Asymptotic-preserving particle-in-cell method for the magnetized Vlasov--Poisson--Fokker--Planck equation

TL;DR

This work develops an asymptotic-preserving stochastic PIC method for the magnetized Vlasov–Poisson–Fokker–Planck equation (MVPFP), addressing high-dimensional phase space and stringent time-step constraints from the Larmor radius. The authors formulate a PIC representation f_h = ∑ α_s δ(x−X_s) δ(v−V_s) and coupled SDEs dX_s = (1/ε) V_s dt, dV_s = (1/ε)[E(X_s) + (b(X_s)/ε) K V_s − (1/τ) V_s] dt + √(2σ/(ετ)) dW, proving asymptotic-preserving properties as ε → 0 and deriving a guiding-center limit dot u = (K E(u))/b_0. They introduce two time integrators, APSI1 and APSI2, that combine semi-implicit schemes with Milstein updates, and establish uniform-in-ε error estimates and weak-order results, supported by numerical benchmarks and a diocotron-instability study demonstrating magnetic confinement and collision effects. The framework enables stable, accurate, long-time simulations in strongly magnetized plasmas, with potential impact on fusion-relevant modeling and multiscale plasma dynamics. Overall, the paper provides rigorous AP analysis, practical AP schemes, and comprehensive computational validation for MVPFP in high-field regimes.

Abstract

In this work, we develop and rigorously analyze a new class of particle methods for the magnetized Vlasov--Poisson--Fokker--Planck system. 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, which we overcome through semi-implicit discretization schemes. 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 extensive numerical experiments that validate the method's effectiveness in long-term simulations. Our findings demonstrate that the proposed numerical framework accurately captures key physical phenomena, particularly the magnetic confinement effects on plasma behavior, while maintaining computational efficiency.

Figures (9)

• Figure 1: Estimates $error_k=\vert x_k^N-u_k^N\vert$ of semi-implicit schemes with time step $\Delta t=\pi/30$ for $\varepsilon=2^{-m}$ and $T=\pi$. (a)(c): APSI1 (\ref{['APSI1']}). (b)(d): APSI2 (\ref{['APSI2']}). (a)(b): $\sigma=\tau=1$. (c)(d): $\sigma=2^{-6},\tau=2^{6}$ (i.e. $\sigma\le\varepsilon\tau$).
• Figure 2: Estimates $error_k=\vert\mathbb{E}(x_k^N)-u_k^N\vert$ of semi-implicit schemes with time step $\Delta t=\pi/30$ for $\varepsilon=2^{-m}$ at final time $T=\pi$ for $10^4$ Brownian paths when $\sigma=\tau=1$. (a)(c): APSI1 (\ref{['APSI1']}). (b)(d): APSI2 (\ref{['APSI2']}). (a)(b): $u^N$ is the numerical solution of \ref{['GCeq1']}. (c)(d): $x(0)=(10,14)^T,v(0)=\mathcal{O}(\varepsilon)$, $u^N$ is the numerical solution of \ref{['GCeq2']}.
• Figure 3: Comparison between the solution of the guiding center model and the expectations of the solution solving by proposed algorithms with time step $\Delta t=\pi/30$ for $\varepsilon=2^{-m}$ at final time $T=\pi$ for $10^4$ Brownian paths when $\sigma=\tau=1$, $x(0)=(10,14)^T$. (a): APSI1 (\ref{['APSI1']}). (b): APSI2 (\ref{['APSI2']}).
• Figure 4: Estimates $error_k=\vert \mathbb{E}(x_k^N)-\mathbb{E}(x_k(T))\vert$ of semi-implicit scheme \ref{['APSI1']} with time step $\Delta t=\frac{\pi}{30}2^{-m}$ at final time $T=\pi$ for $10^4$ Brownian paths when $\sigma=\tau=1$. (a): $\varepsilon=10^{-2}$. (b): $\varepsilon=10^{-4}$.
• Figure 5: Estimates $error_k=\vert \mathbb{E}(x_k^N)-\mathbb{E}(x_k(T))\vert$ of semi-implicit scheme \ref{['APSI2']} with time step $\Delta t=\frac{2\pi}{15}2^{-m}$ at final time $T=\pi$ for $10^4$ Brownian paths when $\sigma=\tau=1$. (a): $\varepsilon=10^{-6}$. (b): $\varepsilon=10^{-8}$.

Theorems & Definitions (21)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Corollary 1
• Theorem 3.1
• proof
• Theorem 3.2