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(R + iS)/[symbol: see text]. The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density rho = magnitude of psi 2 and on the structure of R = [symbol: see text]/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.