Bell John B, Garcia Alejandro L, Williams Sarah A
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Jul;76(1 Pt 2):016708. doi: 10.1103/PhysRevE.76.016708. Epub 2007 Jul 26.
The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several computational fluid dynamics approaches are considered (including MacCormack's two-step Lax-Wendroff scheme and the piecewise parabolic method) and are found to give good results for the variance of momentum fluctuations. However, neither of these schemes accurately reproduces the fluctuations in energy or density. We introduce a conservative centered scheme with a third-order Runge-Kutta temporal integrator that does accurately produce fluctuations in density, energy, and momentum. A variety of numerical tests, including the random walk of a standing shock wave, are considered and results from the stochastic LLNS solver are compared with theory, when available, and with molecular simulations using a direct simulation Monte Carlo algorithm.
朗道 - 栗弗席兹纳维 - 斯托克斯（LLNS）方程通过使用随机通量将热涨落纳入宏观流体动力学。本文研究了完整LLNS方程的显式欧拉离散化。考虑了几种计算流体动力学方法（包括麦科马克两步拉克斯 - 温德罗夫格式和分段抛物线方法），发现它们对于动量涨落的方差给出了良好结果。然而，这些格式都不能准确再现能量或密度的涨落。我们引入了一种带有三阶龙格 - 库塔时间积分器的守恒中心格式，它能准确产生密度、能量和动量的涨落。考虑了各种数值测试，包括驻波激波的随机游走，并将随机LLNS求解器的结果与理论（如果有）以及使用直接模拟蒙特卡罗算法的分子模拟结果进行了比较。
