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

Wang, Xiuhua; Kou, Jisheng; Cai, Jianchao

doi:10.1007/s10915-020-01127-x

Cited by 35 publications

(22 citation statements)

References 73 publications

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“…We now prove that the discrete scheme (2.8) preserves the positivity and the summation constraint S + I + R = 1. The proofs will be carried out using a variational approach, which has been developed to prove the discrete maximum principles in [14,22,23]. We define an auxiliary variable φ = min(φ, 0) for φ ∈ C h .…”

Section: 2mentioning

confidence: 99%

“…Positivity-preserving schemes not only satisfy the intrinsic requirement of the model, but also can eliminate spurious numerical solutions to dramatically improve the accuracy and stability of the long-time simulation. Actually, the preservation of positivity or boundedness is desirable for numerical methods of numerous scientific and engineering problems, for instance, [3,4,14,20,22].…”

mentioning

confidence: 99%

“…By direct calculations, we can obtain the following discrete variational formula [5,13,14,22,23,25,26]…”

mentioning

confidence: 99%

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A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

Kou¹,

Chen²,

Wang³

et al. 2023

CPAA

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In this paper, we propose an efficient numerical method for a comprehensive infection model that is formulated by a system of nonlinear coupling advection-diffusion-reaction equations. Using some subtle mixed explicitimplicit treatments, we construct a linearized and decoupled discrete scheme. Moreover, the proposed scheme is capable of preserving the positivity of variables, which is an essential requirement of the model under consideration. The proposed scheme uses the cell-centered finite difference method for the spatial discretization, and thus, it is easy to implement. The diffusion terms are treated implicitly to improve the robustness of the scheme. A semi-implicit upwind approach is proposed to discretize the advection terms, and a distinctive feature of the resulting scheme is to preserve the positivity of variables without any restriction on the spatial mesh size and time step size. We rigorously prove the unique existence of discrete solutions and positivity-preserving property of the proposed scheme without requirements for the mesh size and time step size. It is worthwhile to note that these properties are proved using the discrete variational principles rather than the conventional approaches of matrix analysis. Numerical results are also provided to assess the performance of the proposed scheme.

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Section: 2mentioning

confidence: 99%

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confidence: 99%

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A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

Kou¹,

Chen²,

Wang³

et al. 2023

CPAA

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“…There have been several bulk free energy functions used for phase-field models in the literature, for instance, the double well potential and logarithmic Flory-Huggins energy potential. From a physical view of point, the logarithmic energy potential is more realistic than the double well potential [52][53][54]; in fact, it can be viewed as a simplified approximation of the Helmholtz free energy [55] determined by realistic equations of state, e.g. van der Waals equation of state and Peng-Robinson equation of state [56], which are widely employed in physics and oil-gas engineering respectively.…”

Section: Basic Model Equationsmentioning

confidence: 99%

Thermodynamically consistent modeling of two-phase incompressible flows in heterogeneous and fractured media

Gao

Kou

Sun

et al. 2020

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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Numerical modeling of two-phase flows in heterogeneous and fractured media is of great interest in petroleum reservoir engineering. The classical model for two-phase flows in porous media is not completely thermodynamically consistent since the energy reconstructed from the capillary pressure does not involve the ideal fluid energy of both phases and attraction effect between two phases. On the other hand, the saturation may be discontinuous in heterogeneous and fractured media, and thus the saturation gradient may be not well defined. Consequently, the classical phase-field models can not be applied due to the use of diffuse interfaces. In this paper, we propose a new thermodynamically consistent energy-based model for two-phase flows in heterogeneous and fractured media, which is free of the gradient energy. Meanwhile, the model inherits the key features of the traditional models of two-phase flows in porous media, including relative permeability, volumetric phase velocity and capillarity effect. To characterize the capillarity effect, a logarithmic energy potential is proposed as the free energy function, which is more realistic than the commonly used double well potential. The model combines with the discrete fracture model to describe two-phase flows in fractured media. The popularly used implicit pressure explicit saturation method is used to simulate the model. Finally, the experimental verification of the model and numerical simulation results are provided.

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“…An appealing feature of this approach is that it leads to linear, easy-to-implement, and energy stable numerical schemes, and this advantage is more notable for numerical simulation of realistic fluids. Due to its excellent features, this approach has been successfully extended to phase-field models [28,29]. In this work, using the EF approach, an efficient, linear, energy stable semiimplicit numerical scheme is constructed for the proposed model of shale gas transport.…”

Section: Introductionmentioning

confidence: 99%

Chemical Potential-Based Modeling of Shale Gas Transport

et al. 2021

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Shale gas plays an increasingly important role in the current energy industry. Modeling of gas flow in shale media has become a crucial and useful tool to estimate shale gas production accurately. The second law of thermodynamics provides a theoretical criterion to justify any promising model, but it has been never fully considered in the existing models of shale gas. In this paper, a new mathematical model of gas flow in shale formations is proposed, which uses gas density instead of pressure as the primary variable. A distinctive feature of the model is to employ chemical potential gradient rather than pressure gradient as the primary driving force. This allows to prove that the proposed model obeys an energy dissipation law, and thus, the second law of thermodynamics is satisfied. Moreover, on the basis of energy factorization approach for the Helmholtz free energy density, an efficient, linear, energy stable semi-implicit numerical scheme is proposed for the proposed model. Numerical experiments are also performed to validate the model and numerical method.

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

Cited by 35 publications

References 73 publications

A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

Thermodynamically consistent modeling of two-phase incompressible flows in heterogeneous and fractured media

Chemical Potential-Based Modeling of Shale Gas Transport

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