A new approach for solving nonlinear algebraic systems with complementarity conditions. Application to compositional multiphase equilibrium problems

• Published in: Mathematics and computers in simulation 2021-12, Vol.190, p.1243-1274
• Main Authors: Vu, Duc Thach Son, Ben Gharbia, Ibtihel, Haddou, Mounir, Tran, Quang Huy
• Format: Article
• Language: English
• Subjects: Complementarity condition; Interior-point methods; Mathematics; Newton’s method; Numerical Analysis; Phase equilibrium; Unified formulation
• ISSN: 0378-4754
• DOI: 10.1016/j.matcom.2021.07.015

We present a new method to solve general systems of equations containing complementarity conditions, with a special focus on those arising in the thermodynamics of multicomponent multiphase mixtures at equilibrium. Indeed, the unified formulation introduced by Lauser et al. (2011) has recently emerged as a promising way to automatically handle the appearance and disappearance of phases in porous media compositional multiphase flows. From a mathematical viewpoint and after discretization in space and time, this leads to a system consisting of algebraic equations and nonlinear complementarity equations. Due to the nonsmoothness of the latter, semismooth and smoothing methods commonly used for solving such a system are often slow or may not converge at all. This observation led us to design a new strategy called NPIPM (NonParametric Interior-Point Method). Inspired from interior-point methods in optimization, the technique we propose has the advantage of avoiding any parameter management while enjoying theoretical global convergence. This is validated by extensive numerical tests, in which we compare NPIPM to the Newton-min method, the standard reference for almost all reservoir engineers and thermodynamicists.