A Lawson-time-splitting extended Fourier pseudospectral method for the Gross-Pitaevskii equation with time-dependent low regularity potential

TL;DR

This work addresses the Gross-Pitaevskii equation with a time-dependent, spatially low-regularity potential by introducing the LTSeFP method, which combines an extended Fourier pseudospectral spatial discretization with a Lawson-time-splitting exponential integrator in time. The authors prove first-order temporal convergence and optimal spatial accuracy under mild regularity assumptions, with CFL-type time-step restrictions, and show the method's computational cost is comparable to the standard TSFP. Numerical experiments across static, space-time separable, and moving potentials validate the theory and demonstrate superior spatial accuracy and practical efficiency. The approach, including the regularity compensation oscillation (RCO) framework, broadens the applicability of efficient spectral methods to GPE-like models with challenging potentials.

Abstract

We propose a Lawson-time-splitting extended Fourier pseudospectral (LTSeFP) method for the numerical integration of the Gross-Pitaevskii equation with time-dependent potential that is of low regularity in space. For the spatial discretization of low regularity potential, we use an extended Fourier pseudospectral (eFP) method, i.e., we compute the discrete Fourier transform of the low regularity potential in an extended window. For the temporal discretization, to efficiently implement the eFP method for time-dependent low regularity potential, we combine the standard time-splitting method with a Lawson-type exponential integrator to integrate potential and nonlinearity differently. The LTSeFP method is both accurate and efficient: it achieves first-order convergence in time and optimal-order convergence in space in $L^2$-norm under low regularity potential, while the computational cost is comparable to the standard time-splitting Fourier pseudospectral method. Theoretically, we also prove such convergence orders for a large class of spatially low regularity time-dependent potential. Extensive numerical results are reported to confirm the error estimates and to demonstrate the superiority of our method.

Figures (10)

• Figure 4.1: Temporal errors of the LTSeFP method for the GPE \ref{['NLSE']} with low regularity potential \ref{['eq:poten_0']}: (a) $L^2$-norm errors and (b) $H^1$-norm errors
• Figure 4.2: Spatial errors of (a) LTSeFP method and (b) LTSFP method for the GPE \ref{['NLSE']} with low regularity potential \ref{['eq:poten_0']}
• Figure 4.3: Temporal errors of the LTSeFP method for the GPE \ref{['NLSE']} with low regularity potential \ref{['eq:poten_1']}: (a) $L^2$-norm errors and (b) $H^1$-norm errors
• Figure 4.4: Spatial errors of (a) LTSeFP method and (b) LTSFP method for the GPE \ref{['NLSE']} with low regularity potential \ref{['eq:poten_1']}
• Figure 4.5: Temporal errors of the LTSeFP method for the GPE \ref{['NLSE']} with low regularity potential \ref{['eq:poten_2']}: (a) $L^2$-norm errors and (b) $H^1$-norm errors

Theorems & Definitions (12)

• Remark 2.1
• Remark 2.2: Fourier pseudospectral method
• Remark 2.3
• Theorem 3.1
• Remark 3.2
• Lemma 3.3: Lemma 3.1 of bao2023_eFP
• Proposition 3.4: Error decomposition
• proof
• Lemma 3.5
• Proposition 3.6: $L^\infty$-conditional $L^2$-stability estimate