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A sixth-order compact time-splitting Fourier pseudospectral method

Weiguo Gao, Zhansi He, Jia Yin

TL;DR

The paper tackles efficiently solving the Dirac equation in 1D/2D without magnetic potentials using high-order time-splitting methods. It introduces a sixth-order compact time-splitting scheme, $S_{6c}$, that exploits a vanishing double commutator to achieve a compact, cost-effective decomposition and uses time-ordering to handle time-dependent potentials. The method delivers sixth-order temporal accuracy and spectral spatial accuracy while conserving discretized mass; numerical tests show substantial efficiency gains over existing fourth- and sixth-order schemes, and the nonrelativistic regime demonstrates a super-resolution property. The work provides a practical high-precision solver applicable to relativistic and nonrelativistic Dirac dynamics with potential relevance to graphene-like systems and strong-field quantum dynamics.

Abstract

In this paper, we propose a novel sixth-order compact time-splitting scheme, denoted as $ S_{6\text{c}}$, for solving the Dirac equation in the absence of external magnetic potentials. This method is easy to implement, and it provides a substantial reduction in computational complexity compared to the existing sixth-order splitting schemes. By incorporating a time-ordering technique, we also extend $S_{6\text{c}}$ to address problems with time-dependent potentials. Comprehensive comparisons with various time-splitting methods show that $S_{6\text{c}}$ exhibits significant advantages in terms of both precision and efficiency. Moreover, numerical results indicate that $S_{6\text{c}}$ maintains the super-resolution property for the Dirac equation in the nonrelativistic regime in the absence of external magnetic potentials.

A sixth-order compact time-splitting Fourier pseudospectral method

TL;DR

The paper tackles efficiently solving the Dirac equation in 1D/2D without magnetic potentials using high-order time-splitting methods. It introduces a sixth-order compact time-splitting scheme, , that exploits a vanishing double commutator to achieve a compact, cost-effective decomposition and uses time-ordering to handle time-dependent potentials. The method delivers sixth-order temporal accuracy and spectral spatial accuracy while conserving discretized mass; numerical tests show substantial efficiency gains over existing fourth- and sixth-order schemes, and the nonrelativistic regime demonstrates a super-resolution property. The work provides a practical high-precision solver applicable to relativistic and nonrelativistic Dirac dynamics with potential relevance to graphene-like systems and strong-field quantum dynamics.

Abstract

In this paper, we propose a novel sixth-order compact time-splitting scheme, denoted as , for solving the Dirac equation in the absence of external magnetic potentials. This method is easy to implement, and it provides a substantial reduction in computational complexity compared to the existing sixth-order splitting schemes. By incorporating a time-ordering technique, we also extend to address problems with time-dependent potentials. Comprehensive comparisons with various time-splitting methods show that exhibits significant advantages in terms of both precision and efficiency. Moreover, numerical results indicate that maintains the super-resolution property for the Dirac equation in the nonrelativistic regime in the absence of external magnetic potentials.
Paper Structure (12 sections, 8 theorems, 97 equations, 3 figures, 9 tables)

This paper contains 12 sections, 8 theorems, 97 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The sixth-order compact time-splitting method ($S_{6\text{c}}$)
  3. Review of the time-splitting methods
  4. The construction of $S_{\text{6c}}$ for the Dirac equation
  5. The full-discretization of $S_{\text{6c}}$
  6. $S_\text{6c}$ with time-independent potentials
  7. $S_\text{6c}$ with time-dependent potentials
  8. Mass conservation
  9. Numerical results
  10. Examples in the classical regime
  11. An example in the nonrelativistic regime
  12. Conclusions and future work

Key Result

Theorem 2.1

If $S(\tau)$ is symmetric, then $\gamma_{2k} = 0$ for all $k$, and $S(\tau)$ is a scheme with even order.

Figures (3)

  • Figure 4.1: Temporal errors for the wave function, the probability density, and the current density with different $T$ for the Dirac equation (\ref{['Dirac1d2d']}) in 2D, with the potential given in (\ref{['V']}) where $\theta(t) \equiv \pi$.
  • Figure 4.2: Temporal errors for the wave function, the probability density, and the current density with different $T$ for the Dirac equation (\ref{['Dirac1d2d']}) in 2D, with the potential given in (\ref{['V']}) where $\theta(t) = \pi + \pi t$.
  • Figure 4.3: Temporal errors for the wave function, the probability density, and the current density with different $T$ for the Dirac equation (\ref{['Dirac1d2d']}) in 2D, with the potential given in (\ref{['V']}) where $\theta(t) = \pi + \pi \cos(\pi t)$.

Theorems & Definitions (14)

  • Definition 2.1
  • Theorem 2.1
  • Definition 2.2: the BCH formula
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.2
  • proof
  • ...and 4 more