Article
Physics, Multidisciplinary
Guoliang He, Yong Zhang
Summary: This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem. The semi-implicit scheme based on Euler method is adopted for time discretization, while the special finite volume scheme is adopted for space discretization. The theoretical analysis results show that under certain conditions, the full discretization proposed here has the characteristics of local stability, and the optimal theoretical and numerical error estimation of velocity and pressure can be obtained.
Article
Mathematics, Applied
Juan Vicente Gutierrez-Santacreu, Marko Antonio Rojas-Medar
Summary: The Navier-Stokes-alpha equations are a type of LES models that aim to capture the influence of small scales on large ones without calculating the entire flow range. The parameter α represents the smallest resolvable scale by the model. When α=0, the classical Navier-Stokes equations for viscous, incompressible, Newtonian fluids are recovered. These equations can also be seen as a regularization of the Navier-Stokes equations, where α stands for the regularization parameter.
PHYSICA D-NONLINEAR PHENOMENA
(2023)
Article
Mathematics, Applied
Yingying Xie, Liuqiang Zhong, Ming Tang
Summary: In this paper, we propose a residual-type error estimator for the modified weak Galerkin (MWG) method of 2D H(curl) elliptic problems. We demonstrate the reliability of this estimator with respect to the approximation error measured in a natural energy norm by decomposing the error into conforming and nonconforming parts. Additionally, we establish the efficiency of the error estimator using standard bubble functions and conduct experiments to verify its performance on both uniform and adaptive meshes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Alex Kaltenbach, Michael Ruzicka
Summary: In this paper, a local discontinuous Galerkin approximation is proposed for fully nonhomogeneous systems of p-Navier--Stokes type. By using the primal formulation, the well-posedness, stability (a priori estimates), and weak convergence of the method are proved. A new discontinuous Galerkin discretization of the convective term is proposed, and an abstract nonconforming theory of pseudomonotonicity, which is applied to the problem, is developed. The approach is also used to treat the p-Stokes problem.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Physics, Mathematical
Jiachuan Zhang, Ran Zhang, Xiaoshen Wang
Summary: Based on auxiliary subspace techniques, this paper presents a posteriori error estimator of nonconforming weak Galerkin finite element method (WGFEM) for the Stokes problem in two and three dimensions. Without the saturation assumption, it is proved that the WGFEM approximation error is bounded by the error estimator up to an oscillation term. The computational cost of the approximation and error problems is considered in terms of the size and sparsity of the system matrix. To reduce the computational cost of the error problem, an equivalent error problem is constructed using diagonalization techniques, which only requires solving two diagonal linear algebraic systems corresponding to the degree of freedom (d.o.f) to obtain the error estimator. Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Mehdi Dehghan, Zeinab Gharibi
Summary: In this article, a weak Galerkin finite element method (WG-FEM) is presented and analyzed for solving the coupled Navier-Stokes/temperature (or Boussinesq) problems. The WG-FEM utilizes discontinuous functions to approximate the velocity, temperature, and the normal derivative of temperature on the boundary, while piecewise constants are used for the pressure approximation. The stability, existence, uniqueness, and error estimates of the WG-FEM are proved in detail. Numerical experiments are conducted to validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Alex Kaltenbach, Michael Ruzicka
Summary: In this paper, we establish convergence rates for the velocity of the local discontinuous Galerkin approximation of systems of p-Navier-Stokes and p-Stokes equations with p \in (2, \infty). The convergence rates are optimal for linear ansatz functions. Numerical experiments are conducted to support the results.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mechanics
Paolo Maremonti, Francesca Crispo, Carlo Romano Grisanti
Summary: This paper focuses on the IBVP of the Navier-Stokes equations and presents results that align with previous findings by the authors. The goal is to assess the potential discrepancy between the energy equality and energy inequality derived for a weak solution.
Article
Mathematics, Applied
Alex Kaltenbach, Michael Ruzicka
Summary: In this paper, we prove the convergence rates of the pressure in the local discontinuous Galerkin approximation for systems of p-Navier-Stokes type and p-Stokes type with p \in (2, \infty ). The proposed method is introduced in Part I of the paper [A. Kaltenbach and M. Ruzv \icv \ka, SIAM J. Numer. Anal., 61 (2023), pp. 1613-1640]. The results are validated through numerical experiments.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Kyungkeun Kang, Jihoon Lee, Michael Winkler
Summary: The Cauchy problem for the chemotaxis-Navier-Stokes system in R-3 is considered, where a globally defined weak solution is constructed under suitable initial data conditions. Additionally, it is shown that a weak solution exists with further regularity features and mass preservation under certain assumptions on the initial data.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Chen Liu, Rami Masri, Beatrice Riviere
Summary: This paper analyzes an energy-stable discontinuous Galerkin algorithm for solving the Cahn-Hilliard-Navier-Stokes equations. By deriving the energy dissipation and stability of the order parameter and providing optimal a priori error estimates, the uniqueness of the solution and mass conservation of the scheme are demonstrated without any regularization of the potential function.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Jingjun Zhao, Wenjiao Zhao, Yang Xu
Summary: This paper introduces a new numerical method for solving the time-dependent incompressible space fractional Navier-Stokes equations. By combining different numerical methods to achieve a fully discrete scheme, the existence, uniqueness, and stability of the solution are proved. Finally, numerical experiments are conducted to validate the effectiveness of the proposed method and the correctness of the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Lin Mu
Summary: In this article, a novel numerical scheme for solving the steady incompressible Navier-Stokes equations is developed and analyzed using the weak Galerkin methods. The algorithm achieves pressure-robustness by employing a divergence-preserving velocity reconstruction operator, which ensures that the velocity error is independent of the pressure and irrotational body force. Error analysis is conducted to establish the convergence rate, and numerical experiments are presented to validate the theoretical conclusions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
T. Tachim Medjo
Summary: In this article, a stochastic version of a coupled Allen-Cahn-Navier-Stokes model in a two- or three-dimensional bounded domain is studied. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. These equations are motivated by the dynamic of binary fluids under the influence of stochastic external forces. The existence of a probabilistic weak solution is proven, relying on a Galerkin approximation as well as some compactness results. In the two-dimensional case, the uniqueness of the weak solutions is proved.
APPLICABLE ANALYSIS
(2022)
Article
Mathematics, Applied
Zhiqiang Cai, Jing Yang
Summary: In this paper, a class of discontinuous Galerkin finite element methods for advection-diffusion-reaction problems is presented, and a priori error estimates are established when the solution is only in H1+s (Omega) space with s is an element of(0, 1/2].
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
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)
