A dynamic domain semi-Lagrangian method for stochastic Vlasov equations

TL;DR

This work introduces a dynamic domain semi-Lagrangian method for stochastic Vlasov equations with transport noise, addressing unbounded velocity support and energy growth by coupling a volume-preserving inverse flow with dynamic velocity-domain adaptation. The method integrates a Lagrange first-order reconstruction on an adaptively enlarging phase-space domain and proves first-order mean-square convergence under suitable regularity, partially resolving a conjecture on stochastic Vlasov solvers. Numerical experiments demonstrate substantial efficiency gains, preservation of positivity and integral invariants, and accurate evolution of mass, momentum, and energy in both linear and Vlasov–Poisson settings. The proposed approach offers a practical, accurate tool for simulating stochastic kinetic models in plasma and astrophysical contexts.

Abstract

We propose a dynamic domain semi-Lagrangian method for stochastic Vlasov equations driven by transport noises, which arise in plasma physics and astrophysics. This method combines the volume-preserving property of stochastic characteristics with a dynamic domain adaptation strategy and a reconstruction procedure. It offers a substantial reduction in computational costs compared to the traditional semi-Lagrangian techniques for stochastic problems. Furthermore, we present the first-order convergence analysis of the proposed method, partially addressing the conjecture in the work [C.-E. Bréhier and D. Cohen, J. Comput. Dyn., 2024] on the convergence order of numerical methods for stochastic Vlasov equations. Several numerical tests are provided to show good performance of the proposed method.

Figures (6)

• Figure 1: Contour plot for Example \ref{['exp:CA']} with the initial density \ref{['eq:TSI']} in the two stream instability problem at times $\{0,1,2,3,4,5\}$: $\sigma\equiv0$(a) and $\sigma(x)=0.5(\cos(2\pi x)+1)$(b).
• Figure 2: Contour plot for Example \ref{['exam:VP']} with the initial density \ref{['eq:TSI']} in the two stream instability problem at times $\{0,6,12,18,24,30\}$: $\sigma\equiv 0$(a) and $\sigma\equiv 0.5$(b).
• Figure 3: (a) Evolution of $L^1(\mathbb T\times[-6,6])$-norm of the numerical solution of the traditional semi-Lagrangian method and (b) Evolution of mass error of Algorithm \ref{['Algo:2']}, for Example \ref{['exp:CA']} with different $\sigma$ and the initial density \ref{['eq:TSI']} in the two stream instability problem along a single sample path.
• Figure 4: Evolution of $L^p(\mathbb{T}\times[-U_n,U_n])$-norms of numerical solutions of Algorithm \ref{['Algo:2']} with \ref{['eq:inv-SS']} and the Euler--Maruyama method for Example \ref{['exp:CA']} with $\sigma(x)=\sin(2\pi x)$ and the initial density \ref{['eq:TSI']} in the two stream instability problem along a single sample path: $p=1$(a) and $p=2$(b).
• Figure 5: Evolution of physical quantities for Example \ref{['exp:CA']} with the initial density \ref{['eq:TSI']} in the two stream instability problem: $E\equiv E_0=1$ and $\sigma\equiv 1$(a), $E\equiv E_0=1$ and $\sigma(x)=0.5(\cos(2\pi x)+1)$(b), and $E(x)=\cos(2\pi x)$ and $\sigma \equiv 1$(c).

Theorems & Definitions (13)

• Remark 2.2
• Proposition 2.3
• proof
• Remark 3.2
• Lemma 4.2
• proof
• Theorem 4.3
• proof
• Remark 4.4
• Corollary 4.5