A high-order energy-conserving semi-Lagrangian discontinuous Galerkin method for the Vlasov-Ampere system

TL;DR

The Vlasov-Ampère system poses a challenge for long-time, high-fidelity simulations due to dimensionality and stiffness. This work develops a high-order energy-conserving semi-Lagrangian DG method (ECSLDG) that combines a semi-Lagrangian spatial discretization with an energy-preserving time integrator and high-order operator splitting to remove the CFL constraint and achieve unconditional stability. The scheme exactly conserves mass and total energy at the fully discrete level, improves Gauss's law enforcement over low-order methods, and attains high-order accuracy in space and time, validated by comprehensive numerical experiments including weak and strong Landau damping and two-stream instabilities. The ECSLDG framework offers robust multiscale plasma simulations and can be extended to multidimensional VA problems and to the full Vlasov-Maxwell system.

Abstract

In this paper, we propose a high-order energy-conserving semi-Lagrangian discontinuous Galerkin(ECSLDG) method for the Vlasov-Ampere system. The method employs a semi-Lagrangian discontinuous Galerkin scheme for spatial discretization of the Vlasov equation, achieving high-order accuracy while removing the Courant-Friedrichs-Lewy (CFL) constraint. To ensure energy conservation and eliminate the need to resolve the plasma period, we adopt an energy-conserving time discretization introduced by Liu et al. [J. Comput. Phys., 492 (2023), 112412]. Temporal accuracy is further enhanced through a high-order operator splitting strategy, yielding a method that is high-order accurate in both space and time. The resulting ECSLDG scheme is unconditionally stable and conserves both mass and energy at the fully discrete level, regardless of spatial or temporal resolution. Numerical experiments demonstrate the accuracy, stability, and conservation properties of the proposed method. In particular, the method achieves more accurate enforcement of Gauss's law and improved numerical fidelity over low-order schemes, especially when using a large CFL number.

Figures (13)

• Figure 3.1: The integral region $\tilde{I}_j(t)\times [t_m,t_{m+1}]$ related to space-time variables.
• Figure 4.1: Weak Landau damping: $N_x=N_v=128$ and $T=5$. $L^1$ and $L^2$ errors of the velocity distribution function $f$, and $L^2$ errors of the electric field $E$ for the ECSLDG method under different CFL numbers.
• Figure 4.2: Weak Landau damping: $N_x=N_v=128$ and $T=50$. Time evolution of the electric energy (a) and the total energy (b).
• Figure 4.3: Strong Landau damping: $N_x=N_v=128$ and $\text{CFL}=10$. Time evolution of the electric energy (a) and the absolute value of the relative deviation of the total energy (b).
• Figure 4.4: Strong Landau damping: $N_x=N_v=128$. Temporal evolution of Gauss law's residuals for the ECSLDG method with different splitting schemes at $\text{CFL}=10$ (a) and $\text{CFL}=20$ (b).

Theorems & Definitions (14)

• Remark 3.1
• Remark 3.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof