Article
Mathematics, Applied
Li Yan, Yao Cheng
Summary: In this paper, we investigated the local discontinuous Galerkin (LDG) method for a third order singularly perturbed problem with different types of boundary layer. We proved an optimal order error estimate in the energy-norm on graded Duran-Shishkin and Duran type meshes, which holds uniformly up to a logarithmic factor. Numerical experiments were conducted to validate our theoretical findings.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2023)
Article
Computer Science, Interdisciplinary Applications
Aditya Kaushik, Vijayant Kumar, Manju Sharma, Nitika Sharma
Summary: This paper introduces a modified graded mesh for solving singularly perturbed reaction-diffusion problems using a recursive generation method. The numerical solution is based on finite element method with polynomials of degree at least p. The parameter uniform convergence of optimal order in epsilon-weighted energy norm is proven. Test examples and comparative analysis with other adaptive meshes validate the effectiveness of the proposed method.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
Vikas Gupta, Sanjay K. Sahoo, Ritesh K. Dubey
Summary: A parameter-uniform fitted mesh finite difference scheme is proposed and analyzed for singularly perturbed interior turning point problems, showing second-order uniform convergence with respect to the singular perturbation parameter. The proposed method is validated through theoretical bounds and numerical experiments, demonstrating competitive results compared to existing methods in literature.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Maria Gabriela Armentano, Ariel L. Lombardi, Cecilia Penessi
Summary: The aim of this paper is to provide robust approximations of singularly perturbed reaction-diffusion equations in two dimensions using finite elements on graded meshes. By appropriately choosing the mesh grading parameter, quasioptimal error estimations for piecewise bilinear elements are obtained using a weighted variational formulation introduced by N. Madden and M. Stynes in Calcolo 58(2) 2021. A supercloseness result is also proven, indicating that the difference between the finite element solution and the Lagrange interpolation of the exact solution, in the weighted balanced norm, is of higher order than the error itself. Numerical examples are presented to demonstrate the good performance of the approach.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Engineering, Multidisciplinary
Aastha Gupta, Aditya Kaushik
Summary: This paper proposes a higher-order numerical approximation scheme to solve singularly perturbed reaction-diffusion boundary value problems, which has fourth-order uniform convergence.
AIN SHAMS ENGINEERING JOURNAL
(2021)
Article
Mathematics, Applied
Sanjay Ku Sahoo, Vikas Gupta
Summary: This article investigates a singularly perturbed convection-diffusion equation with discontinuous convective and source terms. Through the use of specially designed meshes and extrapolation schemes, the problem is successfully solved and the theoretical results are verified.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
D. Avijit, S. Natesan
Summary: This article presents a numerical solution method for a class of singularly perturbed 2D parabolic convection-diffusion-reaction problem, which includes spatial discretization using a piecewise-uniform Shishkin mesh and error estimation with an appropriate stabilization parameter. The stability of the method depends on the choice of time step-length, which is demonstrated through numerical simulations validating the theoretical error estimate.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Swati Yadav, Pratima Rai
Summary: This study aims to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem with a multiple interior turning point. The proposed method shows second-order convergence in time and almost second-order convergence in space up to a logarithmic factor.
ENGINEERING COMPUTATIONS
(2021)
Article
Mathematics, Applied
Kumar Rajeev Ranjan, S. Gowrisankar
Summary: This paper introduces numerical methods for singularly perturbed convection-diffusion problems with a turning point, developing a non-symmetric discontinuous Galerkin finite element method for both boundary layers and cusp-type interior layers. The uniform error estimates in L-2-norm and DG-norm are obtained and confirmed through numerical experiments.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Swati Yadav, Pratima Rai
Summary: In this article, a higher order numerical method is constructed and analyzed for a class of two dimensional parabolic singularly perturbed problem of convection-diffusion type with vanishing convection coefficient. The proposed scheme is proven to be uniformly convergent with respect to parameter epsilon, and exhibits high accuracy according to the numerical results.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics
Chein-Shan Liu, Essam R. El-Zahar, Chih-Wen Chang
Summary: In this paper, a mth-order asymptotic-numerical method is developed to solve a second-order singularly perturbed problem with variable coefficients. The method decomposes the solutions into two independent sub-problems and couples them through a left-end boundary condition. Unlike traditional asymptotic solutions, this method performs asymptotic series solution in the original coordinates, leading to better results.
Article
Mathematics, Applied
K. Aarthika, Ram Shiromani, V. Shanthi
Summary: This paper investigates a two-dimensional singularly perturbed reaction-diffusion equation with a discontinuous source term, and proposes a numerical approach using a hybrid finite difference method and a layer adapted piece-wise uniform Shishkin mesh. The convergence of this method with respect to the perturbation parameter is proven. The numerical results validate the theoretical findings.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Meenakshi Shivhare, P. Pramod Chakravarthy, Devendra Kumar
Summary: This article introduces a quadratic B-spline collocation method on an exponentially graded mesh for solving two-parameter singularly perturbed boundary value problems with twin boundary layers. Numerical experiments demonstrate the better accuracy of the proposed method compared to existing results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Xiaoqi Ma, Jin Zhang
Summary: In this paper, a singularly perturbed convection-diffusion problem with a discontinuous convection is discussed. The interior layer appearing in the solution due to this discontinuity is solved using a streamline diffusion finite element method on Shishkin mesh, and the optimal order of convergence in a modified streamline diffusion norm is derived. Numerical results are presented to validate the theoretical conclusion.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Yao Cheng, Li Yan, Xuesong Wang, Yanhua Liu
Summary: In this study, the local discontinuous Galerkin (LDG) method with a generalized alternating numerical flux is proposed for a one-dimensional singularly perturbed convection-diffusion problem. The double-optimal local maximum-norm error estimate is derived on the quasi-uniform meshes for the first time. Additionally, the discrete maximum principle and global L1-error estimate established in the literature are improved.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Xuesong Wang, Yao Cheng
Summary: The local discontinuous Galerkin (LDG) method on a Shishkin mesh is investigated for a one-dimensional singularly perturbed reaction-diffusion problem. Improved pointwise error estimates are derived based on the discrete Green's function in the regular and layer regions. The convergence rates of the pointwise error for both the LDG approximation to the solution and its derivative are analyzed, showing optimal rates in different domains. Moreover, optimal pointwise error estimates are established when the regular component of the exact solution belongs to the finite element space. Numerical experiments are conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arash Goligerdian, Mahmood Khaksar-e Oshagh
Summary: This paper presents a computational method for simulating more accurate models for population growth with immigration, using integral equations with a delay parameter. The method utilizes Legendre wavelets within the Galerkin scheme as an orthonormal basis and employs the composite Gauss-Legendre quadrature rule for computing integrals. An error bound analysis demonstrates the convergence rate of the method, and various numerical examples are provided to validate the efficiency and accuracy of the technique as well as the theoretical error estimate.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
A. Sreelakshmi, V. P. Shyaman, Ashish Awasthi
Summary: This paper focuses on constructing a lucid and utilitarian approach to solve linear and non-linear two-dimensional partial differential equations. Through testing, it is found that the proposed method is highly applicable and accurate, showing excellent performance in terms of cost-cutting and time efficiency.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Shujiang Tang
Summary: This paper investigates the impact of the structure of local smoothness indicators on the computational performance of the WENO-Z scheme. A new class of two-parameter local smoothness indicators is proposed, which combines the classical WENO-JS and WENO-UD5 schemes and appends the coefficients of higher-order terms. A new WENO scheme, WENO-NSLI, is constructed using the global smoothness indicators of WENO-UD5. Numerical experiments show that the new scheme achieves optimal accuracy and has higher resolution compared to WENO-JS, WENO-Z, and WENO-UD5.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Xue-Feng Duan, Yong-Shen Zhang, Qing-Wen Wang
Summary: This paper addresses a class of constrained tensor least squares problems in image restoration and proposes the alternating direction multiplier method (ADMM) to solve them. The convergence analysis of this method is presented. Numerical experiments show the feasibility and effectiveness of the ADMM method for solving constrained tensor least squares problems, and simulation experiments on image restoration are also conducted.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Wanying Mao, Qifeng Zhang, Dinghua Xu, Yinghong Xu
Summary: In this paper, we derive, analyze, and extensively test fourth-order compact difference schemes for the Rosenau equations in one and two dimensions. These schemes are applied under spatial periodic boundary conditions using the double reduction order method and bilinear compact operator. Our results show that these schemes satisfy mass and energy conservation laws and have unique solvability, unconditional convergence, and stability. The convergence order is four in space and two in time under the D infinity-norm. Several numerical examples are provided to support the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Jeremy Chouchoulis, Jochen Schutz
Summary: This work presents an approximate family of implicit multiderivative Runge-Kutta time integrators for stiff initial value problems and investigates two different methods for computing higher order derivatives. Numerical results demonstrate that adding separate formulas yields better performance in dealing with stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hui Yang, Shengfeng Zhu
Summary: In this paper, shape optimization in incompressible Stokes flows is investigated based on the penalty method for the divergence free constraint at continuous level. Shape sensitivity analysis is performed, and numerical algorithms are introduced. An iterative penalty method is used for solving the penalized state and possible adjoint numerically, and it is shown to be more efficient than the standard mixed finite element method in 2D. Asymptotic convergence analysis and error estimates for finite element discretizations of both state and adjoint are provided, and numerical results demonstrate the effectiveness of the optimization algorithms.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Reza MohammadiArani, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: Lattice Boltzmann method is a powerful solver for fluid flow, but it is challenging to use it to solve other partial differential equations. This paper challenges the LBM to solve the two-dimensional DKS equation by finding a suitable local equilibrium distribution function and proposes a modification for implementing boundary conditions in complex geometries.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arijit Das, Prakrati Kushwah, Jitraj Saha, Mehakpreet Singh
Summary: A new volume and number consistent finite volume scheme is introduced for the numerical solution of a collisional nonlinear breakage problem. The scheme achieves number consistency by introducing a single weight function in the flux formulation. The proposed scheme is efficient and robust, allowing easy coupling with computational fluid dynamics softwares.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
H. Ait el Bhira, M. Kzaz, F. Maach, J. Zerouaoui
Summary: We present an asymptotic method for efficiently computing second-order telegraph equations with high-frequency extrinsic oscillations. The method uses asymptotic expansions in inverse powers of the oscillatory parameter and derives coefficients through either recursion or solving non-oscillatory problems, leading to improved performance as the oscillation frequency increases.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hanen Boujlida, Kaouther Ismail, Khaled Omrani
Summary: This study investigates a high-order accuracy finite difference scheme for solving the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A new compact difference scheme is proposed and the a priori estimates and unique solvability are discussed using the discrete energy method. The unconditional stability and convergence of the difference solution are proved. Numerical experiments demonstrate the accuracy and efficiency of the proposed technique.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Alexander Zlotnik, Timofey Lomonosov
Summary: This paper studies a three-level explicit in time higher-order vector compact scheme for solving initial-boundary value problems for the n-dimensional wave equation and acoustic wave equation with variable speed of sound. By using additional sought functions to approximate second order non-mixed spatial derivatives of the solution, new stability bounds and error bounds of orders 4 and 3.5 are rigorously proved. Generalizations to nonuniform meshes in space and time are also discussed.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Fengli Yin, Yayun Fu
Summary: This paper develops an explicit energy-preserving scheme for solving the coupled nonlinear Schrodinger equation by combining the Lie-group method and GSAV approaches. The proposed scheme is efficient, accurate, and can preserve the modified energy of the system.
APPLIED NUMERICAL MATHEMATICS
(2024)
