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Volume 16, Issue 3
A Physics-Informed Structure-Preserving Numerical Scheme for the Phase-Field Hydrodynamic Model of Ternary Fluid Flows

Qi Hong, Yuezheng Gong & Jia Zhao

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 565-596.

Published online: 2023-08

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  • Abstract

Phase-field models are widely used in studying multiphase flow dynamics. Given the complexity and strong nonlinearity, designing accurate, efficient, and stable numerical algorithms to solve these models has been an active research field for decades. This paper proposes a novel numerical scheme to solve a highly cited and used phase field hydrodynamic model for simulating ternary phase fluid flows. The main novelty is the introduction of a supplementary variable to reformulate the original problem into a constrained optimization problem. This reformulation leads to several advantages for our proposed numerical algorithms compared with many existing numerical techniques for solving this model. First, the developed schemes allow more straightforward calculations for the hydrodynamic phase-field models by solving a few decoupled Helmholtz or Poisson-type systems with a constant precomputable coefficient matrix, remarkably reducing the computational cost. Secondly, the numerical schemes can maintain mass conservation and energy dissipation at the discrete level. Additionally, the developed scheme based on the second-order backward difference formula respects the original energy dissipation law that differs from many existing schemes, such as the IEQ, SAV, and Lagrange multiplier approaches for which a modified energy dissipation law is respected. Furthermore, rigorous proof of energy stability and practical implementation strategies are provided. We conduct adequate 2D and 3D numerical tests to demonstrate the proposed schemes’ accuracy and effectiveness.

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-565, author = {Hong , QiGong , Yuezheng and Zhao , Jia}, title = {A Physics-Informed Structure-Preserving Numerical Scheme for the Phase-Field Hydrodynamic Model of Ternary Fluid Flows}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {3}, pages = {565--596}, abstract = {

Phase-field models are widely used in studying multiphase flow dynamics. Given the complexity and strong nonlinearity, designing accurate, efficient, and stable numerical algorithms to solve these models has been an active research field for decades. This paper proposes a novel numerical scheme to solve a highly cited and used phase field hydrodynamic model for simulating ternary phase fluid flows. The main novelty is the introduction of a supplementary variable to reformulate the original problem into a constrained optimization problem. This reformulation leads to several advantages for our proposed numerical algorithms compared with many existing numerical techniques for solving this model. First, the developed schemes allow more straightforward calculations for the hydrodynamic phase-field models by solving a few decoupled Helmholtz or Poisson-type systems with a constant precomputable coefficient matrix, remarkably reducing the computational cost. Secondly, the numerical schemes can maintain mass conservation and energy dissipation at the discrete level. Additionally, the developed scheme based on the second-order backward difference formula respects the original energy dissipation law that differs from many existing schemes, such as the IEQ, SAV, and Lagrange multiplier approaches for which a modified energy dissipation law is respected. Furthermore, rigorous proof of energy stability and practical implementation strategies are provided. We conduct adequate 2D and 3D numerical tests to demonstrate the proposed schemes’ accuracy and effectiveness.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2023-0007 }, url = {http://global-sci.org/intro/article_detail/nmtma/21958.html} }
TY - JOUR T1 - A Physics-Informed Structure-Preserving Numerical Scheme for the Phase-Field Hydrodynamic Model of Ternary Fluid Flows AU - Hong , Qi AU - Gong , Yuezheng AU - Zhao , Jia JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 565 EP - 596 PY - 2023 DA - 2023/08 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2023-0007 UR - https://global-sci.org/intro/article_detail/nmtma/21958.html KW - Energy stable, phase field, Cahn-Hilliard-Navier-Stokes, supplementary variable method. AB -

Phase-field models are widely used in studying multiphase flow dynamics. Given the complexity and strong nonlinearity, designing accurate, efficient, and stable numerical algorithms to solve these models has been an active research field for decades. This paper proposes a novel numerical scheme to solve a highly cited and used phase field hydrodynamic model for simulating ternary phase fluid flows. The main novelty is the introduction of a supplementary variable to reformulate the original problem into a constrained optimization problem. This reformulation leads to several advantages for our proposed numerical algorithms compared with many existing numerical techniques for solving this model. First, the developed schemes allow more straightforward calculations for the hydrodynamic phase-field models by solving a few decoupled Helmholtz or Poisson-type systems with a constant precomputable coefficient matrix, remarkably reducing the computational cost. Secondly, the numerical schemes can maintain mass conservation and energy dissipation at the discrete level. Additionally, the developed scheme based on the second-order backward difference formula respects the original energy dissipation law that differs from many existing schemes, such as the IEQ, SAV, and Lagrange multiplier approaches for which a modified energy dissipation law is respected. Furthermore, rigorous proof of energy stability and practical implementation strategies are provided. We conduct adequate 2D and 3D numerical tests to demonstrate the proposed schemes’ accuracy and effectiveness.

Qi Hong, Yuezheng Gong & Jia Zhao. (2023). A Physics-Informed Structure-Preserving Numerical Scheme for the Phase-Field Hydrodynamic Model of Ternary Fluid Flows. Numerical Mathematics: Theory, Methods and Applications. 16 (3). 565-596. doi:10.4208/nmtma.OA-2023-0007
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