Article
Mathematics, Applied
Tom Lewis, Aaron Rapp, Yi Zhang
Summary: This paper further analyzes the dual-wind discontinuous Galerkin (DWDG) method for approximating Poisson's problem by examining the relationship between the Laplacian and the discrete Laplacian. The DWDG methods are derived from the DG differential calculus framework, which replaces continuous differential operators with discrete ones. We establish error estimates and explore the relationship between the DWDG approximation and the Ritz projection. Numerical experiments are conducted to validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Bo Dong, Wei Wang
Summary: We develop and analyze a high-order multiscale discontinuous Galerkin method for 2D stationary Schrodinger equations in quantum transport. Numerical results show that the method can capture highly oscillating solutions of Schrodinger equations more effectively.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Jonas Zeifang, Jochen Schutz
Summary: In this paper, an implicit two-derivative deferred correction time discretization approach is proposed and combined with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting numerical method achieves high order accuracy in both space and time, demonstrated by the eighth order accuracy obtained in linear advection and compressible Euler equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Jiming Yang, Jing Zhou
Summary: A two-grid algorithm for discontinuous Galerkin approximations to nonlinear Sobolev equations is proposed, with derived H(1)norm error estimate. The analysis shows that the algorithm achieves asymptotically optimal approximation as long as the mesh sizes satisfy certain conditions. Numerical experiments confirm the efficiency of the algorithm.
NUMERICAL ALGORITHMS
(2021)
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
Physics, Mathematical
Li Zhou, Yunzhang Li
Summary: In this paper, a local discontinuous Galerkin (LDG) method is proposed for the multi-dimensional stochastic Cahn-Hilliard type equation, with stability proven on arbitrary polygonal domains with triangular meshes and sub-optimal error estimate derived for Cartesian meshes. Numerical examples demonstrate the performance of the LDG method.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Helmi Temimi
Summary: In this paper, a novel discontinuous Galerkin (DG) finite element method is developed to solve the Poisson's equation on Cartesian grids. The method first applies the standard DG method in the x-spatial variable, resulting in a system of ordinary differential equations (ODEs) in the y-variable. The method of line is then used to discretize the ODEs. The proposed fully DG scheme utilizes DG methods of different degrees in the x and y variables and achieves optimal convergence rate.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Juan Gutierrez-Jorquera, Florian Kummer
Summary: The study presents a fully coupled solver for steady-state diffusion flames using the low-Mach approximation and a one-step kinetic model. The nonlinear equation system is solved using a Newton-Dogleg method, with initial estimates obtained from a flame-sheet model. Results show that the method effectively simulates the process of flame propagation.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2022)
Article
Mathematics, Applied
Manuel Solano, Felipe M. Vargas
Summary: A high order unfitted hybridizable discontinuous Galerkin method is proposed for numerically solving Oseen equations in a domain Omega with a curved boundary, where the boundary condition is approximated by line integrals and pressure is decomposed to have zero-mean. Stability estimates are provided under assumptions related to the distance between the computational boundary and Omega, showing that the approximations of pressure, velocity, and its gradient are of order h(k+1). Numerical experiments validate the theory and demonstrate the method's performance in solving the incompressible Navier-Stokes equations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Astronomy & Astrophysics
Nils L. Fischer, Harald P. Pfeiffer
Summary: We present a discontinuous Galerkin internal-penalty scheme that is applicable to a wide range of linear and nonlinear elliptic partial differential equations. The scheme can handle various boundary conditions and curved meshes, and achieves compactness by eliminating auxiliary degrees of freedom. The accuracy of the scheme is demonstrated through a suite of numerical test problems, and it has been implemented in the SpECTRE numerical relativity code.
Article
Computer Science, Interdisciplinary Applications
Mustafa E. Danis, Jue Yan
Summary: This study proposes a new formula for the nonlinear viscous numerical flux and extends it to the compressible Navier-Stokes equations using the direct discontinuous Galerkin method with interface correction (DDGIC). The new method simplifies the implementation and enables accurate calculation of physical quantities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Buyang Li, Weifeng Qiu, Zongze Yang
Summary: We propose a linearized semi-implicit and decoupled finite element method for the incompressible Navier-Stokes equations with variable density. The method solves the velocity equation using an H-1-conforming finite element method and the density equation using an upwind discontinuous Galerkin finite element method with post-processed velocity. The proposed method is proven to converge in approximating reasonably smooth solutions in three-dimensional convex polyhedral domains.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Yanren Hou, Yongbin Han, Jing Wen
Summary: This paper analyzes an equal-order hybridized discontinuous Galerkin (HDG) method with a small pressure penalty parameter for the Stokes equations. When the pressure penalty parameter gamma tends to 0, the velocity approximation tends to be H(div)-conforming and exactly divergence-free, but taking the value of gamma too small will cause the over-stabilization of the pressure. A post-processing procedure is provided to obtain a stable pressure approximation.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Chemistry, Multidisciplinary
Huanqin Gao, Jiale Zhang, Hongquan Chen, Shengguan Xu, Xuesong Jia
Summary: This paper presents a high-order DG method for solving the preconditioned Euler equations with explicit or implicit time marching schemes. The practical implementation of a precondition matrix and the employed DG spatial discretization scheme are described in detail. The curved boundary treatment is proposed and verified through numerical simulations.
APPLIED SCIENCES-BASEL
(2022)
Article
Computer Science, Interdisciplinary Applications
Wei Guo, Juntao Huang, Zhanjing Tao, Yingda Cheng
Summary: In this paper, an adaptive sparse grid local discontinuous Galerkin method is proposed to solve Hamilton-Jacobi equations in high dimensions, using multiwavelets for multiresolution. Numerical tests show the method performs well in up to four dimensions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
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)