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Dispersion reducing methods for edge discretizations of the electric vector wave equation

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  • We present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time-domain electric vector wave equation on square meshes based on the mimetic finite difference (MFD) method. We compare this strategy, called M-adaptation, to two other discretizations, also based on square meshes. One is the lowest order Nédélec edge element discretization. The other is a modified quadrature approach (GY-adaptation) proposed by Guddati and Yue for the acoustic wave equation in two dimensions. All three discrete methods use the same edge-based degrees of freedom, while the temporal discretization is performed using the standard explicit Leapfrog scheme. To obtain efficient and explicit time stepping methods, the three schemes are further mass lumped. We perform a dispersion and stability analysis for the presented schemes and compare all three methods in terms of their stability regions and phase error. Our results indicate that the method produced by GY-adaptation and the Nédélec method are both second order accurate for numerical dispersion, but differ in the order of their numerical anisotropy (fourth order, versus second order, respectively). The result of M-adaptation is a discretization that is fourth order accurate for numerical dispersion as well as numerical anisotropy. Numerical simulations are provided that illustrate the theoretical results.
  • Keywords: Nédélec edge elements, Vector wave equation, M-adaptation, Maxwell’s equations, Anisotropy, Dispersion
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  • Bokil, V. A., Gibson, N. L., Gyrya, V., & McGregor, D. A. (2015). Dispersion reducing methods for edge discretizations of the electric vector wave equation. Journal of Computational Physics, 287, 88-109. doi:10.1016/j.jcp.2015.01.042
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  • 287
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  • V. Gyrya’s work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396 and the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research.V.A. Bokil’s and D.A. McGregor’s work was partially supported by the National Science Foundation Grant Number#0811223. D. McGregor received additional support from the T5 group of the Los Alamos National Laboratory.
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