Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An asymptotic preserving scheme for the quantum Liouville-BGK equation

Romain Duboscq, Olivier Pinaud

TL;DR

This work develops an asymptotic preserving scheme for the quantum Liouville-BGK equation in the diffusion limit, enabling time steps $\delta t$ that are independent of the small parameter $\varepsilon$. The method combines a two-step approach: first, compute the local density via a modified quantum drift-diffusion equation with an $\varepsilon$-dependent right-hand side, and second, reconstruct the density operator using an asymptotic expansion around the local quantum Maxwellian $\vartheta[\mathrm{n}]$ that is uniform in $\varepsilon$. A key ingredient is the auxiliary operator $\sigma^{\varepsilon}$, which decouples the microscopic collision dynamics from the drift-diffusion limit, yielding a uniform-in-$\varepsilon$ modified QDD equation that drives the density update $n^{\varepsilon}_{n+1}$. Under plausible well-posedness and stability assumptions for the QLE and QDD equations and for the map $\mathrm{n} \mapsto \vartheta[\mathrm{n}]$, the paper provides informal yet coherent error estimates showing that the AP scheme recovers the diffusion limit as $\varepsilon \to 0$ and remains accurate for finite $\varepsilon$, with a per-iteration cost comparable to non-AP schemes. This framework is applicable to spatial domains of arbitrary dimension, and it offers a practical route to efficiently simulate quantum kinetic dynamics in regimes bridging microscopic and diffusive scales.

Abstract

We are interested in this work in the numerical resolution of the Quantum Liouville-BGK equation, which arises in the derivation of quantum hydrodynamical models from first principles. Such models are often obtained in some asymptotic limits, for instance a diffusion or a fluid limit, and as a consequence the original Liouville equation contains small parameters. A standard method such as a split-step algorithm is then accurate provided the time step is sufficiently small compared to the asymptotic parameter, which is a severe limitation. In the case of the diffusion limit, we propose a numerical method that is accurate for time steps independent of the small parameter, and which captures well both the microscopic dynamics and the diffusion limit. Our approach is substantiated by an informal theoretical error analysis.

An asymptotic preserving scheme for the quantum Liouville-BGK equation

TL;DR

This work develops an asymptotic preserving scheme for the quantum Liouville-BGK equation in the diffusion limit, enabling time steps that are independent of the small parameter . The method combines a two-step approach: first, compute the local density via a modified quantum drift-diffusion equation with an -dependent right-hand side, and second, reconstruct the density operator using an asymptotic expansion around the local quantum Maxwellian that is uniform in . A key ingredient is the auxiliary operator , which decouples the microscopic collision dynamics from the drift-diffusion limit, yielding a uniform-in- modified QDD equation that drives the density update . Under plausible well-posedness and stability assumptions for the QLE and QDD equations and for the map , the paper provides informal yet coherent error estimates showing that the AP scheme recovers the diffusion limit as and remains accurate for finite , with a per-iteration cost comparable to non-AP schemes. This framework is applicable to spatial domains of arbitrary dimension, and it offers a practical route to efficiently simulate quantum kinetic dynamics in regimes bridging microscopic and diffusive scales.

Abstract

We are interested in this work in the numerical resolution of the Quantum Liouville-BGK equation, which arises in the derivation of quantum hydrodynamical models from first principles. Such models are often obtained in some asymptotic limits, for instance a diffusion or a fluid limit, and as a consequence the original Liouville equation contains small parameters. A standard method such as a split-step algorithm is then accurate provided the time step is sufficiently small compared to the asymptotic parameter, which is a severe limitation. In the case of the diffusion limit, we propose a numerical method that is accurate for time steps independent of the small parameter, and which captures well both the microscopic dynamics and the diffusion limit. Our approach is substantiated by an informal theoretical error analysis.

Paper Structure

This paper contains 17 sections, 2 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment.
  3. The AP scheme
  4. Setting of the problem.
  5. General strategy.
  6. The AP scheme.
  7. Numerical methods.
  8. Computational cost.
  9. Some properties the AP scheme.
  10. Diffusion limit.
  11. Theoretical analysis
  12. Preliminaries.
  13. Proof of Lemma \ref{['lem:expsigma']}.
  14. Proof of Theorem \ref{['th']}.
  15. Step 1: neglecting $\varsigma_1^\varepsilon$.
  16. ...and 2 more sections

Key Result

Lemma 2.1

For any $t> s\geq 0$, the following equality holds where $\varsigma^{\varepsilon}_1=O(\varepsilon^2 \wedge (t-s) )$ and $\nabla \cdot j[\varsigma^{\varepsilon}_1]=O(\varepsilon [\varepsilon^2 \wedge (t-s)])$.

Theorems & Definitions (2)

  • Lemma 2.1
  • Theorem 3.2