An Overview of Relativistic Particle Pushers and their Extension to Arbitrary Order Accuracy

This paper provides a comprehensive comparison of explicit relativistic particle pushers used in Particle-in-Cell simulations, demonstrating that a significant class of these schemes can be generalized to arbitrary high-order accuracy and evaluating the performance of their fourth-order variants against second-order counterparts.

The Gist

Imagine you are trying to predict the path of a tiny, super-fast marble (a particle) flying through a storm of invisible winds and magnetic whirlwinds (electric and magnetic fields). This is the daily job of scientists using Particle-in-Cell (PIC) simulations to understand everything from how stars explode to how we might build clean fusion energy.

The problem? These marbles are moving so fast—close to the speed of light—that the usual rules of physics get weird (thanks to Einstein's relativity). If your math is even slightly off, the marble might drift off course, gain fake energy, or spiral out of control, ruining the whole simulation.

This paper is a grand race to find the best "calculator" (called a particle pusher) to track these marbles accurately. The author, H. Schmitz, tested about a dozen different mathematical recipes to see which one keeps the marble on the straight and narrow.

Here is the breakdown of the race using everyday analogies:

1. The Old Reliable vs. The New Challengers

For decades, the Boris Scheme has been the "Toyota Camry" of particle simulations. It's not the fastest or the most luxurious, but it's reliable, easy to drive, and rarely breaks down. It's been the default choice for almost every simulation code.

However, in extreme conditions (like super-strong magnetic storms), the Camry starts to sputter. It drifts off the road. So, scientists invented new cars:

• The "Vay" and "Higuera & Cary" (HC) models: These are like upgraded sports cars. They handle the curves better than the Camry, especially when the forces cancel each other out perfectly.
• The "Proper Time" racers (PL, LiLF, GH): These are like Formula 1 cars that switch to a different clock entirely. Instead of counting seconds on the wall, they count the "heartbeat" of the particle itself. In a perfect, unchanging vacuum, these cars are theoretically perfect. But in a bumpy, changing road (realistic simulations), they sometimes get confused about which second is which.

2. The Race Results (The Test Cases)

The author put these cars through seven different obstacle courses:

• The Spinning Top (Gyromotion): A particle spinning in a magnetic field.
  • Result: The Boris car spins a little too slowly or too fast (a phase error). The HC car spins much closer to the perfect rhythm.
• The Straight Line Test: A particle moving where electric and magnetic forces cancel out perfectly. It should go in a straight line.
  • Result: The Boris car starts to wiggle and drift. The Vay and HC cars stay perfectly straight. The PL car is perfect here too.
• The Magnetic Bottle: A particle trapped between two magnetic mirrors, bouncing back and forth.
  • Result: This tests long-term stability. The Implicit Midpoint (IMP) car (a heavy, slow, but very careful driver) did the best at keeping energy constant, but it's computationally expensive. Surprisingly, the HC car was very stable here too.
• The Laser Wave: A particle being blasted by a massive electromagnetic wave.
  • Result: This is where things get tricky. The "Proper Time" cars (PL, GH) were great at tracking the path but struggled with energy conservation. The HC car held its own well against the chaos.

3. The Big Surprise: The "Fourth Order" Upgrade

One of the paper's coolest discoveries is that you can take the "Boris-like" cars (Boris, HC, GYR, CC) and turbocharge them.

Imagine you have a bicycle (2nd order accuracy). The author showed you can add a special gear system (using a method by Yoshida) to turn that bicycle into a high-speed motorcycle (4th order accuracy).

• The Benefit: If you need extreme precision, these upgraded versions converge to the truth much faster. You can take bigger steps and still get the right answer.
• The Catch: If the road is too bumpy (the physics changes too fast for your step size), even the motorcycle can't save you. You still need to slow down and take smaller steps to see the details.

4. The Verdict: Which Car Should You Drive?

The paper concludes that there is no single "best" car for every situation, but there are clear winners for general use:

• The New Standard: The Higuera & Cary (HC) scheme is the new "Goldilocks." It's almost as fast as the old Boris method but handles difficult physics much better. It's the best all-rounder for most scientists.
• The Specialized Tool: If you are in a perfectly static, unchanging environment, the PL (Proper Time) method is mathematically perfect. But real life is messy, so it's less useful for general simulations.
• The Heavy Lifter: The Implicit Midpoint (IMP) method is the most accurate but requires a lot of computing power. Use it only if you need maximum precision and have the budget for it.
• The Academic Curiosity: The ZZ scheme was found to be the worst performer in almost every test. It's like a car that looks cool but has no engine.

Summary

Think of this paper as a Consumer Reports review for the software that drives our universe simulations. It tells us that while the old Boris method is still okay, the Higuera & Cary method is a smarter, more accurate choice for most modern problems. And if you really need to be precise, you can now upgrade your math to a "4th Order" version that gets you the right answer much faster, provided you don't try to drive too fast through a storm.

Technical Summary

Here is a detailed technical summary of the paper "An Overview of Relativistic Particle Pushers and their Extension to Arbitrary Order Accuracy" by H. Schmitz.

1. Problem Statement

Particle-in-Cell (PIC) simulations are essential for modeling kinetic processes in plasma physics, particularly in relativistic regimes (e.g., astrophysics, fusion, high-energy density research). A critical component of these simulations is the particle pusher, the numerical integrator that updates particle positions and velocities based on electromagnetic fields.

While the Boris scheme has been the industry standard for decades due to its simplicity, second-order accuracy, and volume-preserving properties, it exhibits significant shortcomings in specific relativistic scenarios:

• It fails to conserve constants of motion accurately in high-intensity electromagnetic waves.
• It produces incorrect trajectories in crossed electric and magnetic fields where the net force should vanish (leading to artificial drift).
• It suffers from phase errors in cyclotron motion that depend on the Lorentz factor.

The paper addresses the need for a comprehensive comparison of modern explicit integration schemes and investigates whether these schemes can be extended to higher orders of accuracy to improve numerical fidelity without prohibitive computational costs.

2. Methodology

The author conducted a systematic numerical comparison of 11 different integration schemes, categorized into three main types:

• Type I (Time-centered explicit second-order schemes): Includes the classic Boris, Vay, Higuera & Cary (HC), Chin & Cator (CC), and a Gyro-phase corrected (GYR) variant. These schemes split the update into electric field acceleration and magnetic field rotation.
• Type II (Proper time integration): Schemes that solve equations of motion exactly in the particle's proper time ($\tau$), including Pétri (PL), Li et al. (LiLF), Gordon & Hafizi (GH/GH2), and Zhou & Zhang (ZZ). These often require solving implicit equations for the proper time step $\Delta\tau$.
• Implicit Scheme: The Implicit Midpoint Method (IMP) by Ripperda et al., included as a benchmark for volume preservation and accuracy.

Test Cases:
The schemes were evaluated across seven distinct physical scenarios to test accuracy, stability, and conservation laws:

1. Gyromotion: Constant $B$, $E=0$ (tests phase error and energy conservation).
2. Lorentz-boosted particle at rest: Crossed $E$ and $B$ where forces cancel (tests straight-line propagation).
3. Gyration in a Lorentz-boosted frame: Non-canceling crossed fields (tests drift and period accuracy).
4. Parallel fields with spatial variation: Oscillating in $E$ and gyrating in $B$ (tests conservation of canonical momentum and invariants).
5. Magnetic Bottle: Trapped particle motion (tests long-term stability and adiabatic invariants).
6. Relativistic Plane Wave: Particle acceleration in a high-intensity wave (tests constants of motion and energy gain).
7. Oscillating Electric Field ($E \parallel B$): Time-dependent fields (tests general time-dependent performance).

Higher-Order Extension:
The author applied Yoshida's composition method to extend specific second-order "Boris-like" schemes (Boris, HC, GYR, CC) to fourth-order accuracy. This method constructs higher-order symplectic maps by composing the base second-order map with specific coefficients.

3. Key Contributions

1. Comprehensive Benchmarking: Provided a quantitative comparison of the most relevant relativistic pushers across a wide range of physical regimes, identifying specific strengths and weaknesses of each.
2. Identification of Order Reduction: Demonstrated that Type II schemes (proper time integrators), while exact for constant fields, degrade to first-order accuracy in spatially or temporally varying fields due to temporal alignment issues when calculating the proper time step $\Delta\tau$.
3. Higher-Order Construction: Successfully derived and tested fourth-order variants of explicit Boris-like schemes. This provides a pathway to high-accuracy simulations without switching to computationally expensive implicit methods.
4. Error Analysis: Isolated the sources of errors, distinguishing between phase errors (gyrophase), energy conservation errors, and trajectory drift, showing that "exact" rotation angles (as in GYR) do not necessarily solve all problems if the time step is too large to resolve the physics.

4. Key Results

• Explicit vs. Implicit: The Implicit Midpoint (IMP) scheme generally performed best, preserving invariants to machine precision in most cases. However, it failed in the Magnetic Bottle test regarding magnetic moment conservation and is computationally expensive.
• Best Explicit Performers:
  • Higuera & Cary (HC): Consistently among the best explicit schemes. It offers a modest improvement over Boris in general cases and a substantial improvement in the "force-free" crossed-field scenario.
  • Vay: Excellent for the specific crossed-field scenario it was designed for but not volume-preserving.
  • Pétri (PL) & Variants: Exact for constant fields but suffer from first-order convergence in inhomogeneous fields.
  • Zhou & Zhang (ZZ): Performed exceptionally poorly across almost all tests and is deemed of limited practical value.
• High-Intensity Plane Waves: In strong wave fields, Type II schemes (PL, GH) showed superior conservation of constants of motion ($\gamma - u_x/c$) compared to Type I schemes, but Type I schemes (specifically HC and Boris) performed better in conserving total energy ($\gamma$) in certain regimes.
• Higher-Order Schemes: The fourth-order variants of Boris, HC, GYR, and CC demonstrated fourth-order convergence for small time steps, reducing errors by orders of magnitude compared to their second-order counterparts. However, for large time steps (where the physics is under-resolved), higher-order schemes offered no advantage over second-order schemes.
• Conservation Laws:
  • In the Magnetic Bottle test, most schemes preserved energy, but IMP showed the largest error in magnetic moment conservation.
  • In spatially varying fields, Type II schemes failed to conserve canonical momentum $p_z$ due to first-order convergence, whereas Type I schemes maintained second-order convergence.

5. Significance and Conclusions

This paper provides a critical guide for developers and users of PIC codes:

• Recommendation for General Use: The Higuera & Cary (HC) scheme is recommended as a robust, general-purpose explicit pusher. It offers a significant improvement over the standard Boris scheme with only a marginal increase in computational cost (requiring an extra square root evaluation).
• Limitations of Proper Time Integrators: While Type II schemes are theoretically exact for constant fields, their degradation to first-order accuracy in realistic, time-dependent PIC simulations (where fields vary on the grid) limits their utility compared to well-tuned explicit schemes.
• Path to High Accuracy: For applications requiring extreme precision with small time steps, the fourth-order extensions of Boris-like schemes offer a viable, volume-preserving alternative to implicit methods, which are often too costly for large-scale 3D simulations.
• Contextual Accuracy: The author emphasizes that the accuracy of a PIC simulation depends not just on the pusher, but also on the field interpolation and grid staggering (e.g., Yee grid). A high-order pusher cannot compensate for poor field interpolation or time-step sizes that fail to resolve the underlying physical scales (e.g., gyrofrequency).

In summary, the paper moves beyond the dominance of the Boris scheme by validating Higuera & Cary as a superior general-purpose alternative and establishing a framework for arbitrary-order explicit integration to meet the demands of next-generation high-precision plasma simulations.