Speeding up simulations of relativistic systems using an optimal boosted frame

It can be computationally advantageous to perform computer simulations in a Lorentz boosted frame for a certain class of systems. However, even if the computer model relies on a covariant set of equations, it has been pointed out that algorithmic difficulties related to discretization errors may have to be overcome in order to take full advantage of the potential speedup. We summarize the findings, the difficulties and their solutions, and show that the technique enables simulations important to several areas of accelerator physics that are otherwise problematic, including self-consistent modeling in three-dimensions of laser wakefield accelerator stages at energies of 10 GeV and above.

💡 Research Summary

The paper presents a systematic methodology for accelerating first‑principles simulations of relativistic plasma and beam systems by performing the calculations in an optimally chosen Lorentz‑boosted frame. The authors begin by demonstrating that, for systems containing two or more relativistically moving components, the ratio between the longest and shortest space‑time scales scales as γ² under a Lorentz transformation. Since the number of time steps required by a particle‑in‑cell (PIC) or any Maxwell‑Vlasov solver is proportional to this ratio, moving to a frame where the bulk motion is large (high γ) can reduce the computational cost by a factor of γ².

The paper then addresses three major numerical challenges that arise when discretizing covariant equations in a boosted frame. First, the widely used Boris particle pusher fails to cancel the electric and magnetic forces exactly for ultra‑relativistic beams, leading to errors that grow as γ². The authors derive a modified Lorentz‑force formulation and solve the resulting implicit system analytically, restoring the exact cancellation and eliminating the error.

Second, most input parameters (laser wavelength, plasma density, beam distribution) are defined in the laboratory frame. Transforming them to the boosted frame requires Doppler shifting the laser, drifting the plasma, and possibly stretching the beam longitudinally. The authors implement a “fixed‑lab‑plane injection” scheme in the WARP code: the laser and beam are injected at a plane that is stationary in the lab frame but moves in the boosted frame. To approximate the long‑range space‑charge fields near the injection plane they employ “frozen” drifting macro‑particles.

Third, output data must be transformed back to the laboratory frame for diagnostics. The solution is to place a set of “stations” fixed in the lab coordinates and record field and particle data at regular intervals in the boosted simulation. Because the station locations do not coincide with the boosted grid, interpolation is performed during data collection. A short‑wavelength numerical instability, whose growth rate increases with the boost velocity and with decreasing grid spacing, is mitigated by using low‑dispersion electromagnetic solvers and low‑pass digital filtering.

With these techniques the authors demonstrate dramatic speed‑ups in several accelerator‑physics applications. For laser‑wakefield acceleration (LWFA) they performed full‑3D PIC simulations of a 10 GeV stage at a realistic plasma density (10¹⁷ cm⁻³) using 512 cores on a LBNL cluster. The boosted‑frame run (γ≈130) completed in ~4 hours, whereas an equivalent laboratory‑frame simulation would require on the order of 15 years on the same resources, implying a speed‑up of roughly 10⁴. Similar gains (≈500×) were observed for electron‑cloud instability simulations, where boosted‑frame PIC results matched quasistatic calculations but required far less specialized code development. For free‑electron lasers, the authors note that the number of required time steps scales as 2γ², making boosted‑frame electromagnetic PIC an attractive alternative when eikonal approximations are insufficient. Preliminary studies of coherent synchrotron radiation (CSR) also show that the boosted‑frame approach can capture the characteristic energy‑loss pattern of a short, high‑current beam traversing a dipole.

In conclusion, the authors argue that the non‑invariance of scale ratios makes the choice of simulation frame a dominant factor in computational cost for a broad class of relativistic problems. The numerical difficulties associated with loss of covariance, input/output transformations, and short‑wavelength instabilities have been successfully addressed, and no fundamental “show‑stopper” remains. Boosted‑frame simulations thus provide a viable, often superior, alternative to reduced‑model approaches such as quasistatic or eikonal approximations, enabling fully three‑dimensional, self‑consistent studies of next‑generation accelerator concepts, electron‑cloud effects, FEL physics beyond standard codes, and CSR phenomena.