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High resolution finite volume scheme for the quantum hydrodynamic equations

机译：量子流体动力学方程的高分辨率有限体积方案

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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(-12). 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. Crown Copyright (c) 2008 Published by Elsevier Inc. All rights reserved.
机译：量子流体动力学（QFD）理论有助于纳米技术工程师了解量子力的物理作用。尽管量子流体动力学的控制方程和经典流体力学的控制方程具有相同的形式，但在QFD中必须解决两个数值模拟问题。首先是当概率密度非常小且接近于零时，量子势项变得奇异并在数值模拟中引起发散。第二个是量子波包的时间演化的统一性很重要。精确的数值评估对于各种量子流体系统产生的流场的仿真至关重要。本文开发了一种有限体积方案来解决运动的量子流体力学方程，从而显着提高了该方法的准确性和稳定性。 QFD方程在欧拉方法中以数值方式实现。针对守恒律，建立了三阶改进的Osher-Chakravarthy（MOC）迎风中心有限体积方案，以评估对流项，并使用二阶中心有限体积方案绘制了量子势场。使用显式Runge-Kutta方法执行时间积分，以实现所提出方案的快速收敛。为了满足数值结果可以符合物理现象并避免由于极低的概率密度而发生数值偏差，概率密度的最小值设置必须大于零且小于一定值。当最小概率密度为10（-5）到10（-12）时，在建议的数值方法中找到了最佳值以保持收敛的数值模拟。通过长时间的数值模拟，波包的归一化保持接近统一，并且与1.0的偏差约为10（-4）。为了检查QFD有限差分数值计算，分别针对Eckart屏障和下坡斜坡屏障求解了一维和二维粒子运动。将结果与使用有限差分法获得的相同电势与Schrodinger方程的解进行比较。最后，该新方法被应用于模拟量子纳米射流系统，并在量子计算流体动力学方面提供了更多完整的理论。 Crown版权所有（c）2008，由Elsevier Inc.发行。保留所有权利。

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