Article
Computer Science, Hardware & Architecture
Mohammad Javidi, Mahdi Saedshoar Heris
Summary: In this paper, algorithms using piecewise linear interpolation polynomial are designed and developed to solve partial fractional differential equations involving Caputo derivative. The algorithms are applied to both uniform and non-uniform meshes, and new methods for selecting mesh points are proposed. Error bounds for the proposed methods with uniform and equidistributing meshes are obtained. The numerical method is stable and convergent with an accuracy of O(kappa(2) + h), as demonstrated through numerical examples and a comparative study for different parameter values.
JOURNAL OF SUPERCOMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Yanli Chen, Xue Jiang, Jun Lai, Peijun Li
Summary: This paper investigates the three-dimensional electromagnetic scattering from a large open rectangular cavity embedded in a perfectly electrically conducting infinite ground plane. By introducing a transparent boundary condition and utilizing a fast algorithm involving fast Fourier transform and Gaussian elimination, the paper solves the linear system for cavities filled with either a homogeneous or layered medium, demonstrating superior performance in numerical experiments.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Leijie Qiao, Wenlin Qiu, Da Xu
Summary: This study constructs and analyzes a nonlocal evolution equation with a weakly singular kernel in three-dimensional space, using different numerical methods to ensure stability and convergence. The numerical results confirm the theoretical analysis.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Alexander Zlotnik, Raimondas Ciegis
Summary: The study examines the necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation on non-uniform spatial meshes. It is found that exponential growth in solution norm and excessively strong conditions between time and space steps are required for stability, even in the case of non-uniform time stability.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Yunqing Huang, Jichun Li, Xuancen Yi, Haoke Zhao
Summary: In this paper, a finite element method (FEM) is developed and analyzed for the Drude perfectly matched layer (PML) model. The stability analysis and error estimate for the scheme are established. Numerical results demonstrate the effectiveness of this PML in absorbing outgoing waves in the Drude metamaterial.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Interdisciplinary Applications
Jie Gu, Lijuan Nong, Qian Yi, An Chen
Summary: In this paper, the authors propose effective numerical schemes for the time-fractional Black-Scholes equation. The original equation is converted into an equivalent integral-differential equation and discretized using piecewise linear interpolation for the time-integral term and compact difference formula for the spatial direction. The derived fully discrete compact difference scheme demonstrates second-order accuracy in time and fourth-order accuracy in space, with rigorous proofs of stability and convergence. Furthermore, the authors extend the results to non-uniform meshes for dealing with non-smooth solutions and provide a temporal non-uniform mesh-based compact difference scheme.
FRACTAL AND FRACTIONAL
(2023)
Article
Physics, Mathematical
Shuai Su, Huazhong Tang, Jiming Wu
Summary: This paper introduces an efficient finite volume scheme on general polygonal meshes for calculating the two-dimensional nonequilibrium three-temperature radiation diffusion equations. The scheme does not require nonlinear iteration for linear problems and can ensure the positivity, existence, and uniqueness of the cell-centered solutions obtained on the corrector phase. Numerical experiments demonstrate the accuracy, efficiency, and positivity of the scheme on various distorted meshes.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Xixian Bai, Hongxing Rui
Summary: A fully implicit finite-difference time-domain (FDTD) scheme with second-order space-time accuracy for Maxwell's equations in a Cole-Cole dispersive medium is proposed and analyzed. The proposed scheme is rigorously analyzed and shown to be unconditionally stable with second-order accuracy in both time and space. Numerical examples are provided to validate the theoretical findings.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Qiya Hu
Summary: This paper proves the convergence of the HX preconditioner proposed by Hiptmair and Xu for Maxwell's equations, and establishes extensions of the discrete regular decomposition for edge finite element functions in three-dimensional domains. The new functions defined by the discrete regular decompositions inherit zero degrees of freedom of the edge finite element function in polyhedral domains and possess nearly optimal stability with only a logarithmic factor.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Leijie Qiao, Bo Tang
Summary: This paper presents an L1 implicit difference scheme based on non-uniform meshes for solving the time-fractional Burgers equation. The difficulty caused by the singularity of the exact solution at t = 0 can be overcome with non-uniform meshes. Through the energy method, the paper derives the unconditional stability and optimal convergence rate, with numerical experiments confirming the theoretical estimate.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Qingqing Tian, Xuehua Yang, Haixiang Zhang, Da Xu
Summary: In this paper, an implicit robust difference method is constructed to handle the modified Burgers model with nonlocal dynamic properties. The L1 formula on graded meshes is employed for the fractional derivative. The existence and uniqueness of numerical solutions are proved, and the unconditional stability is derived. A new discrete Gronwall inequality is introduced to improve the stability, and the optimal convergence order in the L-2 energy norm is also obtained. Numerical experiments demonstrate the efficiency of the proposed robust method on uniform and graded meshes.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Puttha Sakkaplangkul, Vrushali A. Bokil
Summary: This paper develops and analyzes finite difference methods for the 3D Maxwell's equations in three different types of linear dispersive media, and investigates their stability and convergence using energy method. Numerical examples confirm the effectiveness and accuracy of the methods.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2021)
Article
Mathematics, Interdisciplinary Applications
Pier Domenico Lamberti, Michele Zaccaron
Summary: This paper proves the spectral stability of the curlcurl operator subject to electric boundary conditions on a cavity under boundary perturbations. The cavities are assumed to be sufficiently smooth with weak restrictions on the strength of the perturbations. The methods involve the construction of suitable Piola-type transformations and the proof of uniform Gaffney inequalities using uniform a priori H-2 estimates for the Poisson problem of the Dirichlet Laplacian. Connections to boundary homogenization problems are also mentioned.
MATHEMATICS IN ENGINEERING
(2022)
Article
Automation & Control Systems
Han-Jing Ren, Bao-Zhu Guo
Summary: This paper investigates the uniformly exponential stability of a semi-discretized coupled system generated by a 1-D wave equation under stabilizing output feedback control. By designing an observer and constructing a suitable Lyapunov function, the exponential stability of the closed-loop system with observer-based feedback is proven. Equivalent semi-discrete finite difference schemes are developed and their uniformly exponential stability is established for both the transformed and original coupled systems. The weak convergence of the discrete solution to the continuous counterpart is also briefly discussed.
SYSTEMS & CONTROL LETTERS
(2022)
Article
Computer Science, Interdisciplinary Applications
Zi-Yun Zheng, Yuan-Ming Wang
Summary: This paper proposes and analyzes a second-order accurate Crank-Nicolson finite difference method for a class of nonlinear partial integro-differential equations with weakly singular kernels. The method approximates the first-order time derivative using a Crank-Nicolson time-stepping technique and treats the singular integral term with a product averaged integration rule. The method is proven to be solvable, stable, and convergent through the discrete energy method, positive semi-definite property, and perturbation technique. Numerical results confirm the theoretical convergence and demonstrate the effectiveness of the method.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Xixian Bai, Hongxing Rui
Summary: A fast and easily implemented algorithm for time fractional Maxwell's system, based on the SOE approximation and FDTD method, achieves high efficiency with no loss in accuracy as shown through numerical experiments in 2D and 3D.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Xixian Bai, Shuang Wang, Hongxing Rui
Summary: Two efficient numerical schemes based on L1 formula and Finite-Difference Time-Domain (FDTD) method are constructed for Maxwell's equations in a Cole-Cole dispersive medium. By rigorously carrying out energy stability and error analysis using the energy method, it is proven that both schemes converge with a certain order. Numerical experiments are performed to confirm the theoretical analysis.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Xixian Bai, Hongxing Rui
Summary: A fully implicit finite-difference time-domain (FDTD) scheme with second-order space-time accuracy for Maxwell's equations in a Cole-Cole dispersive medium is proposed and analyzed. The proposed scheme is rigorously analyzed and shown to be unconditionally stable with second-order accuracy in both time and space. Numerical examples are provided to validate the theoretical findings.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Xixian Bai, Jian Huang, Hongxing Rui, Shuang Wang
Summary: In this paper, a fast second-order finite-difference time-domain (FDTD) algorithm based on the FL2-1(sigma) formula and a weighted approach is proposed for solving the time fractional Maxwell's system. The discretization of the Caputo derivative term is constructed, and a fast second-order fully discrete FDTD algorithm is developed based on this discretization. Numerical examples with analytic solutions are provided to demonstrate the accuracy and efficiency of the proposed algorithm.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
