A Novel Energy Factorization Approach for the Diffuse-Interface Model with Peng--Robinson Equation of State

Abstract

The Peng-Robinson equation of state (PR-EoS) has become one of the most extensively applied equations of state in chemical engineering and petroleum industry due to its excellent accuracy in predicting the thermodynamic properties of a wide variety of materials, especially hydrocarbons. Although great efforts have been made to construct efficient numerical methods for the diffuse interface models with PR-EoS, there is still not a linear numerical scheme that can be proved to preserve the original energy dissipation law. In order to pursue such a numerical scheme, we propose a novel energy factorization (EF) approach, which first factorizes an energy function into a product of several factors and then treats the factors using their properties to obtain the semiimplicit linear schemes. We apply the EF approach to deal with the Helmholtz free energy density determined by PR-EoS, and then propose a linear semi-implicit numerical scheme that inherits the original energy dissipation law. Moreover, the proposed scheme is proved to satisfy the maximum principle in both the time semi-discrete form and the cell-centered finite difference fully discrete form under certain conditions. Numerical results are presented to demonstrate the stability and efficiency of the proposed scheme.

Stabilized Energy Factorization Approach for Allen–Cahn Equation with Logarithmic Flory–Huggins Potential

“…More recently, a novel energy factorization (EF) approach is proposed in [31] to design linear, efficient numerical schemes for the diffuse interface model with PR-EOS. The basic idea of EF is to factorize an energy function/term into a product of several factors, which can be treated in the energy difference by the use of individual prop-erties of each factor.…”

“…It is different from the IEQ/SAV approach that the EF approach never introduces any new independent energy variable, and thus the resultant schemes can preserve the original energy dissipation law. The EF approach has been successfully applied to design the semi-implicit linear schemes for the PR-EOS model [31]. In this paper, we will propose the stabilized EF approach to deal with the logarithmic Flory-Huggins energy potential, and from this, we will propose a linear semi-implicit discrete scheme inheriting the original energy dissipation law.…”

“…For the logarithmic Flory-Huggins energy potential, the implicit Euler scheme was analyzed in [10] under the restricted constraint on the time step size due to the fully implicit treatment for all energy terms, and recently, the discrete maximum principle of the semi-implicit convex splitting scheme was analyzed in [8]. More recently, the maximum principle of a linear numerical scheme for the PR-EOS diffuse interface model has been proved in [31]. However, there are no results regarding the linear semi-implicit scheme with the maximum principle for the phase-field models with the logarithmic Flory-Huggins energy potential.…”

“…This gap will be filled in this paper. The discrete maximum principle of the proposed scheme will be rigorously proved using the discrete variational principle [31]. Appropriate stability strategy plays a crucial role in preserving the discrete maximum principle.…”

“…Discrete function spaces and discrete operators. In this section, we describe the discrete function spaces, discrete operators and discrete variational principle [9,18,22,25,30,31,48,50,51], which are based on the cell-centered finite difference (CCFD) method [2,48]. Here, we consider the two-dimensional case only, but the forms of the three-dimensional case are similar.…”