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

TL;DR

This work tackles the challenge of resolving fine-scale phase-space structures in the 1D+1V Vlasov–Poisson system by representing the solution through backward flow maps and their semigroup composition, instead of relying on dense phase-space grids. It introduces a hybrid semi-Lagrangian scheme that combines Numerical Flow Iteration (NuFI), which iteratively builds accurate, conservative flow maps, with the Characteristic Mapping Method (CMM), which uses compositional submaps for efficient remapping. The key contributions are (i) a hybrid framework that reduces memory and CPU time by replacing long NuFI sequences with CMM submaps at chosen intervals, (ii) rigorous analysis of error sources combining $\mathcal{O}(\tau^2)$ time stepping and interpolation errors $\mathcal{O}(1/N_{\boldsymbol{\chi}}^{\alpha})$, and (iii) numerical benchmarks on Landau damping and the two-stream instability showing preserved invariants, improved non-diffusive behavior, and substantial memory/computational advantages. The approach enables high-resolution, long-time simulations with zoom capability in 1D+1V VP and holds promise for extension to higher dimensions, collisions, and advanced boundary treatments.

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.

Figures (13)

• Figure 1: Illustration of the backward map, as a composition of two submaps. The blue line illustrates one characteristic curve.
• Figure 2: Graphical sketch of the NuFI algorithm. The red arrows indicate the NuFI iteration \ref{['algo:Nufi_iteration']} at which all $(x,v)$ are evaluated in a streamline fashion, without storing $f$.
• Figure 3: Incompressibility error using Lagrange interpolation. a) maximum error as a function of the number of maps. b): Decay of the $L_\infty$-error as a function of the map grid resolution $N_{\boldsymbol{\chi}}$.
• Figure 4: Schematic of the backward flow $(x^*,v^*)=\boldsymbol{\Phi}_0^t(x,v)$ for two consecutive iterations with the hybrid CMM-NuFI method. Given a point in phase space $(x,v)$, the characteristics are traced back to their footpoints $(x^*,v^*)$ using an iterative flowmaps $\boldsymbol{\Psi}$ (green arrows) and compositional flowmaps $\boldsymbol{\chi}$ (blue arrows). The two consecutive iterations visualize the remapping. After four NuFI steps (green), the resulting map is replaced in the next iteration by a single large submap (dark blue) and concatenated with the rest of the submaps.
• Figure 5: Spatial convergence for the Landau damping test case at $T=2$ with $\tau=0.1$ using different interpolation schemes. The reference solution is pure NuFI with a spatial resolution $N_{f}=512$.

Theorems & Definitions (3)

• Theorem 2.1
• Remark 1
• Example 1