Correction
Mathematics
Vladislav Balashov, Alexander Zlotnik
Summary: The proof of Theorem 2 regarding finite-difference equilibrium solutions in the mentioned paper has been corrected.
MATHEMATICAL MODELLING AND ANALYSIS
(2021)
Article
Computer Science, Interdisciplinary Applications
Meiqi Tan, Juan Cheng, Chi-Wang Shu
Summary: Time discretization is crucial for time-dependent partial differential equations (PDEs). This paper discusses different time-marching methods and their limitations. The EIN method, which involves adding and subtracting a large linear highest derivative term, is proposed as a solution for equations with nonlinear high derivative terms.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Multidisciplinary
A. J. A. Ramos, R. Kovacs, M. M. Freitas, D. S. Almeida Junior
Summary: The Guyer-Krumhansl heat equation has important practical applications in heat conduction problems and can effectively describe the thermal behavior of macroscale heterogeneous materials. It is a promising candidate to be the next standard model in engineering, but its mathematical properties need to be thoroughly investigated and understood. This paper presents the basic structure of the equation and focuses on its differences from the Fourier heat equation. Additionally, it proves the well-posedness of a specific initial and boundary value problem and investigates the stability of the solution using a finite difference approach.
APPLIED MATHEMATICAL MODELLING
(2023)
Article
Computer Science, Interdisciplinary Applications
Kaihua Ji, Amirhossein Molavi Tabrizi, Alain Karma
Summary: Phase-field models are commonly used for simulating microstructural pattern formation during alloy solidification. However, the finite-difference method used in these models introduces a spurious lattice anisotropy that can affect the dynamics of solid-liquid interfaces. This study investigates the significant influence of lattice anisotropy on polycrystalline dendritic solidification and proposes isotropic finite-difference approximations to overcome this problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Augusto Cesar Albuquerque-Ferreira, Miguel Urena, Higinio Ramos
Summary: The study proposes an adaptive discretization technique that allows using fewer points for solving partial differential equations, resulting in lower computational cost while maintaining the same accuracy as regular discretization.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Engineering, Chemical
M. Starnoni, C. Manes
Summary: A new multiphase multicomponent framework is presented for modeling dynamic filtration processes at the Darcy-scale, validated through simulations replicating experimental conditions. The model demonstrates accuracy and robustness under various operational conditions.
SEPARATION AND PURIFICATION TECHNOLOGY
(2021)
Article
Computer Science, Interdisciplinary Applications
Nek Sharan, Peter T. Brady, Daniel Livescu
Summary: A systematic approach to obtain high-order EB methods with non-dissipative centered schemes in the interior is discussed in this study. The proposed EB schemes are up to sixth order accurate in the interior and fourth order accurate globally for hyperbolic, parabolic as well as incompletely parabolic problems. Various numerical tests are performed to evaluate the stability and accuracy of the proposed schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Dilberto S. Almeida Junior, Anderson J. A. Ramos, Alberto S. Noe, Mirelson M. Freitas, Pedro T. P. Aum
Summary: This study focuses on analyzing the asymptotic behavior of solutions to a one-dimensional initial boundary value problem associated with isothermal linear theory of swelling porous elastic media. Key results include the well-posedness of the system, exponential stabilization of solutions, and the discretization of equations using a specific numerical scheme. Numerical simulations of solution and total energy are provided to explain the findings.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2021)
Article
Physics, Mathematical
Armando Martinez-Perez, Gabino Torres-Vega
Summary: In this paper, a discrete derivative is used to introduce a time operator for non-relativistic quantum systems with point spectrum. The symmetry requirement on the time operator leads to well-defined time values related to the dynamics of discrete quantum systems. Travel times between hits with the walls for the quantum particle in a box model are found, suggesting a classical analog of time eigenstates, and classical analogs for the Woods-Saxon potential are also proposed.
JOURNAL OF MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Yuyu He, Xiaofeng Wang, Weizhong Dai
Summary: Two coupled and decoupled dissipative finite difference schemes with high-order accuracy are proposed for solving the dissipative generalized symmetric regularized long wave equations in this paper. Dissipation of the discrete energy with different parameters is discussed, and the a priori estimate, existence and uniqueness of numerical solutions, convergence with O(tau 2+h4), and stability of the schemes are proved by the discrete energy method. Numerical examples are provided to support the theoretical analysis.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
F. A. K. E. R. BEN BELGACEM, V. I. V. E. T. T. E. GIRAULT, F. A. T. E. N. JELASS
Summary: This study focuses on the fully discrete finite element approximation of the data completion problem and derives an error bound with respect to mesh-size and regularization parameter. The central technical tools used in the analysis are sharp local finite element estimates derived by Nitsche and Schatz.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Polymer Science
Juliana Bertoco, Antonio Castelo, Luis L. Ferras, Celio Fernandes
Summary: This work introduces a novel numerical method for addressing three-dimensional unsteady free surface flows incorporating integral viscoelastic constitutive equations. The newly developed method has proven effective in handling complex fluid flow scenarios and a semi-analytical solution for the velocity and stress fields of the fluid in a pipe has also been derived.
Article
Mathematics, Applied
Xiaofeng Wang, Wenlong Xue, Yong He, Fu Zheng
Summary: This paper studies Timoshenko beams with interior damping and boundary damping from the viewpoints of control theory and numerical approximation. The uniform exponential stability of the beams is particularly investigated. The meaning of uniform exponential stability in this paper refers to both the classical sense and the stability of semi-discretization systems with respect to the discretized parameter. Various methods, including stability theory, Lyapunov functional method, perturbation theory, spectral analysis, and frequency standard, are involved to investigate the uniform exponential stability of continuous and discrete systems. A new method based on frequency domain characteristics is proposed to verify the uniform exponential stability of semi-discretization systems derived from coupled systems. Numerical simulations demonstrate the effectiveness of the numerical approximating algorithms.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Hyun Geun Lee, Seokjun Ham, Junseok Kim
Summary: In this paper, the authors investigate isotropic finite difference discretizations of the 2D and 3D Laplacian operators and propose benchmark functions for evaluating their isotropy quantitatively. These benchmark functions have analytic solutions for 2D and 3D Laplacian operators, allowing for exact computation of the errors between numerical and analytic solutions.
APPLIED AND COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
B. Adhira, G. Nagamani
Summary: This article focuses on the investigation of finite-time boundedness and exponential (Q, S, R)-dissipative performance for a class of discretized competitive neural networks (CNNs) with time-varying delays. By using the semi-discretization technique, a discrete analog of the continuous-time CNNs is formulated and a state estimator is developed to achieve finite-time exponential (Q, S, R)-dissipative performance. Two novel weighted summation inequalities are proposed to obtain a tighter summation bound. An illustrative example is provided to demonstrate the sustainability and merits of the proposed method.
CHAOS SOLITONS & FRACTALS
(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)