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

Jean-Pierre Minier 1 , Eric Peirano 2 ,  and Sergio Chibbaro 3
  • 1 Electricité de France, MFTT, 6 Quati Watier 78400 France.
  • 2 ADEME-Der, 500 Route des Lucioles 06560 Valbonne France.
  • 3 Dipartimento di fisica and INFN, Università di Cagliari, Cittadella Universitaria Monserrato CA, Italy.
  • 4 E-mail: Jean-Pierre.Minier@edf.fr, eric.peirano@ademe.fr, chibbaro@chi80bk.der.edf.fr

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.

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This quarterly published journal presents original articles on the theory and applications of Monte Carlo and Quasi-Monte Carlo methods. Launched in 1995 the journal covers all stochastic numerics topics with emphasis on the theory of Monte Carlo methods and new applications in all branches of science and technology.