International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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Please note that all times are shown in the time zone of the conference. The current conference time is: 28th Nov 2022, 08:39:38pm CET
MS 30b: advanced numerical methods for electromagnetic problems
The primary objective of this minisymposium is to present new developments in theory and algo-
rithms related to discretization of Maxwell’s equations. Engineered electromagnetic materials show
great promise to revolutionize the field of optical design by providing smaller and lighter devices
that consume less power while also being multi-functional and tunable. Numerical algorithms and
simulation tools will play an important role in turning this promise into practical devices. In this
minisymposium we will bring together researchers with the intent to provide a forum for cross fer-
tilization of ideas. The minisymposium will feature both senior and junior researchers.
4:00pm - 4:30pm
Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems
1Michigan State University, United States of America; 2University of Colorado, Boulder; 3Kansas State University
Fourier continuation is an approach used to create periodic extensions of non-periodic functions in order to obtain highly-accurate Fourier expansions. Fourier Continuation methods have been used in collocation based PDE-solvers and have demonstrated high-order convergence and accurate dispersion relations. Here we use the discrete Fourier extension by Albin et al. to create a basis to be used together with the Discontinuous Galerkin framework and demonstrate the proposed methods high order accuracy for linear hyperbolic problems. In particular we will demonstrate its use for dispersive models of Maxwell's equations.
4:30pm - 5:00pm
High-order accurate schemes for Maxwell’s equations with nonlinear active media and material interfaces
1Rensselaer Polytechnic Institute, United States of America; 2Purdue University, United States of America
In this talk, we will consider nonlinear dispersive models for active multi-level media with material interfaces in 2D and 3D arbitrary geometry. The considered nonlinear multilevel systems in the formulation of rate equations are experimental generalizations of two-level atomic systems, which are equivalent to the two-level classical Maxwell-Bloch systems in the density matrix formulation. We developed efficient and high-order schemes for arbitrary multilevel systems with any number of polarization vectors in high dimensional complex geometry, with or without material interfaces, in the Overture framework using overlapping grids. The developed schemes allow compact stencils in time integration, large CFL numbers, and point-wise update of the numerical solutions. It is also shown that the multilevel systems have bounded growth in the energy estimate. Additionally, numerical evidences, including convergence results and simulations, are provided for planar and curved interfaces in both 2D and 3D. This is joint work with J. W. Banks, G. Kovačič, W. D. Henshaw, D. W. Schwendeman from RPI, and A. V. Kildishev, L. J. Prokopeva from Purdue University.
5:00pm - 5:30pm
On stability and convergence of time-domain Foldy-Lax models
A Foldy-Lax [Foldy 1945], [Lax 1951, 1952] model describes frequency-domain wave scattering by multiple small obstacles of circular shape, in the regime when the size of the obstacles tends to zero. It is based on the representation of the total scattered field as a linear combination of the fields scattered by each of the obstacles; the interaction between the obstacles are taken into account when computing the amplitudes of these fields. The latter necessitates solving a linear system of equations of the size $N\times N$, where $N$ is the number of scatterers.
Frequency-domain Foldy-Lax models have been derived and analyzed in [Liao 2013], [Cassier, Hazard 2014], and extended to obstacles of arbitrary shape in e.g. [Challa, Sini 2016]. However, the stability and convergence analysis of their time-domain counterparts is still in their infancy stage. Up to our knowledge, the only article dedicated to this question is [Barucq et al. 2021], where an asymptotic high-order model for time-domain scattering by a single particle in 3D was analyzed.
This talk is dedicated to stability and convergence analysis of the Foldy-Lax model for scattering by $N$ circular scatterers. We prove that the time-domain counterpart of the 2D Foldy-Lax model considered in [Cassier, Hazard 2014] may be unstable for some geometric configurations with $N\geq 3$ scatterers. In order to stabilize the model, we propose its simple reinterpretation it as a Galerkin discretization of a boundary integral equation with a single-layer boundary integral operator. This procedure yields automatically a stable time-domain model, thanks to some coercivity-like properties of the underlying operator. The price to pay is an extra convolution term occurring in the resulting system of equations, compared to [Cassier, Hazard 2014]. We present the convergence analysis of this model and illustrate our findings with numerical experiments.
5:30pm - 6:00pm
The virtual element method for resistive magnetohydrodynamics
1Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA; 2Universita di Milano-Bicocca; 3Group T-5, Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545 NM, USA
We present a virtual element method (VEM) for the numerical approximation of the electromagnetics subsystem of the resistive magnetohydrodynamics (MHD) model in two spatial dimensions. The major advantages of the virtual element method include great flexibility of polygonal meshes and automatic divergence-free constraint on the magnetic flux field. In this work, we rigorously prove the well-posedness of the method and the solenoidal nature of the discrete magnetic flux field. We also derive stability energy estimates. The design of the method includes three choices for the construction of the nodal mass matrix and criteria for more alternatives. This approach is novel in the VEM literature and allows us to preserve a commuting diagram property. We present a set of numerical experiments that independently validate theoretical results. The numerical experiments include the convergence rate study, energy estimates and verification of the divergence-free condition on the magnetic flux field. All these numerical experiments have been performed on triangular, perturbed quadrilateral and Voronoi meshes. Finally, we demonstrate the development of the VEM method on a numerical model for Hartmann flows as well as in the case of magnetic reconnection. Our results have been published in Comput. Methods Appl. Mech. Engrg. 381 (2021) 113815.
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