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Fourth-order uniformly accurate integrators with long time near conservations for the nonlinear Dirac equation in the nonrelativistic regime

Lina Wang, Bin Wang, Jiyong Li

TL;DR

The paper tackles the nonlinear Dirac equation in the nonrelativistic regime, where temporal oscillations scale as $O(\varepsilon^{2})$ and standard time stepping becomes inefficient. By combining a two-scale formulation with exponential integrators, the authors derive two fourth-order uniformly accurate schemes, SEP-TS4 (symmetric) and EEP-TS4 (explicit), and provide a rigorous uniform accuracy analysis. They prove fourth-order temporal convergence independent of $\varepsilon$ and, for the symmetric scheme, long-time near conservation of energy and mass via a modulated Fourier expansion; the methods are extendable to higher dimensions and magnetic potentials. Numerical experiments in 1D and 2D Dirac equations validate the theory, showing spectral spatial accuracy, fourth-order temporal accuracy, and robust long-time behavior, including soliton dynamics. The work offers practically efficient, structure-preserving tools for highly oscillatory Dirac dynamics with potential broad applications in quantum simulations and related fields.

Abstract

In this paper, we propose two novel fourth-order integrators that exhibit uniformly high accuracy and long-term near conservations for solving the nonlinear Dirac equation (NLDE) in the nonrelativistic regime. In this regime, the solution of the NLDE exhibits highly oscillatory behavior in time, characterized by a wavelength of O($\varepsilon^{2}$) with a small parameter $\varepsilon>0$. To ensure uniform temporal accuracy, we employ a two-scale approach in conjunction with exponential integrators, utilizing operator decomposition techniques for the NLDE. The proposed methods are rigorously proved to achieve fourth-order uniform accuracy in time for all $\varepsilon\in (0,1]$. Furthermore, we successfully incorporate symmetry into the integrator, and the long-term near conservation properties are analyzed through the modulated Fourier expansion. The proposed schemes are readily extendable to linear Dirac equations incorporating magnetic potentials, the dynamics of traveling wave solutions and the two/three-dimensional Dirac equations. The validity of all theoretical ndings and extensions is numerically substantiated through a series of numerical experiments.

Fourth-order uniformly accurate integrators with long time near conservations for the nonlinear Dirac equation in the nonrelativistic regime

TL;DR

The paper tackles the nonlinear Dirac equation in the nonrelativistic regime, where temporal oscillations scale as and standard time stepping becomes inefficient. By combining a two-scale formulation with exponential integrators, the authors derive two fourth-order uniformly accurate schemes, SEP-TS4 (symmetric) and EEP-TS4 (explicit), and provide a rigorous uniform accuracy analysis. They prove fourth-order temporal convergence independent of and, for the symmetric scheme, long-time near conservation of energy and mass via a modulated Fourier expansion; the methods are extendable to higher dimensions and magnetic potentials. Numerical experiments in 1D and 2D Dirac equations validate the theory, showing spectral spatial accuracy, fourth-order temporal accuracy, and robust long-time behavior, including soliton dynamics. The work offers practically efficient, structure-preserving tools for highly oscillatory Dirac dynamics with potential broad applications in quantum simulations and related fields.

Abstract

In this paper, we propose two novel fourth-order integrators that exhibit uniformly high accuracy and long-term near conservations for solving the nonlinear Dirac equation (NLDE) in the nonrelativistic regime. In this regime, the solution of the NLDE exhibits highly oscillatory behavior in time, characterized by a wavelength of O() with a small parameter . To ensure uniform temporal accuracy, we employ a two-scale approach in conjunction with exponential integrators, utilizing operator decomposition techniques for the NLDE. The proposed methods are rigorously proved to achieve fourth-order uniform accuracy in time for all . Furthermore, we successfully incorporate symmetry into the integrator, and the long-term near conservation properties are analyzed through the modulated Fourier expansion. The proposed schemes are readily extendable to linear Dirac equations incorporating magnetic potentials, the dynamics of traveling wave solutions and the two/three-dimensional Dirac equations. The validity of all theoretical ndings and extensions is numerically substantiated through a series of numerical experiments.

Paper Structure

This paper contains 12 sections, 2 theorems, 98 equations, 17 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Formulation of the method
  3. Transformation of the equation
  4. Two-scale formulation and its initial value
  5. Full discretization
  6. Uniform accuracy and long-time analysis
  7. Error estimates
  8. Long-time analysis
  9. Numerical experiments
  10. One-dimensional Dirac equation
  11. Two-dimensional Dirac equation
  12. Conclusion

Key Result

Theorem 3.1

\newlabeltheo-10 (Uniform accuracy) Under the assumptions stated above, there exist $t_{0}>0$, $h_{0}>0$ and $\tau_{0}>0$ independent of $\varepsilon$ such that for any $0< \varepsilon \leq 1$, when $0<\triangle t\leq t_{0}$, $0<\triangle x\leq h_{0}$ and $0<\triangle \tau\leq \tau_{0}$ ,the globa where $\nu>8+d/2$ and $m_{0}\geq 1$.

Figures (17)

  • Figure 1: The solution $\phi_{1}(t=2,x)$ and $\phi_{1}(t,x=0)$ of the Dirac equation \ref{['equ-1']} with $d=1$ for different $\varepsilon$ (Re($\phi_{1}$) denotes the real part of $\phi_{1}$).
  • Figure 1: Problem 1. Spatial error of NLDE \ref{['equ-1']} in 1D at $t=1$ under different $\varepsilon$ in $\tau$-direction (left) and in $x$-direction (right).
  • Figure 2: Problem 1. Temporal error of NLDE \ref{['equ-1']} in 1D at $t=1$ under different $\varepsilon$.
  • Figure 3: Problem 1. Energy error (top) and mass error (bottom) of NLDE \ref{['equ-1']} in 1D under different $\varepsilon$ and $\triangle t$.
  • Figure 4: Problem 2. Temporal error of NLDE \ref{['equ-3-6-3']} in 1D at $t=1$ under different $\varepsilon$.
  • ...and 12 more figures

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 3.1
  • Remark 3.2
  • Proof 1
  • Theorem 3.3
  • Proof 2