A Conservative Cascade Semi-Lagrangian Method for Solving the Vlasov Equation

TL;DR

This work tackles the challenge of solving the Vlasov equation with strict conservation in high-dimensional phase space by applying the conservative cascade semi-Lagrangian (CCSL) method. It analyzes the method's second-order spatial accuracy and identifies geometric backtracking as the dominant error source, then introduces two enhancements: a freestream-preserving correction that enforces exact volume conservation under divergence-free flows and a maximum-principle limiter that preserves positivity while suppressing spurious oscillations. The enhanced CCSL achieves superior mass and L1 conservation, maintains positivity, and demonstrates long-time stability across linear transport, guiding-center dynamics, and relativistic Vlasov–Maxwell tests, outperforming CSL and BSL schemes. The results indicate that the improved CCSL framework provides robust, high-fidelity plasma kinetic simulations with good parallel scalability, enabling more reliable kinetic modeling in plasma physics applications.

Abstract

The cascade remapping method, originally proposed by Nair et al. (2002) for atmospheric modeling, enables efficient and mass conservative semi Lagrangian (SL) transport through successive one dimensional remapping. While widely used in geophysical flows, its application to plasma kinetics remains limited. To exploit its potential advantages in conservation and scalability, this work applies the conservative cascade semi Lagrangian (CCSL) scheme to the Vlasov equation and related plasma models. A consistency analysis shows that the scheme attains second order spatial accuracy, with the dominant error arising from the geometric approximation of the backtracked region. Moreover, two improvements are introduced: a freestream preserving correction that ensures exact volume conservation, and a maximum principle limiter that suppresses spurious oscillations while maintaining positivity and mass conservation. Numerical tests, including linear advection, guiding center, and relativistic Vlasov Maxwell models, confirm the high accuracy, robustness, and long term stability of the improved CCSL method. Compared with the conservative semi Lagrangian (CSL) and the backward semi Lagrangian (BSL) schemes, it better preserves physical invariants under divergence free conditions, providing a robust and efficient framework for high-fidelity plasma kinetic simulations with good parallel scalability.

Figures (14)

• Figure 1: Schematic diagram of the CCSL method for 2D case.
• Figure 2: Schematic diagram of the backtracked cells (red), the clipped and spliced true intermediate cells (blue), and the true backtracked region (black).
• Figure 3: Coordinate transformation in 2D CCSL method
• Figure 4: CCSL scheme grid configuration with standard and reconstructed coordinates.
• Figure 5: Swirling deformation flow experiment with complex initial condition Eq. \ref{['Eq. discontinuous initial']} ($N_x = N_y = 160,\;\Delta t=0.03125$, $\text{CFL}=5.0$), corresponding to times $t = 0$, $t = 1$, and $t = 2$

Theorems & Definitions (13)

• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof