Article
Mathematics, Applied
Chen Cui, Jiaqi Liu, Yuchang Mo, Shuying Zhai
Summary: An effective operator splitting scheme for solving the nonlocal Allen-Cahn equation with a Lagrange multiplier is studied. The original equation is discretized into a nonlinear equation, a nonlocal equation and a Lagrange multiplier equation, which are then solved analytically and numerically to verify the algorithm's validity in terms of mass conservation and convergence.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Yue Yu, Jiansong Zhang, Rong Qin
Summary: In this paper, the authors study a class of time-fractional phase-field models, including the Allen-Cahn and Cahn-Hilliard equations. Two explicit time-stepping schemes are proposed based on the exponential scalar auxiliary variable (ESAV) approach, where the fractional derivative is discretized using L1 and L1+ formulas respectively. These novel schemes allow for the decoupled computations of the phase variable phi and the auxiliary variable R. Furthermore, the schemes exhibit energy dissipation law on general nonuniform meshes, which is inherent in the continuous level. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the proposed methods.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Interdisciplinary Applications
Eylem Ozturk, Joseph L. Shomberg
Summary: In this study, a viscous Cahn-Hilliard phase-separation model with memory and a nonlocal fractional Laplacian operator in the chemical potential is examined. The existence of global weak solutions is proven using a Galerkin approximation scheme. Continuous dependence estimate provides uniqueness of weak solutions and existence of a compact connected global attractor in the weak energy phase space.
FRACTAL AND FRACTIONAL
(2022)
Article
Engineering, Multidisciplinary
Revanth Mattey, Susanta Ghosh
Summary: A physics informed neural network (PINN) is a method that incorporates the physics of a system into a neural network's loss function by satisfying the system's boundary value problem. To address the accuracy issue for highly non-linear and higher-order time-varying partial differential equations, a novel backward compatible PINN (bc-PINN) scheme is proposed, which solves the PDE sequentially over successive time segments using a single neural network and re-trains the network to satisfy the already obtained solutions for previous time segments. Two techniques, using initial conditions and transfer learning, are introduced to improve the accuracy and efficiency of the bc-PINN scheme.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Azer Khanmamedov
Summary: This paper considers the hyperbolic relaxation of the 2D Cahn-Hilliard/Allen-Cahn equation with cubic nonlinearity of class C-2, proving that the semigroup generated by weak solutions possesses a global attractor. It also improves previously obtained results regarding global attractors for the 2D Cahn-Hilliard equation with the inertial term.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Zehra Sen, Azer Khanmamedov
Summary: This paper discusses the initial-boundary value problem for the hyperbolic relaxation of the Cahn-Hilliard/Allen-Cahn equation with a proliferation term in a smooth two-dimensional domain. By assuming appropriate conditions on the nonlinearities, the existence, uniqueness, and regularity (in time) of the energy solution of the problem are proven. The continuous dependence of the energy solution on the initial data is established by verifying the validity of the energy equality. Under additional conditions on the nonlinearities, it is also shown that the associated semigroup possesses a global attractor.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Dong Li, Chaoyu Quan
Summary: This study proves the energy stability of a standard operator-splitting method for the Cahn-Hilliard equation, establishing a uniform bound of Sobolev norms for the numerical solution and convergence of the splitting approximation. This is the first energy stability result for the operator-splitting method in the context of the Cahn-Hilliard equation, and the analysis can be extended to many other models.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Weixin Ma, Yongxing Shen
Summary: This paper presents a new implementation strategy for solving the Allen-Cahn and Cahn-Hilliard equations using the proper generalized decomposition (PGD) method for parametric studies. The strategy includes a mixed formulation and a new data structure, significantly improving computational efficiency compared to traditional PGD formulations and implementations.
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2021)
Article
Engineering, Mechanical
Junxiang Yang, Yibao Li, Chaeyoung Lee, Hyun Geun Lee, Soobin Kwak, Youngjin Hwang, Xuan Xin, Junseok Kim
Summary: In this paper, an explicit conservative Saul'yev finite difference scheme for the Cahn-Hilliard equation is presented. The proposed scheme has four main merits and outperforms in computational experiments.
INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES
(2022)
Article
Physics, Mathematical
Colby L. Wight, Jia Zhao
Summary: This paper focuses on using deep neural networks to design an improved Physics Informed Neural Network (PINN) for automatically solving the Allen-Cahn and Cahn-Hilliard equations. Various techniques and sampling strategies are proposed to enhance the efficiency and accuracy of the PINN in solving phase field equations, allowing for a wider applicability to a broader class of PDE problems without restriction on the explicit form of the PDEs.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Hao Chen, Hai-Wei Sun
Summary: This paper introduces fully discrete numerical methods for solving multidimensional Allen-Cahn equations with anisotropic spatial fractional Riesz derivatives using the Strang splitting technique. The proposed methods are proven to preserve the discrete maximum principle unconditionally and have second-order accuracy in both time and space, with the use of fast algorithms for practical implementation. Numerical examples validate the theoretical analysis and demonstrate the efficiency of the proposed methods.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Zhengguang Liu, Xiaoli Li, Jian Huang
Summary: Two accurate and efficient linear algorithms are proposed for the time fractional Cahn-Hilliard and Allen-Cahn equations with general nonlinear bulk potential, demonstrating unconditional energy stability and numerical simulations verify their accuracy and efficiency in 2D and 3D.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Hao Chen, Hai-Wei Sun
Summary: This paper presents a numerical method for solving the multidimensional Allen-Cahn equations with theoretical proof of preserving the discrete maximum principle and error estimation. In practical computations, the algorithm can be implemented by computing linear systems and matrix exponentials of Toeplitz matrices to reduce complexity. Numerical examples demonstrate the effectiveness and efficiency of the proposed scheme in two and three spatial dimensions.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Soobin Kwak, Junxiang Yang, Junseok Kim
Summary: In this study, a novel conservative Allen-Cahn equation with a curvature-dependent Lagrange multiplier is proposed, which demonstrates superior structure-preserving properties. The model has minimal dynamics of motion by mean curvature and only exhibits smoothing properties of the interface transition layer, making it suitable for modeling conservative phase-field applications such as two-phase fluid flows. Computational tests confirm the superior performance of the proposed CAC equation in terms of structure preservation.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Xingchun Xu, Yanwei Hu, Yurong He, Jiecai Han, Jiaqi Zhu
Summary: In this work, three modified lattice Boltzmann (LB) schemes with temporal or spatial difference-based correction term are proposed to accurately simulate the conservative Allen-Cahn equation. Through high-order truncation analysis, the dominant error terms of these modified LB schemes at third and fourth order are extracted for the first time. Comparative studies among different models are performed in terms of accuracy, convergence rate, boundedness, and efficiency. The proposed model shows superiority in reducing numerical errors and maintaining boundedness over a wide range of relaxation time, which is consistent with theoretical analysis. Additionally, the locality of the modified model is improved by reorganizing the collision step and adding a correction step after the streaming process.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
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)