The Hermite-Taylor Correction Function Method for Maxwell’s Equations
Communications on Applied Mathematics and Computation - Trang 1-25 - 2023
Tóm tắt
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.
Tài liệu tham khảo
Abraham, D.S., Marques, A.N., Nave, J.C.: A correction function method for the wave equation with interface jump conditions. J. Comput. Phys. 353, 281–299 (2018)
Assous, F., Ciarlet, P., Labrunie, S.: Mathematical Foundations of Computational Electromagnetism. Springer International Publishing, New York (2018)
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. 161, 218–249 (2000)
Balsara, D.S., 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. 376, 1108–1137 (2019)
Beznosov, O., Appelö, D.: Hermite-discontinuous Galerkin overset grid methods for the scalar wave equation. Commun. Appl. Math. Comput. 3, 391–418 (2021)
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. 48, 1530–1554 (2010)
Chen, X., Appelö, D., Hagstrom, T.: A hybrid Hermite-discontinuous Galerkin method for hyperbolic systems with application to Maxwell’s equations. J. Comput. Phys. 257, 501–520 (2014)
Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)
Fan, G.X., Liu, Q.H., Hesthaven, J.S.: Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media. IEEE Trans. Geosci. Remote Sens. 40, 1366–1373 (2002)
Galagusz, R., Shirokoff, D., Nave, J.C.: A Fourier penalty method for solving the time-dependent Maxwell’s equations in domains with curved boundaries. J. Comput. Phys. 306, 167–198 (2016)
Goodrich, J., Hagstrom, T., Lorenz, J.: Hermite methods for hyperbolic initial-boundary value problems. Math. Comp. 75, 595–630 (2005)
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)
Hazra, A., Chandrashekar, P., Balsara, D.S.: Globally constraint-preserving FR/DG scheme for Maxwell’s equations at all orders. J. Comput. Phys. 394, 298–328 (2019)
Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 186–221 (2002)
Law, Y.M., Marques, A.N., Nave, J.C.: Treatment of complex interfaces for Maxwell’s equations with continuous coefficients using the correction function method. J. Sci. Comput. 82(3), 56 (2020)
Law, Y.M., Nave, J.C.: FDTD schemes for Maxwell’s equations with embedded perfect electric conductors based on the correction function method. J. Sci. Comput. 88(3), 72 (2021)
Law, Y.M., Nave, J.C.: High-order FDTD schemes for Maxwell’s interface problems with discontinuous coefficients and complex interfaces based on the correction function method. J. Sci. Comput. 91(1), 26 (2022)
Liang, D., Yuan, Q.: The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell’s equations. J. Comput. Phys. 243, 344–364 (2013)
Lindell, I., Sihvola, A.: Boundary Conditions in Electromagnetics. John Wiley & Sons, New Jersey (2019)
Loya, A.A., Appelö, D., Henshaw, W.D.: Hermite methods for the wave equation: compatibility and interface conditions (2022) (in preparation)
Marques, A.N., Nave, J.C., Rosales, R.R.: A correction function method for Poisson problems with interface jump conditions. J. Comput. Phys. 230, 7567–7597 (2011)
Marques, A.N., Nave, J.C., Rosales, R.R.: High order solution of Poisson problems with piecewise constant coefficients and interface jumps. J. Comput. Phys. 335, 497–515 (2017)
Marques, A.N., Nave, J.C., Rosales, R.R.: Imposing jump conditions on nonconforming interfaces for the correction function method: a least squares approach. J. Comput. Phys. 397, 108869 (2019)
Namiki, T.: A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microw. Theory Technol. 47, 2003–2007 (1999)
Tan, E.L., Heh, D.Y.: ADI-FDTD method with fourth order accuracy in time. IEEE Microw. Wirel. Compon. Lett. 18, 296–298 (1999)
Xie, Z., Chan, C.H., Zhang, B.: An explicit fourth-order staggered finite-difference time-domain method for Maxwell’s equations. J. Comput. Appl. Math. 147, 75–98 (2002)
Yang, B., Gottlieb, D., Hesthaven, J.S.: Spectral simulations of electromagnetic wave scattering. J. Comput. Phys. 134, 216–230 (1997)
Yee, K.S.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14(3), 302–307 (1966)
Zheng, F., Chen, Z., Zhang, J.: A finite-difference time-domain method without the Courant stability conditions. IEEE Microw. Wirel. Compon. Lett. 9, 441–443 (1999)