A Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of State

Xiaolin Fan, Jisheng Kou, Zhonghua Qiao, Shuyu Sun

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This paper presents a componentwise convex splitting scheme for numerical simulation of multicomponent two-phase fluid mixtures in a closed system at constant temperature, which is modeled by a diffuse interface model equipped with the Van der Waals and the Peng-Robinson equations of state (EoS). The Van der Waals EoS has a rigorous foundation in physics, while the Peng-Robinson EoS is more accurate for hydrocarbon mixtures. First, the phase field theory of thermodynamics and variational calculus are applied to a functional minimization problem of the total Helmholtz free energy. Mass conservation constraints are enforced through Lagrange multipliers. A system of chemical equilibrium equations is obtained which is a set of second-order elliptic equations with extremely strong nonlinear source terms. The steady state equations are transformed into a transient system as a numerical strategy on which the scheme is based. The proposed numerical algorithm avoids the indefiniteness of the Hessian matrix arising from the second-order derivative of homogeneous contribution of total Helmholtz free energy; it is also very efficient. This scheme is unconditionally componentwise energy stable and naturally results in unconditional stability for the Van der Waals model. For the Peng-Robinson EoS, it is unconditionally stable through introducing a physics-preserving correction term, which is analogous to the attractive term in the Van der Waals EoS. An efficient numerical algorithm is provided to compute the coefficient in the correction term. Finally, some numerical examples are illustrated to verify the theoretical results and efficiency of the established algorithms. The numerical results match well with laboratory data.
Original languageEnglish (US)
Pages (from-to)B1-B28
Number of pages1
JournalSIAM Journal on Scientific Computing
Issue number1
StatePublished - Jan 19 2017

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The work of the first and fourth authors was supported by funding from King Abdullah University of Science and Technology (KAUST). The work of the second author was partially supported by National Natural Science Foundation of China (11301163). The work of the third author was partially supported by Hong Kong Research Grant Council GRF grants 509213, 15302214 and NSFC/RGC joint research scheme N HKBU204/12.


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