A Hybrid semi-Lagrangian Flow Mapping Approach for Vlasov Systems: Combining Iterative and Compositional Flow Maps

TL;DR

This paper introduces a hybrid semi-Lagrangian scheme for Vlasov systems that combines iterative and compositional flow maps to improve accuracy, efficiency, and conservation in numerical simulations.

Contribution

The paper presents a novel hybrid method that merges NuFI and CMM techniques, enhancing computational efficiency and conservation in semi-Lagrangian Vlasov solvers.

Findings

The hybrid scheme reduces storage requirements compared to traditional methods.

It maintains high accuracy and conservation properties in numerical experiments.

The approach effectively balances memory usage and computational cost.

Abstract

We propose a hybrid semi-Lagrangian scheme for the Vlasov--Poisson equation that combines the Numerical Flow Iteration (NuFI) method with the Characteristic Mapping Method (CMM). Both approaches exploit the semi-group property of the underlying diffeomorphic flow, enabling the reconstruction of solutions through flow maps that trace characteristics back to their initial positions. NuFI builds this flow map iteratively, preserving symplectic structure and conserving invariants, but its computational cost scales quadratically with time. Its advantage lies in a compact, low-dimensional representation depending only on the electric field. In contrast, CMM achieves low computational costs when remapping by composing the global flow map from explicitly stored submaps. The proposed hybrid method merges these strengths: NuFi is employed for accurate and conservative local time stepping, while CMM efficiently propagates the solution through submap composition. This approach reduces storage requirements, maintains accuracy, and improves structural properties. Numerical experiments demonstrate the effectiveness of the scheme and highlight the trade-offs between memory usage and computational cost. We benchmark against a semi-Lagrangian predictor-corrector scheme used in modern gyrokinetic codes, evaluating accuracy and conservation properties.