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On the approximation of the von Neumann equation in the semiclassical limit. Part II : numerical analysis

Francis Filbet, François Golse

Abstract

This paper is devoted to the numerical analysis of the Hermite spectral method proposed in [14], which provides, in the semiclassical limit, an asymptotic preserving approximation of the von Neumann equation. More precisely, it relies on the use of so-called Weyl's variables to effectively address the stiffness associated to the equation. Then by employing a truncated Hermite expansion of the density operator, we successfully manage this stiffness and provide error estimates by leveraging the propagation of regularity in the exact solution.

On the approximation of the von Neumann equation in the semiclassical limit. Part II : numerical analysis

Abstract

This paper is devoted to the numerical analysis of the Hermite spectral method proposed in [14], which provides, in the semiclassical limit, an asymptotic preserving approximation of the von Neumann equation. More precisely, it relies on the use of so-called Weyl's variables to effectively address the stiffness associated to the equation. Then by employing a truncated Hermite expansion of the density operator, we successfully manage this stiffness and provide error estimates by leveraging the propagation of regularity in the exact solution.

Paper Structure

This paper contains 17 sections, 9 theorems, 188 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Quantum dynamics in Weyl's variables
  3. Semiclassical limit
  4. Hermite Spectral method
  5. Hermite approximation of the von Neumann equation
  6. Hermite approximation of the semiclassical limit equation
  7. Preliminary results
  8. Analysis of the semiclassical model
  9. Propagation of regularity for the semiclassical limit equation
  10. Proof of Theorem \ref{['th:01']}
  11. Proof of Theorem \ref{['th:errorSC']}
  12. Analysis of the von Neumann equation
  13. Propagation of regularity on the von Neumann equation
  14. Proof of Theorem \ref{['th:errorvN']}
  15. Numerical simulations
  16. ...and 2 more sections

Key Result

Proposition 1.1

Consider $R^\hbar$ the solution to the von Neumann equation vNvW. Then, we have for each $t\geq 0$,

Figures (1)

  • Figure 5.1: $L^2$ error in log scale.

Theorems & Definitions (15)

  • Proposition 1.1
  • proof
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 5 more