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

## Abstract

The paper proposes a numerical technique within the Lagrangian description for propagating the quantum fluid dynamical (QFD) equations in terms of the Madelung field variables R and S, which are connected to the wave function via the transformation {psi}=exp{l_brace}(R+iS)/({Dirac_h}/2{pi})(right brace). The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density {rho}=|{psi}|{sup 2} and on the structure of R=({Dirac_h}/2{pi})/2 ln {rho} generally being simpler and smoother than {rho}. The spatially smooth functions R and S are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme. Examples of two-dimensional model systems show that the method rivals, in both efficiency and accuracy, the split-operator and Chebychev expansion methods. The results on a three-dimensional model system indicates that the present method is superior to the existing ones, especially, for its low storage requirement and its uniform accuracy. The advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool. (c) 2000 The American Physical Society.

- Authors:

- Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009 (United States)
- Department of Mathematics, Koc University, 80860 Istanbul, (Turkey)

- Publication Date:

- OSTI Identifier:
- 20216398

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

- Additional Journal Information:
- Journal Volume: 61; Journal Issue: 5; Other Information: PBD: May 2000; Journal ID: ISSN 1063-651X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; QUANTUM FLUIDS; LAGRANGIAN FUNCTION; INTERPOLATION; NUMERICAL ANALYSIS; FLOW MODELS; WAVE FUNCTIONS; DISTRIBUTION FUNCTIONS; THEORETICAL DATA

### Citation Formats

```
Hu, Xu-Guang, Ho, Tak-San, Rabitz, Herschel, and Askar, Attila.
```*Solution of the quantum fluid dynamical equations with radial basis function interpolation*. United States: N. p., 2000.
Web. doi:10.1103/PhysRevE.61.5967.

```
Hu, Xu-Guang, Ho, Tak-San, Rabitz, Herschel, & Askar, Attila.
```*Solution of the quantum fluid dynamical equations with radial basis function interpolation*. United States. doi:10.1103/PhysRevE.61.5967.

```
Hu, Xu-Guang, Ho, Tak-San, Rabitz, Herschel, and Askar, Attila. Mon .
"Solution of the quantum fluid dynamical equations with radial basis function interpolation". United States. doi:10.1103/PhysRevE.61.5967.
```

```
@article{osti_20216398,
```

title = {Solution of the quantum fluid dynamical equations with radial basis function interpolation},

author = {Hu, Xu-Guang and Ho, Tak-San and Rabitz, Herschel and Askar, Attila},

abstractNote = {The paper proposes a numerical technique within the Lagrangian description for propagating the quantum fluid dynamical (QFD) equations in terms of the Madelung field variables R and S, which are connected to the wave function via the transformation {psi}=exp{l_brace}(R+iS)/({Dirac_h}/2{pi})(right brace). The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density {rho}=|{psi}|{sup 2} and on the structure of R=({Dirac_h}/2{pi})/2 ln {rho} generally being simpler and smoother than {rho}. The spatially smooth functions R and S are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme. Examples of two-dimensional model systems show that the method rivals, in both efficiency and accuracy, the split-operator and Chebychev expansion methods. The results on a three-dimensional model system indicates that the present method is superior to the existing ones, especially, for its low storage requirement and its uniform accuracy. The advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool. (c) 2000 The American Physical Society.},

doi = {10.1103/PhysRevE.61.5967},

journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},

issn = {1063-651X},

number = 5,

volume = 61,

place = {United States},

year = {2000},

month = {5}

}