scalability, time complexity, and parallelization efficiency up to thousands of cores are also discussed in this work.

Figure 1

(a) The 3D simulation domain in wand-pic. The 2D transverse plane is partitioned into square subdomains. In the longitudinal direction, the adaptive step size is shown by the red ticks which are denser at the back of the bubble. (b) The multigrid approach shown is a hierarchy of grids in one subdomain with different mesh sizes.

Figure 2

Comparison of the simulation results from vlpl-3D and wand-pic. (a) Plasma bubbles from vlpl-3D (upper half) and wand-pic (lower half). (b) Longitudinal on-axis wakefield $E_z(\xi ,\mathbf{r}=0)$ from VLPL-3D and wand-pic ($k$). Maximum step size refinement level $k$: step size refinement proceeds until the adaptive step size $d\xi$ is reduced by $k$ from the original step size. (c) Transverse focusing wakefields $F_{\perp x}(x)=E_x-B_y$ at the center of the bubble in the $x\text{−}z$ plane. (d) Normalized runtime of wand-pic ($k$) for different levels of step size refinement $k$.

Figure 3

Direct laser acceleration of an injected electron bunch by a leading (pump) and trailing (DLA) pulses: comparison between vlpl-3D [top of (a) and (b) panels] and wand-pic [bottom of (a) and (c) panels]. (a) Densities of the plasma electrons and externally injected electrons (red). (b) and (c) Distribution of the injected electrons in the $(W_{\parallel },W_{\perp })$ space of the work done by the longitudinal ($W_{\parallel }$) and transverse ($W_{\perp }$) electric fields. Color-coding: by relativistic factor $\gamma$, insets: electron energy spectra. The dashed black curve in (b) encloses the $(W_{\parallel },W_{\perp })$ space from the lower-resolution vlpl-3D simulation. Laser parameters: peak powers $P_{\mathrm{pump}}=P_{DLA}=21\mathrm{TW}$, wavelengths ${\lambda }_{\mathrm{pump}}={\lambda }_{\mathrm{DLA}}=0.8\mu \mathrm{m}$, durations ${\tau }_{\mathrm{pump}}=16.6\mathrm{fs}$and ${\tau }_{\mathrm{DLA}}=9.4\mathrm{fs}$, spot sizes $w_{\mathrm{pump}}=8.7\mu \mathrm{m}$ and $w_{\mathrm{DLA}}=5.5\mu \mathrm{m}$. Plasma parameters: density $n_0=7.7\times {10}^{18}{\mathrm{cm}}^{-3}$ and length $z=1.15\mathrm{mm}$. Grid/step size for high (low) vlpl-3D resolutions: $\delta x=\delta y={\lambda }_L/5,\delta z\approx c\delta t={\lambda }_L/100$ ($\delta x=\delta y={\lambda }_L/5,\delta z\approx c\delta t={\lambda }_L/50$).

Figure 4

3D nonaxisymmetric wand-pic simulation of an ultrashort bunch interacting with a laser pulse in a phase-dependent manner. (a) Color-coded plasma and electron bunch densities at the start of the interaction ($ct=0\mathrm{mm}$). Red curve: laser pulse envelope. (b) Same as (a), but at $ct=z_{\mathrm{fin}}=24.2\mathrm{mm}$, with a purple curve denoting the position where the transverse wake $W_x=0$ and black dashed line denoting the axis of laser propagation. (c) Bunch electrons in the ($W_{\parallel },W_{\perp }$) space at $ct=z_{\mathrm{fin}}$. Inset: electron energy spectrum at $z=z_{\mathrm{fin}}$. (d) Solid lines: transverse wakefields $F_{x,y}$ at the back of the plasma bubble ($(z-ct)/{\lambda }_L=30$) as a function of the propagation distance $ct$. Different colors represent different initial CEPs ${\varphi }_0\equiv {\varphi }_{\mathrm{CEP}}(t=0)$: ${\varphi }_0=0$ (black), ${\varphi }_0=\pi /2$ (blue), and ${\varphi }_0=\pi$ (red). Laser pulse, electron bunch, and plasma parameters: see text. Simulation parameters: $\delta x=\delta y=\delta z=c\delta t=0.1k_p^{-1}$, subcycling number $N_{\mathrm{sub}}=50$.

Figure 5

Nonuniform grids and its application. (a) A nine-points stencil of a nonuniform 2D grids. (b) A common beam-driven simulation. A Gaussian driver bunch is used: ${\sigma }_x^d={\sigma }_y^d=0.5k_p^{-1}$, ${\sigma }_z^d=2.5k_p^{-1}$, and peak density $=20n_0$. (c) Grid size $\delta x$ as a function of $x$ (same for the $\delta y$). There are three main regions: (I) plasma region, (II) bubble region, and (III) beam region. (d) 2D electron density distribution at $k_p(z-ct)=8$, overlaid with the transverse grids. To make the grids visually clear, we enlarged the grid size by 5 times, the actual grids used in the simulation are 5 times denser everywhere.

Figure 6

Benchmarking of wand-pic on Lonestar 5 Architecture (Xeon E5-2690 v3). (a) Strong scaling of wand-pic, transverse problem size (number of cells) is fixed at $N_{\mathrm{grid}}=1000\times 1000$. (b) The algorithm time complexity of wand-pic: the number of cores fixed at $N_{\mathrm{c}}=64$, the problem size $N_{\mathrm{grid}}=n\times N_0$ is varied ($N_0=320\times 320$). (c) Weak scaling of wand-pic for fixed $N_{\mathrm{grid}}/N_{\mathrm{c}}=20\times 20$.