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An extended Fourier pseudospectral method for the Gross-Pitaevskii equation with low regularity potential

Weizhu Bao, Bo Lin, Ying Ma, Chushan Wang

TL;DR

This work introduces an extended Fourier pseudospectral (eFP) method for the Gross-Pitaevskii equation with low regularity potentials, achieving FS-like accuracy while retaining the computational efficiency of Fourier pseudospectral methods. The key idea is to approximate the nonlinearity by Fourier interpolation and the potential by Fourier projection, enabling fast FFT-based evaluation of the discrete updates. The authors prove optimal $L^2$ and $H^1$ error bounds for the time-splitting eFP scheme under modest regularity assumptions and establish that spatial convergence is optimal with respect to the regularity of the exact solution, while a CFL-type time-step restriction is necessary. Numerical results corroborate the theory, showing superior spatial accuracy for rough potentials and consistent temporal convergence when coupled with time-splitting schemes, with the approach easily adaptable to other temporal integrators. This method offers a robust, efficient tool for simulating Bose-Einstein condensates and related quantum systems with rough potentials.

Abstract

We propose and analyze an extended Fourier pseudospectral (eFP) method for the spatial discretization of the Gross-Pitaevskii equation (GPE) with low regularity potential by treating the potential in an extended window for its discrete Fourier transform. The proposed eFP method maintains optimal convergence rates with respect to the regularity of the exact solution even if the potential is of low regularity and enjoys similar computational cost as the standard Fourier pseudospectral method, and thus it is both efficient and accurate. Furthermore, similar to the Fourier spectral/pseudospectral methods, the eFP method can be easily coupled with different popular temporal integrators including finite difference methods, time-splitting methods and exponential-type integrators. Numerical results are presented to validate our optimal error estimates and to demonstrate that they are sharp as well as to show its efficiency in practical computations.

An extended Fourier pseudospectral method for the Gross-Pitaevskii equation with low regularity potential

TL;DR

This work introduces an extended Fourier pseudospectral (eFP) method for the Gross-Pitaevskii equation with low regularity potentials, achieving FS-like accuracy while retaining the computational efficiency of Fourier pseudospectral methods. The key idea is to approximate the nonlinearity by Fourier interpolation and the potential by Fourier projection, enabling fast FFT-based evaluation of the discrete updates. The authors prove optimal and error bounds for the time-splitting eFP scheme under modest regularity assumptions and establish that spatial convergence is optimal with respect to the regularity of the exact solution, while a CFL-type time-step restriction is necessary. Numerical results corroborate the theory, showing superior spatial accuracy for rough potentials and consistent temporal convergence when coupled with time-splitting schemes, with the approach easily adaptable to other temporal integrators. This method offers a robust, efficient tool for simulating Bose-Einstein condensates and related quantum systems with rough potentials.

Abstract

We propose and analyze an extended Fourier pseudospectral (eFP) method for the spatial discretization of the Gross-Pitaevskii equation (GPE) with low regularity potential by treating the potential in an extended window for its discrete Fourier transform. The proposed eFP method maintains optimal convergence rates with respect to the regularity of the exact solution even if the potential is of low regularity and enjoys similar computational cost as the standard Fourier pseudospectral method, and thus it is both efficient and accurate. Furthermore, similar to the Fourier spectral/pseudospectral methods, the eFP method can be easily coupled with different popular temporal integrators including finite difference methods, time-splitting methods and exponential-type integrators. Numerical results are presented to validate our optimal error estimates and to demonstrate that they are sharp as well as to show its efficiency in practical computations.
Paper Structure (12 sections, 7 theorems, 48 equations, 7 figures)

This paper contains 12 sections, 7 theorems, 48 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A time-splitting extended Fourier pseudospectral method
  3. The FS method and the FS method with quadrature
  4. An extended Fourier pseudospectral (eFP) method
  5. Optimal error bounds for the eFP method
  6. Main results
  7. Proof of the optimal $L^2$-norm error bound
  8. Proof of the optimal $H^1$-norm error bound
  9. Numerical results
  10. Spatial errors
  11. Temporal errors
  12. Conclusion

Key Result

Theorem 3.1

Assume that $V \in L^\infty(\Omega)$ and $\psi \in C([0, T]; H^2_\text{\rm per}(\Omega)) \cap C^1([0, T]; L^2(\Omega))$. There exists $h_0 > 0$ sufficiently small such that when $0 < h < h_0$ and $\tau \leq h^2/\pi$, we have In addition, if $V \in W^{1, 4}(\Omega) \cap H^1_\text{\rm per}(\Omega)$ and $\psi \in C([0, T]; H^3_\text{\rm per}(\Omega)) \cap C^1([0, T]; H^1(\Omega))$, we have

Figures (7)

  • Figure 1: Examples of low regularity potential: (a) square-well potential and power-law potential with order $0.75$ in 1D, and (b) square-well potential combined with a harmonic potential in 2D.
  • Figure 2: Spatial errors in $L^2$- and $H^1$-norm of the eFP method for the GPE \ref{['NLSE']} with $L^\infty$-potential $V = V_1$.
  • Figure 3: Spatial errors in $L^2$- and $H^1$-norm of the eFP method for the GPE \ref{['NLSE']} with $W^{1, 4}$-potential $V = V_2$.
  • Figure 4: Spatial errors in $L^2$- and $H^1$-norm of the eFP method for the GPE \ref{['NLSE']} with $H^2$-potential $V = V_3$.
  • Figure 5: Spatial errors in $L^2$- and $H^1$-norm of the eFP method for the GPE \ref{['NLSE']} with $H^3$-potential $V = V_4$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Theorem 3.1
  • Remark 3.1: Optimal spatial convergence
  • Lemma 3.1
  • Proposition 3.1: Local truncation error
  • Proposition 3.2: Stability
  • Proposition 3.3: Local truncation error
  • Proposition 3.4: Stability
  • Remark 3.2
  • Remark 3.3: Generalization to the Strang splitting
  • Theorem 3.2
  • ...and 1 more