Article
Mathematics, Applied
Hailiang Liu, Peimeng Yin
Summary: This paper introduces novel discontinuous Galerkin schemes for the Cahn-Hilliard equation, designed by integrating the mixed DG method with the Invariant Energy Quadratization approach. The resulting IEQ-DG schemes are shown to be unconditionally energy dissipative and can be efficiently solved, demonstrating good performance in terms of efficiency, accuracy, and preservation of solution properties.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Tao Tang, Xu Wu, Jiang Yang
Summary: In this paper, a fully discrete method for phase-field gradient flows is constructed and analyzed, which utilizes the extrapolated Runge-Kutta with scalar auxiliary variable (RK-SAV) method in time and discontinuous Galerkin (DG) method in space. A novel technique is proposed to decouple the system, reducing the problem to solving several elliptic scalar problems with constant coefficients independently. The method demonstrates arbitrarily high order in both time and space, as rigorously demonstrated for the Allen-Cahn equation and the Cahn-Hilliard equation. Theoretical results are verified through numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Chaobao Huang, Na An, Xijun Yu
Summary: In this paper, a fully discrete semi-implicit stabilized scheme for the time-fractional Cahn-Hilliard equation is investigated. The scheme adopts the nonuniform fast L1 scheme in time, the mixed finite element method in space, and the stabilization technique for the derivative of the double-well potential. The boundedness of L∞-norm of the computed solution Uhn is obtained using the temporal-spatial splitting technique. An unconditional optimal error estimate is given without certain temporal restrictions dependent on the spatial mesh size. The proposed scheme preserves the modified discrete energy dissipation property.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Xiaoli Li, Jie Shen
Summary: In this paper, we propose first- and second-order time discretization schemes for the Cahn-Hilliard-Navier-Stokes system based on the multiple scalar auxiliary variables approach and (rotational) pressure-correction. These schemes are linear, fully decoupled, unconditionally energy stable, and only require solving a sequence of elliptic equations with constant coefficients at each time step. We provide a rigorous error analysis for the first-order scheme, establishing optimal convergence rates for all relevant functions in different norms. We also verify our theoretical results through numerical experiments.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
G. Deugoue, B. Jidjou Moghomye, T. Tachim Medjo
Summary: In this paper, we consider a stochastic model for the motion of an incompressible isothermal mixture of two immiscible non-Newtonian fluids perturbed by a multiplicative noise of Gaussian and Levy type. The model consists of the stochastic Oldroyd model of order one, coupled with a stochastic nonlocal Cahn-Hilliard model. We establish the global existence and uniqueness of strong probabilistic solution, and show that the sequence of Galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Nan Zheng, Xiaoli Li
Summary: In this paper, several efficient numerical schemes are constructed based on two types of scalar auxiliary variable approaches for the Cahn-Hilliard-Brinkman system. The temporal discretizations are implemented using first-order, second-order, and higher-order methods. The energy stability analyses for the constructed schemes are rigorously derived, and various numerical simulations are conducted to demonstrate the accuracy and performance of the schemes.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Thermodynamics
Xiao-Rong Kang, Yan-Mei Wu, Ke-Long Cheng
Summary: This paper presents a second order numerical scheme for the Cahn-Hilliard equation with a Fourier pseudo-spectral approximation in space. An additional Douglas-Dupont regularization term is introduced for energy stability. A linear iteration algorithm is proposed to solve the non-linear system, and its efficiency is verified through numerical simulations.
Article
Physics, Mathematical
Lizhen Chen, Zengyan Zhang, Jia Zhao
Summary: In this paper, a new class of linear time-integration schemes for phase-field models are developed by introducing extra free parameters to further stabilize the schemes and improve their accuracy. The proposed schemes are generic, computationally cost-effective, and guarantee the existence and uniqueness of solutions in each time step. The numerical tests demonstrate that the proposed schemes are accurate and efficient, providing insights in developing numerical approximations for general phase field models.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
R. Altmann, C. Zimmer
Summary: This paper focuses on time stepping schemes for the Cahn-Hilliard equation with three types of dynamic boundary conditions. The proposed first and second order schemes are both mass-conservative and energy-dissipative. They allow different spatial discretizations in the bulk and on the boundary, which leads to computational advantages in terms of refinements on the boundary without mesh adaptation in the interior. Numerical experiments demonstrate the effectiveness of the proposed schemes.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Junxiang Yang, Junseok Kim
Summary: Linear, decoupled, and energy dissipation-preserving schemes for ternary Cahn-Hilliard (CH) fluid models using a modified scalar auxiliary variable (MSAV) approach were proposed in this study. The MSAV method simplifies calculations and improves efficiency by solving a set of linear equations through a step-by-step process. Additionally, a fast linear multigrid algorithm was employed to accelerate convergence, with various numerical tests confirming the good performance of the proposed methods.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Engineering, Multidisciplinary
Wei Li, Martin Z. Bazant, Juner Zhu
Summary: Recent advances in scientific machine learning have led to new insights in modeling pattern-forming systems. This article introduces a physics-informed operator neural network framework called Phase-Field DeepONet, which predicts the dynamic responses of systems governed by gradient flows of free-energy functionals. By incorporating the minimizing movement scheme, the framework optimizes and controls the evolution of the system's total free energy, enabling fast real-time predictions of pattern-forming dynamical systems.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Computer Science, Interdisciplinary Applications
Yakun Li, Wenkai Yu, Jia Zhao, Qi Wang
Summary: The paper introduces the thermodynamically consistent Cahn-Hilliard-Extended-Darcy (CHED) model to describe transient motion of a binary incompressible fluid flow in porous media. A series of linear, second-order numerical algorithms for the CHED model based on the energy quadratization strategy are developed. The numerical algorithms respect energy-dissipation-rate and volume conservation property at a discrete level, leading to unconditional energy stability. Various numerical tests are conducted to validate the model.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Rui Chen, Shuting Gu
Summary: In this paper, two fully discrete time-marching schemes, including the first order and the second order schemes, are presented for the Cahn-Hilliard equation. The proposed method is based on an improved Invariant Energy Quadratization method. The two schemes are shown to be linear and energy stable for the original energy, and the well-posedness and energy stability of the discrete problems are proven. Extensive numerical experiments are conducted to verify the convergence, robustness, and energy stability of these schemes.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Jia Zhao, Daozhi Han
Summary: The CHNS system is traditionally solved using numerical extrapolation for coupling terms, but this paper proposes a new strategy to reformulate it into a constraint gradient flow formulation, revealing reversible and irreversible structures. The proposed operator splitting schemes reduce computational costs and guarantee the thermodynamic laws of the CHNS system at the discrete level, making them a significant improvement over other approaches.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Xiangling Chen, Lina Ma, Xiaofeng Yang
Summary: In this article, error estimates for two second-order numerical schemes for solving the viscous Cahn-Hilliard equation are derived and the theoretical predictions of the convergence rate and energy stability of the algorithms are verified through numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mechanics
Adam Janecka, Josef Malek, Vit Prusa, Giordano Tierra
Article
Biophysics
Kiran Kumari, Burkhard Duenweg, Ranjith Padinhateeri, J. Ravi Prakash
BIOPHYSICAL JOURNAL
(2020)
Article
Mathematics, Applied
Francisco Guillen-Gonzalez, Maria Angeles Rodriguez-Bellido, Giordano Tierra
Summary: In this study, a new model was proposed to represent the interaction between flows and vesicle membranes with liquid crystalline phases. A new numerical scheme was introduced to approximate the model, demonstrating good performance and showcasing the dynamics of such vesicle membranes through several numerical results.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Physics, Condensed Matter
Aaron Brunk, Burkhard Duenweg, Herbert Egger, Oliver Habrich, Maria Lukacova-Medvid'ova, Dominic Spiller
Summary: A new model for viscoelastic phase separation is proposed based on a systematically derived two-fluid model with dissipative effects. The model is consistent with the second law of thermodynamics and has been studied for well-posedness in two space dimensions, showing good agreement between numerical simulations and physical experiments, as well as qualitative agreement with mesoscopic simulations.
JOURNAL OF PHYSICS-CONDENSED MATTER
(2021)
Article
Physics, Condensed Matter
Luca Tubiana, Hideki Kobayashi, Raffaello Potestio, Burkhard Duenweg, Kurt Kremer, Peter Virnau, Kostas Daoulas
Summary: By utilizing molecular knots as indicators to validate the ability of algorithms to equilibrate high-molecular-weight polymer melts, two state-of-the-art algorithms demonstrate excellent consistency, paving the way for studying topological properties of polymer melts beyond the time and length scales accessible to traditional brute-force molecular dynamics simulations.
JOURNAL OF PHYSICS-CONDENSED MATTER
(2021)
Article
Physics, Condensed Matter
Dominic Spiller, Aaron Brunk, Oliver Habrich, Herbert Egger, Maria Lukacova-Medvid'ova, Burkhard Duenweg
Summary: The model is based on a coarse-grained molecular model, with well-defined theoretical basis and momentum conservation equation. It considers both equilibrium and non-equilibrium thermodynamics, deriving a rheological constitutive equation that differs from the standard Oldroyd-B model. More investigation is needed to determine if the model can fully reproduce viscoelastic phase separation.
JOURNAL OF PHYSICS-CONDENSED MATTER
(2021)
Article
Mechanics
Aritra Santra, B. Duenweg, J. Ravi Prakash
Summary: A multiparticle Brownian dynamics simulation algorithm is used to describe the static behavior of associative polymer solutions considering pairwise excluded volume interactions. Predictions for the fractions of stickers bound by intrachain and interchain associations are obtained and validated by systematic comparison with scaling relations. The simulations confirm the predictions of scaling theory across a wide range of parameter values.
JOURNAL OF RHEOLOGY
(2021)
Article
Engineering, Multidisciplinary
Mireille El Haddad, Giordano Tierra
Summary: In this work, a thermodynamically consistent model based on phase field theory is derived to represent two-phase flows with different densities. Three linear numerical schemes are introduced to decouple the system computation in an energy-stable manner. Numerical results are presented to demonstrate the validity of the model and the well behavior of the proposed schemes.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Chemistry, Physical
Philipp Baumli, Lukas Hauer, Emanuela Lorusso, Azadeh Sharifi Aghili, Katharina Hegner, Maria D'Acunzi, Jochen S. Gutmann, Burkhard Duenweg, Doris Vollmer
Summary: External shear flows of oil lead to linear dehydration and shrinkage of hydrogel due to finite solubility of water in oil continuously removing water from the hydrogel by diffusion. The flow advects the water-rich oil, as demonstrated by numerical solutions, and shear does not cause gel shrinkage for water-saturated oils or non-solvents. The solubility of water in the oil will tune the dehydration dynamics.
Article
Mathematics, Applied
Francisco Guillen-Gonzalez, Maria Angeles Rodriguez-Bellido, Giordano Tierra
RESULTS IN APPLIED MATHEMATICS
(2020)
Article
Physics, Fluids & Plasmas
Dominic Spiller, Burkhard Duenweg
Article
Chemistry, Physical
Aritra Santra, Kiran Kumari, Ranjith Padinhateeri, B. Duenweg, J. Ravi Prakash
Proceedings Paper
Mathematics, Applied
Maria Lukacova-Medvid'ova, Burkhard Duenweg, Paul Strasser, Nikita Tretyakov
MATHEMATICAL ANALYSIS OF CONTINUUM MECHANICS AND INDUSTRIAL APPLICATIONS II
(2018)
Article
Mathematics, Applied
Junfeng Cao, Ke Chen, Huan Han
Summary: This paper proposes a two-stage image segmentation model based on structure tensor and fractional-order regularization. In the first stage, fractional-order regularization is used to approximate the Hausdorff measure of the MS model. The solution is found using the ADI scheme. In the second stage, thresholding is used for target segmentation. The proposed model demonstrates superior performance compared to state-of-the-art methods.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Dylan J. Oliver, Ian W. Turner, Elliot J. Carr
Summary: This paper discusses a projection-based framework for numerical computation of advection-diffusion-reaction (ADR) equations in heterogeneous media with multiple layers or complex geometric structures. By obtaining approximate solutions on a coarse grid and reconstructing solutions on a fine grid, the computational cost is significantly reduced while accurately approximating complex solutions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Nathan V. Roberts, Sean T. Miller, Stephen D. Bond, Eric C. Cyr
Summary: In this study, the time-marching discontinuous Petrov-Galerkin (DPG) method is applied to the Vlasov equation for the first time, using backward Euler for a Vlasov-Poisson discretization. Adaptive mesh refinement is demonstrated on two problems: the two-stream instability problem and a cold diode problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Yizhi Sun, Zhilin Sun
Summary: This work investigates the convexity of a specific class of positive definite probability measures and demonstrates the preservation of convexity under multiplication and intertwining product. The study reveals that any integrable function on an interval with a polynomial expansion of fast absolute convergence can be decomposed into a pair of positive convex interval probabilities, simplifying the study of interval distributions and discontinuous probabilistic Galerkin schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Bhagwan Singh, Komal Jangid, Santwana Mukhopadhyay
Summary: This paper examines the prediction of bending characteristics of nanoscale materials using the Moore-Gibson-Thompson thermoelasticity theory in conjunction with the nonlocal strain gradient theory. The study finds that the stiffness of the materials can be affected by nonlocal and length-scale parameters, and the aspect ratios of the beam structure play a significant role in bending simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Guoliang Wang, Bo Zheng, Yueqiang Shang
Summary: This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method, and the post-processing technique. The algorithm generates a global continuous approximate solution using the partition of unity method and improves the smoothness of the solution by adding an extra coarse grid correction step. It has good parallel performance and is validated through theoretical error estimates and numerical test examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Hao Xu, Zeng-Qi Wang
Summary: Fluid flow control problems are crucial in industrial applications, and solving the optimal control of Navier-Stokes equations is challenging. By using Oseen's approximation and matrix splitting preconditioners, we can efficiently solve the linear systems and improve convergence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhengya Yang, Xuejuan Chen, Yanping Chen, Jing Wang
Summary: This paper focuses on the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for spatial discretization and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. As a result, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Several numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed numerical scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
