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

TL;DR

This paper introduces an improved conservative cascade semi-Lagrangian method for solving the Vlasov equation, enhancing accuracy, stability, and conservation properties in plasma kinetic simulations.

Contribution

It adapts and improves the cascade remapping scheme for plasma models, incorporating volume preservation and positivity limiters, demonstrating superior invariants preservation and scalability.

Findings

Achieves second order spatial accuracy.

Ensures exact volume conservation with correction.

Maintains positivity and suppresses oscillations.

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.