Article
Physics, Multidisciplinary
Tomasz P. Stefanski, Jacek Gulgowski
Summary: In this paper, the formulation of time-fractional electrodynamics using the Riemann-Silberstein vector is derived, allowing for modeling of energy dissipation and dynamics of electromagnetic systems with memory. The properties of TF Maxwell's equations are analyzed from the perspective of classical electrodynamics, and classical solutions for wave-propagation problems are derived assuming various symmetries. The results are supported by numerical simulations and analysis, with discussion on the relationship between TF Schrodinger equation and TF electrodynamics.
Article
Physics, Fluids & Plasmas
V. E. Moiseenko, O. Agren
Summary: A new form of time-harmonic Maxwell's equations is proposed for numerical modeling, featuring positive differential operators, absence of curl operators, and leading operators including Laplacian and gradient of divergence. This new form demonstrates applicability through a simple example.
PLASMA PHYSICS AND CONTROLLED FUSION
(2021)
Article
Computer Science, Interdisciplinary Applications
Long Yuan, Wenxiu Gong
Summary: This paper focuses on Trefftz discretizations of the time-dependent Maxwell equations in anisotropic media in three-dimensional domains. A class of space-time Trefftz DG methods is proposed, which includes the Trefftz variational formulation from Egger et al. (2015). Error estimates of the approximate solutions with respect to the meshwidth and the condition number of the coefficient matrices are proved. Furthermore, a global Trefftz DG method combined with local DG methods is proposed to solve the time-dependent linear nonhomogeneous Maxwell equations in anisotropic media. Numerical results validate the theoretical results and demonstrate the high accuracy of the resulting approximate solutions.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
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
Mathematics, Applied
Gang Chen, Peter Monk, Yangwen Zhang
Summary: The concept of M-decomposition is introduced for constructing Hybridizable Discontinuous Galerkin (HDG) methods to approximate solutions of elliptic PDEs and Maxwell's equations on unstructured meshes. The methods show optimal error estimate and superconvergence with any choice of spaces having M-decomposition and rich enough auxiliary spaces, even for non-coercive problems, as confirmed by numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Damian Trofimowicz, Tomasz P. Stefanski, Jacek Gulgowski, Tomasz Talaska
Summary: This paper presents the application of control engineering methods in modeling and simulating signal propagation in time-fractional electrodynamics. By simulating signal propagation in electromagnetic media using Maxwell's equations with fractional-order constitutive relations in the time domain, the equations in time-fractional electrodynamics can be considered as a continuous-time system of state-space equations in control engineering. Analytical solutions are derived for electromagnetic-wave propagation in the time-fractional media based on state-transition matrices, and discrete time zero-order-hold equivalent models are developed and their analytical solutions are derived. The proposed models yield the same results as other reference methods, but are more flexible in terms of the number of simulation scenarios that can be tackled due to the application of the finite-difference scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Kaiwen Shi, Xinlong Feng, Haiyan Su
Summary: In this article, a first-order decoupling penalty method is proposed for solving the 2D/3D time-dependent incompressible magnetohydrodynamic equations in a convex domain. By applying a penalty term to the constraint divu = 0, the saddle point problem is transformed into two smaller problems. The method is then validated through numerical tests and the error estimate is derived. The spatial discretization for the tests is done using Lagrange finite element.
NUMERICAL ALGORITHMS
(2023)
Article
Engineering, Multidisciplinary
T. Chaumont-Frelet, S. Lanteri, P. Vega
Summary: This study investigates the impact of electron motion in metallic nanostructures on the finite element discretizations of Maxwell's equations. A novel residual-based error estimator is proposed and shown to be reliable and efficient in driving a mesh adaptive process. Numerical examples demonstrate the quality of the proposed estimator and the potential computational savings offered by mesh adaptivity.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(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
Lucrezia Cossetti, Rainer Mandel
Summary: This work investigates the L-p - L-q-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator, and derives an L-p - L-q-type Limiting Absorption Principle for this operator. New results for Helmholtz systems with zero order non-Hermitian perturbations were utilized in the analysis. Additionally, an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems was provided.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Physics, Mathematical
Xiang Wang, Jichun Li, Zhiwei Fang
Summary: This article examines three quadrature methods for quickly solving stochastic time-dependent Maxwell's equations with uncertain properties, providing mathematical analysis of error estimates for single level Monte Carlo, multi-level Monte Carlo, and quasi-Monte Carlo methods, supported by numerical experiments.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Benjamin Doerich, Marlis Hochbruck
Summary: In this paper, two exponential integrators are proposed for a class of quasilinear wave-type equations. A well-posedness result is deduced based on the analysis and error analysis. Numerical examples are included to confirm the theoretical findings.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Interdisciplinary Applications
Philip Freese
Summary: In this study, the finite element heterogeneous multiscale method is applied to dispersive first-order time-dependent Maxwell systems. An analytic homogenization result is used to show that the effective system includes additional dispersive effects. A thorough study of the micro problems, including H-2 and micro error estimates, is provided. Finally, a semidiscrete error estimate for the method is proven.
MULTISCALE MODELING & SIMULATION
(2022)
Article
Mathematics, Applied
Ying Liang, Hua Xiang, Shiyang Zhang, Jun Zou
Summary: In this paper, a new family of preconditioners for saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell's equations in three dimensions is proposed. It is shown that the preconditioned conjugate gradient method is applicable for wave numbers smaller than a critical value, while methods like preconditioned MINRES can be used for larger wave numbers. The spectral behaviors of resulting preconditioned systems are analyzed and compared, with numerical experiments demonstrating and comparing the efficiencies of these preconditioners.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Li Zhang, Maohua Ran, Hanyue Zhang
Summary: In this research, a two-dimensional ADI-FDTD method for Maxwell's equations without sources and charges is proposed. The method shows fourth-order accuracy in time and is approximately energy-preserving. Numerical experiments confirm the theoretical results.
