TY - JOUR
T1 - A Fourier penalty method for solving the time-dependent Maxwell's equations in domains with curved boundaries
AU - Galagusz, Ryan
AU - Shirokoff, David
AU - Nave, Jean Christophe
N1 - Funding Information:
The authors would like to thank Mark Lyon, Dmitry Kolomenskiy and Kai Schneider for numerous enlightening conversations. This research was partly supported through the NSERC Discovery and Discovery Accelerator Supplement grants of the third author. This work was supported by a grant from the Simons Foundation (# 359610 , David Shirokoff).
Publisher Copyright:
© 2015 Elsevier Inc.
PY - 2016/2/1
Y1 - 2016/2/1
N2 - We present a high order, Fourier penalty method for the Maxwell's equations in the vicinity of perfect electric conductor boundary conditions. The approach relies on extending the smooth non-periodic domain of the equations to a periodic domain by removing the exact boundary conditions and introducing an analytic forcing term in the extended domain. The forcing, or penalty term is chosen to systematically enforce the boundary conditions to high order in the penalty parameter, which then allows for higher order numerical methods. We present an efficient numerical method for constructing the penalty term, and discretize the resulting equations using a Fourier spectral method. We demonstrate convergence orders of up to 3.5 for the one-dimensional Maxwell's equations, and show that the numerical method does not suffer from dispersion (or pollution) errors. We also illustrate the approach in two dimensions and demonstrate convergence orders of 2.5 for transverse magnetic modes and 1.5 for the transverse electric modes. We conclude the paper with numerous test cases in dimensions two and three including waves traveling in a bent waveguide, and scattering off of a windmill-like geometry.
AB - We present a high order, Fourier penalty method for the Maxwell's equations in the vicinity of perfect electric conductor boundary conditions. The approach relies on extending the smooth non-periodic domain of the equations to a periodic domain by removing the exact boundary conditions and introducing an analytic forcing term in the extended domain. The forcing, or penalty term is chosen to systematically enforce the boundary conditions to high order in the penalty parameter, which then allows for higher order numerical methods. We present an efficient numerical method for constructing the penalty term, and discretize the resulting equations using a Fourier spectral method. We demonstrate convergence orders of up to 3.5 for the one-dimensional Maxwell's equations, and show that the numerical method does not suffer from dispersion (or pollution) errors. We also illustrate the approach in two dimensions and demonstrate convergence orders of 2.5 for transverse magnetic modes and 1.5 for the transverse electric modes. We conclude the paper with numerous test cases in dimensions two and three including waves traveling in a bent waveguide, and scattering off of a windmill-like geometry.
KW - Active penalty method
KW - Fourier continuation
KW - Fourier methods
KW - Maxwell equations
KW - Sharp mask function
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U2 - 10.1016/j.jcp.2015.11.031
DO - 10.1016/j.jcp.2015.11.031
M3 - Article
AN - SCOPUS:84949034163
SN - 0021-9991
VL - 306
SP - 167
EP - 198
JO - Journal of Computational Physics
JF - Journal of Computational Physics
ER -