Nyström Type Exponential Integrators for Strongly Magnetized Charged Particle Dynamics

TL;DR

The paper tackles stiff, strongly magnetized charged particle pushing in PIC simulations and introduces Nyström-type exponential integrators that directly integrate the second-order Newtonian equation by partitioning into position and velocity. By employing Lagrange-Sylvester interpolation to efficiently evaluate $\varphi_k$-functions and constructing second- and third-order Nyström-type schemes (EPRKN2 and EPRKN3), the authors achieve notable speedups over standard exponential integrators and Boris/Buneman pushers across 2D and 3D test problems, including electric potential wells, gyroradius, and grad-$B$ drift scenarios. The results demonstrate improved computational efficiency while maintaining or improving accuracy, especially in strongly magnetized regimes, highlighting the practical potential of Nyström exponential integrators for PIC simulations. The work also outlines future directions toward geometric property preservation, better nonlinear quadrature, and broader field configurations to further enhance long-time accuracy and applicability. Overall, Nyström-type exponential integrators offer a promising, scalable alternative for efficient charged-particle dynamics in magnetized plasmas.

Abstract

Solving for charged particle motion in electromagnetic fields (i.e. the particle pushing problem) is a computationally intensive component of particle-in-cell (PIC) methods for plasma physics simulations. This task is especially challenging when the plasma is strongly magnetized due numerical stiffness arising from the wide range of time scales between highly oscillatory gyromotion and long term macroscopic behavior. A promising approach to solve these problems is by a class of methods known as exponential integrators that can solve linear problems exactly and are A-stable. This work extends the standard exponential integration framework to derive Nyström-type exponential integrators that integrates the Newtonian equations of motion as a second-order differential equation directly. In particular, we derive second-order and third-order Nyström-type exponential integrators for strongly magnetized particle pushing problems. Numerical experiments show that the Nyström-type exponential integators exhibit significant improvement in computation speed over the standard exponential integrators.

Figures (12)

• Figure 1: Gyromotion in a uniform magnetic field.
• Figure 2: Drift motion from an electric field perpendicular $\bm{E}$ to the $\bm{B}$ field.
• Figure 3: Grad-$B$ drift motion
• Figure 4: Results for particle position, 2D potential well test problems with magnetic field $\bm{B} = 100\,\hat{\bm{z}}$: reference solution orbits (first row) and precision diagrams (second row). Boris/Buneman step sizes are $h = 10^{-3}$, $10^{-4}$, $10^{-5}$, $10^{-6}$ for the quadratic potential problem and $h = 10^{-4}$, $10^{-5}$, $10^{-6}$, $10^{-7}$ for the cubic/quartic potential problems. Exponential integrators step sizes are $h =$ 100, 10, 1, $10^{-1}$ for the quadratic potential problem and $h = 10^{-2}$, $10^{-3}$, $10^{-4}$, $10^{-5}$ for the cubic/quartic potential problems.
• Figure 5: Results for particle velocity, 2D potential well test problems with magnetic field $\bm{B} = 100\,\hat{\bm{z}}$: reference solution orbits (first row) and precision diagrams (second row). Boris/Buneman step sizes are $h = 10^{-3}$, $10^{-4}$, $10^{-5}$, $10^{-6}$ for the quadratic potential problem and $h = 10^{-4}$, $10^{-5}$, $10^{-6}$, $10^{-7}$ for the cubic/quartic potential problems. Exponential integrators step sizes are $h =$ 100, 10, 1, $10^{-1}$ for the quadratic potential problem and $h = 10^{-2}$, $10^{-3}$, $10^{-4}$, $10^{-5}$ for the cubic/quartic potential problems.

Theorems & Definitions (2)

• Remark 1
• Remark 2