A Afzal et al., Solving Maxwell's Equations with Modern C++ and SYCL: A Case Study, Ieeexplore.Ieee.Org, pp. 1-8.
In scientific computing, unstructured meshes are a crucial foundation for the simulation of real-world physical phenomena. Compared to regular grids, they allow resembling the computational domain with a much higher accuracy, which in turn leads to more efficient computations.
There exists a wealth of supporting libraries and frameworks that aid programmers with the implementation of applications working on such grids, each built on top of existing paral- lelization technologies. However, many approaches require the programmer to introduce a different programming paradigm into their application or provide different variants of the code. SYCL is a new programming standard providing a remedy to this dilemma by building on standard C++17 with its so-called single-source approach: Programmers write standard C++ code and expose parallelism using C++17 keywords. The application is then transformed into a concrete implementation by the SYCL implementation. By encapsulating the OpenCL ecosystem, dif- ferent SYCL implementations enable not only the programming of CPUs but also of heterogeneous platforms such as GPUs or other devices. For the first time, this paper showcases a SYCL- based solver for the nodal Discontinuous Galerkin method for Maxwell’s equations on unstructured meshes. We compare our solution to a previous C-based implementation with respect to programmability and performance on heterogeneous platforms
(web, pdf)
Ulf Andersson, Time-Domain Methods for the Maxwell Equations, , .
The most widespread time-domain method for the numerical simulation of the Maxwell equations is the finite-difference time-domain method (FD-TD). It has been widely used for electromagnetic simulation, for instance in radar cross sec- tion computations and electromagnetic compatibility investigations. The FD-TD method is second-order accurate and very efficient for simple geometries. A ma- jor drawback with the FD-TD method is its inability to accurately handle curved boundaries. Such boundaries are approximated with so-called staircasing to fit into the Cartesian FD-TD grid. Staircasing introduces errors that destroy the second- order accuracy of the FD-TD method.
We present three different methodologies to tackle the errors caused by stair- casing. They are parallelization, hybridization with unstructured grids, and regu- larization of material interfaces.
By using parallel computers it is possible to lower the staircasing errors by using a grid with many cells. We examine the scale-up and speed-up properties of the FD-TD method and demonstrate that it can be used to solve gigantic problems. This is shown by a one-billion-cell computation of an aircraft.
We also present a new hybridization strategy. We hybridize FD-TD with meth- ods for unstructured tetrahedral grids. On the unstructured grid we use either an explicit finite volume method or an implicit finite element method, depending one the size of the smallest tetrahedron in the unstructured grid. The implicit method is used on grids with tetrahedra that are much smaller than the hexahedra in the FD-TD grid. Otherwise the explicit method is used.
In two dimensions, our hybrid methods are second-order accurate and stable. This is demonstrated by extensive numerical experimentation.
In three dimensions, our hybrid methods have been successfully used on realistic geometries such as a generic aircraft model. The methods show super-linear conver- gence for a vacuum test case. However, they are not second-order accurate. This is shown to be caused by the interpolation when sending values from the FD-TD grid to the unstructured grid.
Our hybrid methods have been implemented in a code package that is used in an industrial environment.
The hybridization strategy is successful but can be expensive in terms of memory and arithmetic operations needed per cell in the grids. We present a new regular- ization procedure for material interfaces that restore second-order accuracy without adding any extra memory or arithmetic operations during the timestepping. By replacing the discontinuous material function with a properly chosen continuous function prior to the discretization, we can restore second-order accuracy. This is shown for a circular dielectric cylinder for the TMz polarization of the Maxwell equations.
(pdf)
Anon. (2009), Lecture 5 Motion of a charged particle in a magnetic field, , 2009 pp. 1-49.
(pdf)
Chris Blair, Ma432 Classical Field Theory, , 2010 pp. 1-34.
(pdf)
Charles T Sebens, Electromagnetism as Quantum Physics, Arxiv.Org, quant-ph (2019) 365-389.
One can interpret the Dirac equation either as giving the dynamics for a classical field or a quantum wave function. Here I examine whether Maxwell's equations, which are standardly interpreted as giving the dynamics for the classical electromagnetic field, can alternatively be interpreted as giving the dynamics for the photon's quantum wave function. I explain why this quantum interpretation would only be viable if the electromagnetic field were sufficiently weak, then motivate a particular approach to introducing a wave function for the photon (following Good, 1957). This wave function ultimately turns out to be unsatisfactory because the probabilities derived from it do not always transform properly under Lorentz transformations. The fact that such a quantum interpretation of Maxwell's equations is unsatisfactory suggests that the electromagnetic field is more fundamental than the photon.
Published in: Foundations of Physics 49(4) (2019) 365-389
(web, pdf)
Anon. (2009), Motion of a charged particle in a magnetic field , , 2009 pp. 1-58.
(pdf)
A I Arbab, Complex Maxwell's equations, Chinese Physics B, 22 (2013) 030301-6.
A unified complex model of Maxwell’s equations is presented. The wave nature of the electromagnetic field vector is related to the temporal and spatial distributions and the circulation of charge and current densities. A new vacuum solution is obtained, and a new transformation under which Maxwell’s equations are invariant is proposed. This transformation extends ordinary gauge transformation to include charge-current as well as scalar-vector potential. An electric dipole moment is found to be related to the magnetic charges, and Dirac’s quantization is found to determine an uncertainty relation expressing the indeterminacy of electric and magnetic charges. We generalize Maxwell’s equations to include longitudinal waves. A formal analogy between this formulation and Dirac’s equation is also discussed.
(web, pdf)
IEEE Standards Board, IEEE Standard Definitions of Terms for Radio Wave Propagation, , .
(web, pdf)
Roberto De Luca et al., Feynman's different approach to electrodynamics, Arxiv.Org, 2019.
We discuss a previously unpublished description of electromagnetism developed by Richard P. Feynman in the 1960s. Though similar to the existing approaches deriving electromagnetism from special relativity, the present one extends a long way towards the derivation of Maxwell's equations with minimal physical assumptions (in particular, without postulating Coulomb's law). Homogeneous Maxwell's equations are, indeed, derived by following a route different to the standard one, i.e. first introducing electromagnetic potentials in order to write down a relativistic invariant action, which is just the inverse approach to the usual one. Also, Feynman's derivation of the Lorentz force exclusively follows from its linearity in the charge velocity and from relativistic invariance. The obvious historical significance of such approach is then complemented by its possible relevance for didactics, providing a novel way to develop the foundations of electromagnetism, which can fruitfully supplement usual treatments.
(web, pdf)
Marco Di Mauro et al., Some insight into Feynman's approach to electromagnetism, Arxiv Preprint, 2020 2001.09069, physics.hist-ph.
We retrace an ab initio relativistic derivation of Maxwell's equations that was developed by Feynman in unpublished notes, clarifying the analogies and the differences with analogous treatments present in the literature. Unlike the latter, Feynman's approach stands out because it considers electromagnetic potentials as primary, reflecting his ideas about the quantum foundations of electromagnetism. Some considerations about the foundations of special relativity, which are naturally suggested by this approach, are given in appendix.
(web, pdf)
Vera Hartenstein and Mario Hubert, When Fields Are Not Degrees of Freedom, Arxiv.Org, 2018 1809.08486v1, physics.hist-ph.
We show that in the Maxwell-Lorentz theory of classical electrodynamics most initial values for fields and particles lead to an ill-defined dynamics, as they exhibit singularities or discontinuities along light-cones. This phenomenon suggests that the Maxwell equations and the Lorentz force law ought rather to be read as a system of delay differential equations, that is, differential equations that relate a function and its derivatives at different times. This mathematical reformulation, however, leads to physical and philosophical consequences for the ontological status of the electromagnetic field. In particular, fields cannot be taken as independent degrees of freedom, which suggests that one should not add them to the ontology.
(web, pdf)
I Orfanid, Maxwell’s Equations, , 2016 pp. 1-18.
(pdf)
Andrzej Wolski, Theory of electromagnetic fields, , 2011.
We discuss the theory of electromagnetic fields, with an emphasis on aspects relevant to radiofrequency systems in particle accelerators. We begin by reviewing Maxwell's equations and their physical significance. We show that in free space, there are solutions to Maxwell's equations representing the propagation of electromagnetic fields as waves. We introduce electromagnetic potentials, and show how they can be used to simplify the calculation of the fields in the presence of sources. We derive Poynting's theorem, which leads to expressions for the energy density and energy flux in an electromagnetic field. We discuss the properties of electromagnetic waves in cavities, waveguides and transmission lines.
(web, pdf)
Thomas Yu, Lagrangian Formulation of the Electromagnetic Field , , 2012 pp. 1-15.
(pdf)