Article
Mathematics
J. J. H. Brouwers
Summary: This article describes the Langevin and diffusion equations for passively marked fluid particles in turbulent flow with spatially varying and anisotropic statistical properties. The solutions are obtained through a power series expansion and can be directly implemented in computational fluid dynamics codes.
Article
Computer Science, Interdisciplinary Applications
Ruben Sevilla
Summary: The hybridisable discontinuous Galerkin (HDG) method, proposed by Cockburn and co-workers, is popular for reducing the global number of coupled degrees of freedom compared to other DG methods. This work introduces a dual time stepping (DTS) approach to solve the global system of equations in the HDG formulation of convection-diffusion problems, presenting a proof of the existence and uniqueness of the steady state solution. The stability limit and optimal choice for the dual time step are derived, and different time marching approaches are compared for convection-dominated problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Stephen Metcalfe, Siva Nadarajah
Summary: In this work, a new quasi-optimal test norm for discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation is proposed. The theoretical analysis shows that the proposed test norm leads to favorable scalings between the target norm and the energy norm. Numerical experiments are conducted to confirm the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Natalia Kopteva, Richard Rankin
Summary: The symmetric interior penalty discontinuous Galerkin method and its weighted averages version are applicable on shape-regular nonconforming meshes for solving singularly perturbed semilinear reaction-diffusion equations. Residual-type a posteriori error estimates in maximum norm are given, with error cons...
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Liuqiang Zhong, Yue Xuan, Jintao Cui
Summary: This paper studies a two-grid method based on discontinuous Galerkin discretization for the convection-diffusion-reaction equation. The algorithm solves the original nonsymmetric problem on a coarse grid and the corresponding positive definite diffusion problem on a fine grid. The two-grid algorithm essentially transforms the DG solution of the convection-diffusion-reaction equation into an approximation for the DG solution of the diffusion equation.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Jiansong Zhang, Jiang Zhu, Hector Andres Vargas Poblete, Maosheng Jiang
Summary: A new hybrid mixed finite element method is proposed for solving the convection-diffusion-reaction equation with local exponential fitting technique. The convergence of the method is analyzed and a priori error estimate is derived. Numerical results are provided to confirm the theoretical analysis.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Hamdullah Yucel
Summary: This study focuses on goal-oriented a posteriori error estimates for the numerical approximation of Dirichlet boundary control problem on a two dimensional convex polygonal domain, using the local discontinuous Galerkin method for discretization. Primal-dual weighted error estimates are derived for the objective functional, with numerical examples presented to demonstrate the performance of the estimator.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Hamid Safdari, Majid Rajabzadeh, Moein Khalighi
Summary: A local discontinuous Galerkin method is proposed for solving a nonlinear convection-diffusion equation with a fractional diffusion, nonlinear diffusion, and nonlinear convection term, achieving higher accuracy using Spline interpolations. Compared to the direct Galerkin method, this proposed method is demonstrated to be suitable for general fractional convection-diffusion problems, significantly improving stability and providing a convergence order of O(h(k+1)), where k represents the degree of polynomials.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Huihui Cao, Yunqing Huang, Nianyu Yi
Summary: This paper investigates the adaptive direct discontinuous Galerkin method for second order elliptic equations in two dimensions, introducing a numerical flux with general weighted averages and proper weights for interface problems. In addition, it proposes a residual-type a posteriori error estimator and establishes global upper bounds and local lower bounds for errors in the DG norm. Several numerical examples are conducted to confirm the reliability and efficiency of the proposed error estimator and method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Leszek Feliks Demkowicz, Nathan Roberts, Judit Munoz-Matute
Summary: This paper studies both conforming and non-conforming versions of the practical DPG method for the convection-reaction problem. It determines that the common approach of constructing a local Fortin operator for DPG stability analysis is not feasible for this problem. Instead, a new approach based on direct proof of discrete stability is developed, along with the introduction of a subgrid mesh. The argument is supported by mathematical analysis and numerical experiments.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Gautam Singh, Srinivasan Natesan
Summary: The parabolic convection-diffusion-reaction problem is discretized using the NIPG method in space and the DG method in time. Piecewise Lagrange interpolation at Gauss points is used to improve the order of convergence, and the error bound in the discrete energy norm is estimated. The study demonstrates superconvergence properties of the DG method with (k+1)-order convergence in space and (l+1)-order convergence in time, with numerical results confirming the theoretical findings.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Jiaqi Li, Leszek Demkowicz
Summary: Building upon the standard Discontinuous Petrov-Galerkin (DPG) method in Hilbert spaces, this study generalizes the approach to Banach spaces. Numerical experiments on a 1D convection-dominated diffusion problem demonstrate that the Banach-based method yields solutions less affected by the Gibbs phenomenon. H-adaptivity is implemented using an error representation function as an indicator of error. Published by Elsevier Ltd.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Sergiy Reutskiy, Ji Lin, Bin Zheng, Jiyou Tong
Summary: This paper presents a new semi-analytical method based on the B-spline approximation for modeling transfer in anisotropic inhomogeneous mediums. Numerical examples demonstrate the high accuracy of the proposed method in solving 2D convection-diffusion-reaction problems.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2021)
Article
Mathematics, Applied
Meng Li, Xianbing Luo
Summary: In this work, we propose a new ensemble Monte Carlo hybridizable discontinuous Galerkin algorithm for simulating the convection-diffusion equation with random diffusion coefficients. The algorithm reduces computation costs compared to the traditional Monte Carlo HDG method and achieves optimal convergence rates in both space and time. Numerical experiments confirm the accuracy of our theoretical results.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Haitao Leng
Summary: In this paper, a hybridizable discontinuous Galerkin method with divergence-free and H(div)-conforming velocity field is proposed for the stationary incompressible Navier-Stokes equations. The pressure-robustness, which ensures that the a priori error estimates of the velocity are independent of the pressure error, is satisfied. Additionally, an efficient and reliable a posteriori error estimator is derived for the L-2 errors in the velocity gradient and pressure, under a smallness assumption. Numerical examples are provided to demonstrate the pressure-robustness and the performance of the obtained a posteriori error estimator.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Martin Cermak, Frederic Hecht, Zuqi Tang, Martin Vohralik
NUMERISCHE MATHEMATIK
(2018)
Article
Mathematics, Applied
J. Papez, Z. Strakos, M. Vohralik
NUMERISCHE MATHEMATIK
(2018)
Article
Engineering, Multidisciplinary
Martin Vohralik, Soleiman Yousef
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2018)
Article
Mathematics, Applied
Patrik Daniel, Alexandre Ern, Iain Smears, Martin Vohralik
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2018)
Article
Mathematics, Applied
Sarah Ali Hassan, Caroline Japhet, Martin Vohralik
ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
(2018)
Article
Mathematics, Applied
Patrick Ciarlet, Martin Vohralik
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2018)
Article
Mathematics, Applied
Alexandre Ern, Iain Smears, Martin Vohralik
IMA JOURNAL OF NUMERICAL ANALYSIS
(2019)
Article
Mathematics, Applied
Sarah Ali Hassan, Caroline Japhet, Michel Kern, Martin Vohralik
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
Eric Cances, Genevieve Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralik
NUMERISCHE MATHEMATIK
(2018)
Article
Mathematics, Applied
Jan Blechta, Josef Malek, Martin Vohralik
IMA JOURNAL OF NUMERICAL ANALYSIS
(2020)
Article
Mathematics, Applied
Alexandre Ern, Martin Vohralik
MATHEMATICS OF COMPUTATION
(2020)
Article
Engineering, Multidisciplinary
Patrik Daniel, Alexandre Ern, Martin Vohralik
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2020)
Article
Computer Science, Interdisciplinary Applications
Ibtihel Ben Gharbia, Jad Dabaghi, Vincent Martin, Martin Vohralik
COMPUTATIONAL GEOSCIENCES
(2020)
Article
Mathematics, Applied
Gouranga Mallik, Martin Vohralik, Soleiman Yousef
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Alexandre Ern, Iain Smears, Martin Vohralik
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)