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<p>In this work, we develop and analyze a new class of particle methods for the Vlasov–Poisson–Fokker–Planck system with a strongly magnetized, slowly varying external magnetic field under the asymptotic scaling known as the maximal ordering scaling (MOS). The proposed approach addresses two fundamental challenges: (1) the curse of dimensionality, which we mitigate through particle methods while preserving the system’s asymptotic properties, and (2) the temporal step size limitation imposed by the small Larmor radius in strong magnetic fields and the linear Lenard-Bernstein collision operator, which we overcome through semi-implicit Milstein discretization schemes in the MOS case. The computational cost of these methods is on par with that of the explicit scheme. We establish the theoretical foundations of our method, proving its asymptotic-preserving characteristics and uniform convergence through rigorous mathematical analysis. These theoretical results are complemented by numerical experiments that validate the method’s effectiveness in the MOS regime considered in this work. Our findings demonstrate that the proposed numerical framework accurately captures the leading-order guiding-center dynamics, in particular the dominant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(E \times B\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>×</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> drift behavior that governs the asymptotic limit, while maintaining computational efficiency.</p>

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Asymptotic-Preserving Particle-In-Cell Method for the Magnetized Vlasov–Poisson–Fokker–Planck Equation

  • 期刊论文

  • 发表日期：

    2026年6月22日

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  • Anjiao Gu,
  • Xiaojiang Zhang

摘要

In this work, we develop and analyze a new class of particle methods for the Vlasov–Poisson–Fokker–Planck system with a strongly magnetized, slowly varying external magnetic field under the asymptotic scaling known as the maximal ordering scaling (MOS). The proposed approach addresses two fundamental challenges: (1) the curse of dimensionality, which we mitigate through particle methods while preserving the system’s asymptotic properties, and (2) the temporal step size limitation imposed by the small Larmor radius in strong magnetic fields and the linear Lenard-Bernstein collision operator, which we overcome through semi-implicit Milstein discretization schemes in the MOS case. The computational cost of these methods is on par with that of the explicit scheme. We establish the theoretical foundations of our method, proving its asymptotic-preserving characteristics and uniform convergence through rigorous mathematical analysis. These theoretical results are complemented by numerical experiments that validate the method’s effectiveness in the MOS regime considered in this work. Our findings demonstrate that the proposed numerical framework accurately captures the leading-order guiding-center dynamics, in particular the dominant \(E \times B\) E × B drift behavior that governs the asymptotic limit, while maintaining computational efficiency.