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Weak first- and second-order numerical schemes for stochastic differential equations appearing in Lagrangian two-phase flow modeling

机译：拉格朗日两相流建模中出现的随机微分方程的弱一阶和二阶数值格式

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摘要

Weak first- and second-order numerical schemes are developed to integrate the stochastic differential equations that arise in mean-field - pdf methods (Lagrangian stochastic approach) for modeling polydispersed turbulent two-phase flows. These equations present several challenges, the foremost being that the problem is characterized by the presence of different time scales that can lead to stiff equations, when the smallest time-scale is significantly less than the time-step of the simulation. The numerical issues have been detailed by Minier [Monte Carlo Meth. and Appl. 7 295-310, (2000)] and the present paper proposes numerical schemes that satisfy these constraints. This point is really crucial for physical and engineering applications, where various limit cases can be present at the same time in different parts of the domain or at different times. In order to build up the algorithm, the analytical solutions to the equations are first carried out when the coefficients are constant. By freezing the coefficients in the analytical solutions, first and second order unconditionally stable weak schemes are developed. A prediction/correction method, which is shown to be consistent for the present stochastic model, is used to devise the second-order scheme. A complete numerical investigation is carried out to validate the schemes, having included also a comprehensive study of the different error sources. The final method is demonstrated to have the required stability, accuracy and efficiency.
机译：开发了弱的一阶和二阶数值方案，以积分在均场pdf方法（拉格朗日随机方法）中出现的用于建模多分散湍流两相流的随机微分方程。这些方程式提出了一些挑战，最重要的是，当最小时标明显小于仿真的时间步长时，问题的特征在于存在不同的时标，这些时标会导致僵化的方程式。数值问题由Minier [Monte Carlo Meth。和应用。 7 295-310，（2000）]，本论文提出了满足这些约束的数值方案。对于物理和工程应用程序，这一点确实至关重要，因为在域的不同部分或不同时间可以同时出现各种极限情况。为了建立算法，当系数恒定时，首先对方程进行解析解。通过冻结解析解中的系数，可以开发出一阶和二阶无条件稳定的弱格式。被证明与当前随机模型一致的预测/校正方法被用于设计二阶方案。进行了完整的数值研究以验证方案，其中还包括对不同误差源的全面研究。最终方法被证明具有所需的稳定性，准确性和效率。

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