The positivity-preserving high-order semi-Lagrangian spectral volume method for Vlasov-Poisson equations

TL;DR

The paper addresses the challenge of simulating the Vlasov-Poisson system with high fidelity by introducing a high-order semi-Lagrangian spectral-volume (SLSV) method built on operator splitting. The method fuses semi-Lagrangian characteristic tracing with a spectral-volume discretization, and extends to 2D via Strang splitting, while coupling with a Poisson solver and a positivity-preserving limiter. Key contributions include unconditional stability, local mass conservation, positivity preservation, and high-order spatial accuracy demonstrated through extensive 1D/2D tests and VP benchmarks such as Landau damping and two-stream instabilities, including long-time stability at large time steps. The results show that SLSV offers robust, accurate, and efficient long-time VP simulations, with good conservation properties and the ability to handle complex nonlinear phenomena in plasma dynamics.

Abstract

In this paper, a novel high order semi-Lagrangian (SL) spectral volume (SV) method is proposed and studied for nonlinear Vlasov-Poisson (VP) simulations via operator splitting. The proposed algorithm combines both advantages of semi-Lagrangian and spectral volume approaches, exhibiting strong stability, robustness under large time steps, arbitrary high-order accuracy in space, local mass conservation, and positivity preservation. Numerical study of the SLSV method applied to the one-dimensional and two-dimensional transport equations, the Vlasov-Poisson system, the classical benchmark problems including Landau damping and two-stream instabilities is conducted, confirming the effectiveness, accuracy, and robustness of our algorithm in addressing complex nonlinear phenomena.

Figures (19)

• Figure 1: \newlabelFig:upstream0 Schematic illustration for one-dimensional SLSV schemes.
• Figure 1: \newlabelFig:oper_split0 Schematic illustration of the 2D SLSV scheme via Strang splitting with $k = 2$.
• Figure 1: \newlabelFig:con initial0 Plots of the initial profle. The mesh of $400 \times 400$ is used.
• Figure 2: \newlabelFig:rigid cone0 Rigid body rotation: Plots of the numerical solutions of SLSV schemes with initial data Figure \ref{['Fig:con initial']} at a mesh of $160 \times 160$ and $T = 12\pi$. ${\mathcal{Q}}^2$ SLSV-split + WENO limiter is used with CFL = 2.2 (left) and CFL = 10.2 (right).
• Figure 3: \newlabelFig:rigid cone cut0 Rigid body rotation: Plots of 1D cuts of the numerical solution for SLSV schemes with initial data Figure \ref{['Fig:con initial']} at a mesh of $160\times160$ and $T = 12\pi$. Left: numerical solution at $x = 0 + \pi/100$. Right: numerical solution at $y = \pi/2 + \pi/100$.

Theorems & Definitions (9)

• Remark 3.1
• Example 5.1
• Example 5.2
• Example 5.3
• Example 5.4
• Example 5.5
• Example 5.6
• Example 5.7
• Example 5.8