The Hermite-Taylor Correction Function Method for Maxwell’s Equations

Abstract

The Hermite-Taylor method, introduced in 2005 by Goodrich et al. is highly efficient and accurate when applied to linear hyperbolic systems on periodic domains. Unfortunately, its widespread use has been prevented by the lack of a systematic approach to implementing boundary conditions. In this paper we present the Hermite-Taylor correction function method (CFM), which provides exactly such a systematic approach for handling boundary conditions. Here we focus on Maxwell’s equations but note that the method is easily extended to other hyperbolic problems.

Keywords

Hermite method / Correction function method (CFM) / Maxwell’s equations / High order / Boundary conditions / 35Q61 / 65M70

References

[1] Abraham DS, Marques AN, Nave JC. A correction function method for the wave equation with interface jump conditions. J. Comput. Phys.. 2018, 353: 281-299

[2] Assous F, Ciarlet P, Labrunie S. Mathematical Foundations of Computational Electromagnetism. 2018, New York, Springer International Publishing

[3] Assous F, Ciarlet P, Segré J. Numerical solution to time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method. J. Comput. Phys.. 2000, 161: 218-249

[4] Balsara DS, Käppeli R. von Neumann stability analysis of globally constraint-preserving DGTD and PNPM schemes for the Maxwell equations using multidimensional Riemann solvers. J. Comput. Phys.. 2019, 376: 1108-1137

[5] Beznosov O, Appelö D. Hermite-discontinuous Galerkin overset grid methods for the scalar wave equation. Commun. Appl. Math. Comput.. 2021, 3: 391-418

[6] Chen W, Li X, Liang D. Energey-conserved splitting finite-difference time-domain methods for Maxwell’s equations in three dimensions. SIAM J. Numer. Anal.. 2010, 48: 1530-1554

[7] Chen X, Appelö D, Hagstrom T. A hybrid Hermite-discontinuous Galerkin method for hyperbolic systems with application to Maxwell’s equations. J. Comput. Phys.. 2014, 257: 501-520

[8] Cockburn B, Shu C-W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput.. 2001, 16(3): 173-261

[9] Fan GX, Liu QH, Hesthaven JS. Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media. IEEE Trans. Geosci. Remote Sens.. 2002, 40: 1366-1373

[10] Galagusz R, Shirokoff D, Nave JC. A Fourier penalty method for solving the time-dependent Maxwell’s equations in domains with curved boundaries. J. Comput. Phys.. 2016, 306: 167-198

[11] Goodrich J, Hagstrom T, Lorenz J. Hermite methods for hyperbolic initial-boundary value problems. Math. Comp.. 2005, 75: 595-630

[12] Hagstrom, T., Appelö, D.: Solving PDEs with Hermite interpolation. In: Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014: Selected papers from the ICOSAHOM Conference, June 23–27, 2014, Salt Lake City, Utah, USA, pp. 31–49. Springer (2015)

[13] Hazra A, Chandrashekar P, Balsara DS. Globally constraint-preserving FR/DG scheme for Maxwell’s equations at all orders. J. Comput. Phys.. 2019, 394: 298-328

[14] Hesthaven JS, Warburton T. Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell’s equations. J. Comput. Phys.. 2002, 181: 186-221

[15] Law YM, Marques AN, Nave JC. Treatment of complex interfaces for Maxwell’s equations with continuous coefficients using the correction function method. J. Sci. Comput.. 2020, 82(3): 56

[16] Law YM, Nave JC. FDTD schemes for Maxwell’s equations with embedded perfect electric conductors based on the correction function method. J. Sci. Comput.. 2021, 88(3): 72

[17] Law YM, Nave JC. High-order FDTD schemes for Maxwell’s interface problems with discontinuous coefficients and complex interfaces based on the correction function method. J. Sci. Comput.. 2022, 91(1): 26

[18] Liang D, Yuan Q. The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell’s equations. J. Comput. Phys.. 2013, 243: 344-364

[19] Lindell I, Sihvola A. Boundary Conditions in Electromagnetics. 2019, New Jersey, John Wiley & Sons

[20] Loya, A.A., Appelö, D., Henshaw, W.D.: Hermite methods for the wave equation: compatibility and interface conditions (2022) (in preparation)

[21] Marques AN, Nave JC, Rosales RR. A correction function method for Poisson problems with interface jump conditions. J. Comput. Phys.. 2011, 230: 7567-7597

[22] Marques AN, Nave JC, Rosales RR. High order solution of Poisson problems with piecewise constant coefficients and interface jumps. J. Comput. Phys.. 2017, 335: 497-515

[23] Marques AN, Nave JC, Rosales RR. Imposing jump conditions on nonconforming interfaces for the correction function method: a least squares approach. J. Comput. Phys.. 2019, 397 108869

[24] Namiki T. A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microw. Theory Technol.. 1999, 47: 2003-2007

[25] Tan EL, Heh DY. ADI-FDTD method with fourth order accuracy in time. IEEE Microw. Wirel. Compon. Lett.. 1999, 18: 296-298

[26] Xie Z, Chan CH, Zhang B. An explicit fourth-order staggered finite-difference time-domain method for Maxwell’s equations. J. Comput. Appl. Math.. 2002, 147: 75-98

[27] Yang B, Gottlieb D, Hesthaven JS. Spectral simulations of electromagnetic wave scattering. J. Comput. Phys.. 1997, 134: 216-230

[28] Yee KS. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag.. 1966, 14(3): 302-307

[29] Zheng F, Chen Z, Zhang J. A finite-difference time-domain method without the Courant stability conditions. IEEE Microw. Wirel. Compon. Lett.. 1999, 9: 441-443

Funding

National Science Foundation(NSF-2208164)

RIGHTS & PERMISSIONS

Shanghai University