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Solution of the quantum fluid dynamical equations with radial basis function interpolation

Hu1, Ho, Rabitz

  • 1Department of Chemistry, Princeton University, New Jersey 08544-1009, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|October 14, 2000
PubMed
Summary

A new numerical technique for quantum fluid dynamics (QFD) uses Madelung variables for efficient and accurate simulations. This method offers low storage and uniform accuracy, outperforming existing approaches for complex systems.

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Area of Science:

  • Quantum mechanics
  • Computational fluid dynamics
  • Numerical analysis

Background:

  • The quantum fluid dynamical (QFD) equations describe quantum systems using fluid-like variables.
  • Traditional methods for solving QFD equations can be computationally intensive and require significant storage.
  • The Madelung transformation offers an alternative representation of the wave function.

Purpose of the Study:

  • To develop a novel numerical technique for propagating the QFD equations.
  • To utilize the Madelung field variables (R and S) for enhanced computational efficiency and accuracy.
  • To provide a practical computational tool for higher-dimensional quantum systems.

Main Methods:

  • A Lagrangian description is employed for numerical propagation.

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  • The technique leverages the properties of Madelung variables, where R and S are smoother than probability density.
  • Multivariate radial basis function interpolation is used for implementing the numerical scheme.

    • Main Results:

    • The proposed method demonstrates comparable or superior efficiency and accuracy to split-operator and Chebyshev expansion methods in 2D models.
    • In 3D models, the new method shows superiority, particularly in low storage requirements and uniform accuracy.
    • The technique is well-suited for spatially smooth functions R and S.

    Conclusions:

    • The developed numerical technique offers a robust and efficient approach for solving QFD equations.
    • The method's advantages are expected to increase with system dimensionality.
    • This provides a promising computational tool for advanced quantum fluid dynamics simulations.