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Uniform error estimate of an asymptotic preserving scheme for the Lévy-Fokker-Planck equation

Weiran Sun, Li Wang

TL;DR

This work proves a uniform-in-ε error estimate for an asymptotic preserving scheme for the Lévy-Fokker-Planck equation with a fat-tailed equilibrium. By a micro-macro decomposition f = η M + g with η(t,x,v) = h(t,x+εv), and a two-regime analysis (Regime I and Regime II) in relation to Δt and ε, the authors derive regime-specific energy estimates and relaxations that accommodate nonlocal tail effects. They establish a strong L^2 convergence rate to the fractional diffusion limit when ε is small and provide explicit, ε-independent error bounds for the semi-discrete scheme across regimes, with extensions to any dimension and the whole fractional power range. The results advance AP methodology for fractional kinetic models by addressing tail-induced singularities through weighted energies and careful commutator control, yielding practical, robust uniform accuracy across parameter regimes.

Abstract

We establish a uniform-in-scaling error estimate for the asymptotic preserving scheme proposed in \cite{XW21} for the Lévy-Fokker-Planck (LFP) equation. The main difficulties stem from not only the interplay between the scaling and numerical parameters but also the slow decay of the tail of the equilibrium state. We tackle these problems by separating the parameter domain according to the relative size of the scaling $ε$: in the regime where $ε$ is large, we design a weighted norm to mitigate the issue caused by the fat tail, while in the regime where $ε$ is small, we prove a strong convergence of LFP towards its fractional diffusion limit with an explicit convergence rate. This method extends the traditional AP estimates to cases where uniform bounds are unavailable. Our result applies to any dimension and to the whole span of the fractional power.

Uniform error estimate of an asymptotic preserving scheme for the Lévy-Fokker-Planck equation

TL;DR

This work proves a uniform-in-ε error estimate for an asymptotic preserving scheme for the Lévy-Fokker-Planck equation with a fat-tailed equilibrium. By a micro-macro decomposition f = η M + g with η(t,x,v) = h(t,x+εv), and a two-regime analysis (Regime I and Regime II) in relation to Δt and ε, the authors derive regime-specific energy estimates and relaxations that accommodate nonlocal tail effects. They establish a strong L^2 convergence rate to the fractional diffusion limit when ε is small and provide explicit, ε-independent error bounds for the semi-discrete scheme across regimes, with extensions to any dimension and the whole fractional power range. The results advance AP methodology for fractional kinetic models by addressing tail-induced singularities through weighted energies and careful commutator control, yielding practical, robust uniform accuracy across parameter regimes.

Abstract

We establish a uniform-in-scaling error estimate for the asymptotic preserving scheme proposed in \cite{XW21} for the Lévy-Fokker-Planck (LFP) equation. The main difficulties stem from not only the interplay between the scaling and numerical parameters but also the slow decay of the tail of the equilibrium state. We tackle these problems by separating the parameter domain according to the relative size of the scaling : in the regime where is large, we design a weighted norm to mitigate the issue caused by the fat tail, while in the regime where is small, we prove a strong convergence of LFP towards its fractional diffusion limit with an explicit convergence rate. This method extends the traditional AP estimates to cases where uniform bounds are unavailable. Our result applies to any dimension and to the whole span of the fractional power.
Paper Structure (10 sections, 17 theorems, 290 equations, 2 figures)

This paper contains 10 sections, 17 theorems, 290 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Strong asymptotic limit
  3. Two technical lemmas
  4. Well-posedness
  5. Error estimate
  6. Regime I: kinetic regime with $\varepsilon^{2s} \geq \Delta t^{2s\beta}$
  7. Regime II: diffusive and intermediate regime with $\varepsilon^{2s} \leq \Delta t^{ 2s\beta}$
  8. Appendix
  9. Appendix A
  10. Appendix B

Key Result

Theorem 1

Let $f$ and $\rho$ be the solutions to eqn:111 and eqn: limit_system respectively. Let $t_0 > 0$ be a fixed constant. Then for any $t \geq t_0$, we have where $C_{s,t_0}$, $C$ and $C_{t_0}$ are three constants that do not depend on $\varepsilon$.

Figures (2)

  • Figure 1: Illustration of AP schemes.
  • Figure 2: Two regimes separated by the relation between $\varepsilon^{2s}$ and $\Delta t$. Left: linear scale. Right: log scale.

Theorems & Definitions (37)

  • Remark 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • proof : Proof of Theorem \ref{['APlimit']}
  • Proposition 1
  • Lemma 3
  • ...and 27 more