Scientific Achievement

The particle-in-cell (PIC) and Monte Carlo Collisions (MCC) methods are essential in numerical simulations of physical systems, including fusion and particle accelerators. While both methods can independently conserve energy—either exactly or nearly—combining them has resulted in anomalous numerical heating. This recently identified phenomenon artificially increases the kinetic energy (and temperature) of simulated particles, reducing the accuracy of simulations.

Researchers from the Advanced Modeling Program (AMP) in the Accelerator Technology & Applied Physics (ATAP) Division at the Department of Energy’s Lawrence Berkeley National Laboratory (Berkeley Lab), in collaboration with colleagues from the Department of Energy’s Lawrence Livermore National Laboratory, have explored the origins of this heating and established how to couple the PIC and MCC methods to prevent it.

Significance and Impact

Particle-in-cell loop with Monte Carlo collisions (MCC). The operations occur sequentially clockwise around the loop and from top to bottom in the “push particles” inset. In the standard configuration (top), the MCC step usually occurs before or after pushing the particles’ velocities. In the proposed configuration (bottom), the MCC step occurs either in the middle of the velocities push or in the middle of the positions push.

Computer simulations are essential for advancing fusion. For instance, in inertial fusion energy (IFE), energy from tens to hundreds of high-power lasers must be transferred (either directly or indirectly) to a millimeter-sized pellet containing a mixture of deuterium and tritium for fusion to occur. However, this process involves complex laser-plasma interactions and plasma physics that must be resolved accurately, posing significant computational challenges even for the most advanced simulation codes running on the largest supercomputers.

IFE simulations employ various numerical methods that are periodically enhanced to address the demands of modeling these fundamental processes with greater accuracy across broader spatial and temporal scales. To achieve this, as well as to validate reduced models, the researchers performed first-principles simulations using advanced PIC codes—such as the award-winning PIC code WarpX employed in this study—to leverage the PIC method to capture the “long-range” plasma physics and the MCC method to account for “short-range” Coulomb collisions.

This new approach to combining the two methods provides an innovative approach to preventing anomalous numerical heating, paving the way for more accurate and cost-effective simulations that advance fusion research.

Research Details

Understanding the origin of anomalous numerical heating 

Using a simplified model of single particle motion in a harmonic oscillator, the researchers reproduced the anomalous numerical heating effect, explained its origin, and proposed a solution. The insight that led to the understanding and proposed solution was based on a key observation: within the PIC iteration loop, the MCC step typically occurred before or after updating the particles’ velocity and position. This results in the algorithm’s breakdown of time-centering, a cornerstone of energy preservation for time-dependent methods.

The solution

Based on this observation, the team proposed a solution that employs Strang splitting, a numerical method based on operator splitting for solving differential equations, to integrate the MCC operation into the midpoint of the particle velocity or position updates. Unlike the standard “unsplit” approach, where the MCC operation occurs before or after the velocity update, the proposed modified PIC-MCC loop is time-centered, a method associated with energy conservation.

Verification

A detailed analysis of the various options for the PIC-MCC loop was conducted using the simple harmonic oscillator model. This demonstrated that energy is precisely conserved with the newly proposed centered placements of the MCC step but not in other configurations. Simulations with a single particle in a harmonic potential well were then extended to an ensemble of 1000 independent oscillators, confirming the findings. Final validation was achieved by analyzing a uniform warm plasma’s complete PIC-MCC loop and PIC-MCC simulations.

Application 

The new algorithm was used to model a magnetically-driven piston collisional shock in one dimension, where a time-dependent magnetic field is applied at one end of a plasma column, creating a shock that propagates toward the opposite end. With the standard PIC-MCC implementation, the spurious numerical heating necessitates increasing the number of particles per cell (ppc) to 4000 or more for convergence to a physical result. Conversely, the new PIC-MCC implementation considerably reduces numerical noise, allowing for accurate results with only 1000 ppc, resulting in a quarter of the computational cost.

This new method will be applied to modeling spherical implosions in the context of IFE and used to validate advanced exactly energy-conserving implicit PIC-MCC methods implemented in WarpX. It will allow simulations on longer time scales than is possible with the standard PIC-MCC method. These simulations will shed new light on key kinetic processes that can help diagnose and design more efficient IFE targets to deliver adequate fusion energy output from future fusion power plants.

Contact: Jean-Luc Vay

Researchers: Olga Shapoval, Rémi Lehe, and Axel Huebl (Berkeley Lab); and Justin Angus and David Grote (Lawrence Livermore National Laboratory)

Funding: The work presented here was supported by the KISMET collaboration, a project of the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, and Office of Fusion Energy Sciences, Scientific Discovery Through Advanced Computing (SciDAC) program.

Publication: Jean-Luc Vay, Justin Angus, Olga Shapoval, Rémi Lehe, David Grote, and Axel Huebl. “Energy-preserving coupling of explicit particle-in-cell with Monte Carlo collisions,” Phys. Rev. E 111, 025306 (2025). https://doi.org/10.1103/PhysRevE.111.025306

For more information on ATAP News articles, contact caw@lbl.gov.