High resolution finite volume scheme for the quantum hydrodynamic equations

The theory of quantum fluid dynamics (QFD) helps nanotechnology engineers to understand the physical effect of quantum forces. Although the governing equations of quantum fluid dynamics and classical fluid mechanics have the same form, there are two numerical simulation problems must be solved in QFD. The first is that the quantum potential term becomes singular and causes a divergence in the numerical simulation when the probability density is very small and close to zero. The second is that the unitarity in the time evolution of the quantum wave packet is significant. Accurate numerical evaluations are critical to the simulations of the flow fields that are generated by various quantum fluid systems. A finite volume scheme is developed herein to solve the quantum hydrodynamic equations of motion, which significantly improve the accuracy and stability of this method. The QFD equation is numerically implemented within the Eulerian method. A third-order modified Osher-Chakravarthy (MOC) upwind-centered finite volume scheme was constructed for conservation law to evaluate the convective terms, and a second-order central finite volume scheme was used to map the quantum potential field. An explicit Runge-Kutta method is used to perform the time integration to achieve fast convergence of the proposed scheme. In order to meet the numerical result can conform to the physical phenomenon and avoid numerical divergence happening due to extremely low probability density, the minimum value setting of probability density must exceed zero and smaller than certain value. The optimal value was found in the proposed numerical approach to maintain a converging numerical simulation when the minimum probability density is 10^-^5 to 10^-^1^2. The normalization of the wave packet remains close to unity through a long numerical simulation and the deviations from 1.0 is about 10^-^4. To check the QFD finite difference numerical computations, one- and two-dimensional particle motions were solved for an Eckart barrier and a downhill ramp barrier, respectively. The results were compared to the solution of the Schrodinger equation, using the same potentials, which was obtained using by a finite difference method. Finally, the new approach was applied to simulate a quantum nanojet system and offer more intact theory in quantum computational fluid dynamics.

[1]  J. H. Weiner,et al.  Particle Method for the Numerical Solution of the Time‐Dependent Schrödinger Equation , 1971 .

[2]  Herschel Rabitz,et al.  Multivariate radial basis interpolation for solving quantum fluid dynamical equations , 2002 .

[3]  D. Bohm.  PROOF THAT PROBABILITY DENSITY APPROACHES //psi//$sup 2$ IN CAUSAL INTERPRETATION OF THE QUANTUM THEORY , 1953 .

[4]  R. Wyatt,et al.  Quantum Wave Packet Dynamics with Trajectories , 1999 .

[5]  N. Grün,et al.  Trajectory method for the time-dependent Schrödinger and Thomas-Fermi equations☆ , 1991 .

[6]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[7]  Herschel Rabitz,et al.  Multidimensional wave packet dynamics within the fluid dynamical formulation of the Schrodinger equation , 1998 .

[8]  S. Osher,et al.  Very High Order Accurate TVD Schemes , 1986 .

[9]  R. Wyatt,et al.  Wavepacket dynamics on arbitrary Lagrangian–Eulerian grids: Application to an Eckart barrier , 2003 .

[10]  Attila Askar,et al.  Quantum fluid dynamics in the Lagrangian representation and applications to photodissociation problems , 1999 .

[11]  N. Grün,et al.  Solution of the time-dependent Schrödinger equation with a trajectory method and application to H+-H scattering , 1982 .

[12]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[13]  H. Chiu,et al.  Aging and mechanical properties of NR/BR blends , 2006 .

[14]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[15]  San-Yih Lin,et al.  An upwind finite-volume scheme with a triangular mesh for conservation laws , 1990 .

[16]  D. Bohm,et al.  The Aharonov-Bohm effect and the quantum potential , 1982 .

[17]  San-Yih Lin,et al.  Upwind finite-volume method with a triangular mesh for conservation laws , 1993 .

[18]  B. Kendrick.  A new method for solving the quantum hydrodynamic equations of motion , 2003 .

[19]  Attila Askar,et al.  Time‐Dependent Perturbation Calculations Based on the Hydrodynamic Analogy to Quantum Mechanics , 1971 .

[20]  H. Rabitz,et al.  Alternating direction implicit technique and quantum evolution within the hydrodynamical formulation of Schrödinger's equation , 1998 .

[21]  Ioana Banicescu,et al.  Solving the hydrodynamic formulation of quantum mechanics: A parallel MLS method , 2001 .

[22]  R. Montgomery,et al.  On the fluid approximation to a nonlinear Schrödinger equation , 1993 .

[23]  Askar,et al.  Solution of the quantum fluid dynamical equations with radial basis function interpolation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  D. Bohm.  A SUGGESTED INTERPRETATION OF THE QUANTUM THEORY IN TERMS OF "HIDDEN" VARIABLES. II , 1952 .

[25]  R. Wyatt.  Quantum wave packet dynamics with trajectories: Application to reactive scattering , 1999 .

[26]  N. Zettili.  Quantum Mechanics: Concepts and Applications , 2001 .