Article
Mathematics, Applied
Maurice S. Fabien
Summary: This paper presents an energy-stable hybridizable interior penalty discontinuous Galerkin method for the Allen-Cahn equation, proving its energy stability, existence, and uniqueness. The method's effectiveness and convergence are numerically verified through several examples.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Mariam Al-Maskari, Samir Karaa
Summary: This paper focuses on the strong approximation problem of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time: a Caputo fractional derivative of order alpha E (0, 1) and a Riemann-Liouville fractional integral operator of order gamma E [0, 1] applied to a Gaussian noise. The model is approximated using a standard piecewise linear finite element method (FEM) in space and the classical Grunwald-Letnikov method in time, with the noise handled through L2-projection. Spatially semidiscrete and fully discrete schemes are analyzed, and strong convergence rates are obtained by exploiting the temporal Holder continuity property of the solution. Numerical experiments are conducted to illustrate the theoretical results.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Jian Li, Jiyao Zeng, Rui Li
Summary: In this paper, the discontinuous finite volume element method (DFVEM) is used to solve the Allen-Cahn equation with strong nonlinearity. The proposed method combines DFVEM in space and the backward Euler method in time. The energy stability, unique solvability, and error estimates of the scheme are derived. Numerical experiments demonstrate the effectiveness of the method in capturing phase transition dynamics and ensuring stability during long-term simulations.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Huanrong Li, Zhengyuan Song
Summary: This paper focuses on a reduced-order finite element method for the Allen-Cahn equation. It first derives a traditional finite element formulation and establishes a novel reduced order FE formulation using proper orthogonal decomposition technique. Numerical experiments are provided to confirm the validity of the novel ROFE formulation.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Huanrong Li, Dongmei Wang
Summary: This paper investigates a modified finite volume element (MFVE) method for solving the nonlinear phase field Allen-Cahn model with a small perturbation parameter. A traditional finite volume element (FVE) algorithm is proposed for the phase field Allen-Cahn equation, and error estimations for the FVE solutions are derived. Optimal basis functions are obtained using the proper orthogonal decomposition approach, and a reduced order MFVE method is established. Theoretical results are verified through numerical tests, which also show the superior performance of the MFVE algorithm in terms of CPU running time.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Chaeyoung Lee, Yongho Choi, Junseok Kim
Summary: This paper presents an explicit finite difference method for the Allen-Cahn equation, using an alternating direction explicit method for the diffusion term to allow for a larger time step size compared to explicit methods, resulting in improved stability and preservation of intrinsic properties of the AC equation.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Chaeyoung Lee, Hyundong Kim, Sungha Yoon, Sangkwon Kim, Dongsun Lee, Jinate Park, Soobin Kwak, Junxiang Yang, Jian Wang, Junseok Kim
Summary: A stable numerical scheme for the Allen-Cahn equation with high-order polynomial free energy is proposed in this paper. The method is theoretically proven to be unconditionally stable and computationally shown to be robust and accurate. The effect of the order of the double-well potential on the dynamics of the AC equation is investigated, highlighting the potential of the proposed method for modeling various interfacial phenomena.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Applied
Liang Wu, Mejdi Azaieza, Tomas Chacon Rebollo, Chuanju Xu
Summary: In this paper, an efficient reduced-order finite element method is proposed and analyzed for the parametrized Allen-Cahn equation. The equation is first discretized using a stabilized semi-implicit scheme in time and a finite element method in space for a given parameter. Then, a reduced basis is constructed using proper orthogonal decomposition (POD) based on a set of snapshots. The main contribution of this work lies in the error analysis of the reduced-order model, where an error estimate is derived for the first time for the parametrized Allen-Cahn equation, considering the impact of the diffusion parameter.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Youngjin Hwang, Junxiang Yang, Gyeongyu Lee, Seokjun Ham, Seungyoon Kang, Soobin Kwak, Junseok Kim
Summary: In this study, a fast and efficient finite difference method is proposed for solving the AC equation on cubic surfaces. The method unfolds the cubic surface domain in 3D space into 2D space and applies appropriate boundary conditions on the planar sub-domains to calculate numerical solutions. Numerical experiments show the effectiveness of the proposed algorithm.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2024)
Article
Mathematics, Applied
Danxia Wang, Yanan Li, Hongen Jia
Summary: In this paper, a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential is presented. The method consists of two steps, solving the equation on coarse and fine grids respectively. The energy stabilities and convergence properties of the method are discussed and validated through numerical examples.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Jundong Feng, Yingcong Zhou, Tianliang Hou
Summary: In this paper, a new linear second-order finite difference scheme for Allen-Cahn equations is proposed, with stability and discrete maximum principle. Numerical experiments validate the theoretical results.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Xiuping Wang, Fuzheng Gao, Jintao Cui, Zhengjia Sun
Summary: In this article, a weak Galerkin finite element method and a nonuniform two-step backward differentiation formula scheme are utilized to solve the Allen-Cahn equation. The numerical scheme incorporates a discrete weak gradient operator on discontinuous piecewise polynomials. The energy stability analysis and optimal order error estimates of L-2-norm are derived, and numerical experiments confirm the theoretical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Huanrong Li, Dongmei Wang, Zhengyuan Song, Fuchen Zhang
Summary: This paper proposes a reduced-order finite element method based on the POD technique for simulating the Allen-Cahn phase field model, with error estimates of the solutions provided. Numerical results demonstrate the effectiveness of the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Yaoyao Chen, Yunqing Huang, Nianyu Yi
Summary: This paper carries out error analysis for a totally decoupled, linear, and unconditionally energy stable finite element method to solve the Cahn-Hilliard-Navier-Stokes equations. The a priori error analysis is derived for the phase field, velocity field, and pressure variable in the fully discrete scheme.
APPLIED MATHEMATICS AND COMPUTATION
(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
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)