Abstract

A new multidomain pseudospectral frequency-domain (PSFD) method based on the Legendre polynomials with penalty scheme is developed for numerically modeling electromagnetic wave scattering problems. The primary aim of the proposed method is to more accurately analyzing scattering and coupling problems in plasmonics, in which optical waves interact with nanometer-sized metallic structures. Using light scattering by a silver circular cylinder as a first example, the formulated method is demonstrated to achieve numerical accuracy in near-field calculations on the order of 10<sup>-9</sup> with respect to a unity field strength of the incident wave with excellent exponentially convergent behavior in numerical accuracy. Then, scattering by a dielectric square cylinder and that by several coupled metallic structures involving circular cylinders, square cylinders, or dielectric coated cylinders are examined to provide high-accuracy coupled near-field results.

© 2012 IEEE

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  1. S. A. Maier, H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, (2005) Art. ID 011101.
  2. M. Moskovits, "Surface-enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985).
  3. P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, D. W. Pohl, "Resonant optical antennas," Science 308, 1607-1609 (2005).
  4. S. A. Maier, P. G. Kik, H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
  5. J. P. Kottmann, O. J. F. Martin, "Retardation-induced plasmon resonances in coupled nanoparticles," Opt. Lett. 26, 1096-1098 (2001).
  6. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  7. R. L. Chern, X. X. Liu, C. C. Chang, "Particle plasmons of metal nanospheres: Application of multiple scattering approach," Phys. Rev. E 76, (2007) Art. ID 016609.
  8. R. Gomez-Medina, M. Laroche, J. J. Saenz, "Extraordinary optical reflection from sub-wavelength cylinder arrays," Opt. Exp. 14, 3730-3737 (2006).
  9. A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  10. M. Y. Ng, W. C. Liu, "Local-field confinement in three-pair arrays of metallic nanocylinders," Opt. Exp. 14, 4504-4513 (2006).
  11. J. P. Kottmann, O. J. F. Martin, "Plasmon resonant coupling in metallic nanowires," Opt. Exp. 8, 655-663 (2001).
  12. M. W. Chen, Y. F. Chau, D. P. Tsai, "Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres," Plasmonics 3, 157-164 (2008).
  13. R. Rodriguez-Oliveros, J. A. Sanchez-Gil, "Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surfaces," Opt. Exp. 19, 12208-12219 (2011).
  14. J. P. Kottmann, O. J. F. Martin, D. R. Smith, S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, (2001) Art. ID 235402.
  15. E. Prodan, C. Radloff, N. J. Halas, P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419-422 (2003).
  16. B. Yang, D. Gottlieb, J. S. Hesthaven, "Spectral simulations of electromagnetic wave scattering," J. Comput. Phys. 134, 216-230 (1997).
  17. B. Yang, J. S. Hesthaven, "A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution," IEEE Trans. Antennas Propagat. 47, 132-141 (1999).
  18. J. S. Hesthaven, P. G. Dinesen, J. P. Lynov, "Spectral collocation time-domain modeling of diffractive optical elements," J. Comput. Phys. 155, 287-306 (1999).
  19. G. Zhao, Q. H. Liu, "The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media," IEEE Trans. Antennas Propagat. 52, 742-749 (2004).
  20. Q. H. Liu, "A pseudospectral frequency-domain (PSFD) method for computational electromagnetics," IEEE Antennas Wireless Propagat. Lett. 1, 131-134 (2002).
  21. P. J. Chiang, C. P. Yu, H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, (2007) Art. ID 026703.
  22. P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, H. C. Chang, "Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations," IEEE J. Quantum Electron. 44, 56-66 (2008).
  23. C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, K. A. Feng, "A Legendre pseudospectral penalty scheme for solving time-domain Maxwell's equations," J. Sci. Comput. 36, 351-390 (2008).
  24. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
  25. S. Abarbanel, D. Gottlieb, "A mathematical analysis of the PML method," J. Comput. Phys. 134, 357-363 (1997).
  26. S. Abarbanel, D. Gottlieb, "On the construction and analysis of absorbing layers in CEM," Appl. Numer. Math. 27, 331-340 (1998).
  27. L. Rayleigh, "The dispersal of light by a dielectric cylinder," Philos. Mag. 36, 365-376 (1918).
  28. J. R. Wait, "Scattering of plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189-195 (1955).
  29. W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).
  30. S. Dey, R. Mittra, "A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects," IEEE Microw. Guided Wave Lett. 7, 273-275 (1997).
  31. Y. Liu, C. D. Sarris, G. V. Eleftheriades, "Triangular-mesh-based FDTD analysis of two-dimensional plasmonic structures supporting backward waves at optical frequencies," J. Lightw. Technol. 25, 938-946 (2007).
  32. Y. Zhao, Y. Hao, "Finite-difference time-domain study of guided modes in nano-plasmonic waveguides," IEEE Trans. Antennas Propagat. 55, 3070-3077 (2007).
  33. W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending-function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).
  34. P. B. Johnson, R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
  35. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  36. N. N. Rao, Elements of Engineering Electromagnetics (Prentice-Hall, 2004).
  37. J. B. Andersen, V. V. Solodukhov, "Field behavior near a dielectric wedge," IEEE Trans. Antennas Propagat. 26, 598-602 (1978).
  38. W. S. Don, A. Solomonoff, "Accuracy and speed in computing the Chebyshev collocation derivative," SIAM J. Sci. Comput. 16, 1253-1268 (1995).
  39. J. S. Hesthaven, "A stable penalty method for the compressible Navier-Stokes equations: III. Multidimensional domain decomposition schemes," SIAM J. Sci. Comput. 20, 62-93 (1998).
  40. J. S. Hesthaven, T. Warburton, "Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell's equations," J. Comput. Phys. 181, 186-221 (2002).
  41. D. Funaro, D. Gottlieb, "A new method of imposing boundary-conditions in pseudospectral approximations of hyperbolic-equations," Math. Computation 51, 599-613 (1988).
  42. J. S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems (Cambridge Univ., 2007).
  43. D. Gottlieb, M. Gunzburger, E. Turkel, "On numerical boundary treatment of hyperbolic systems for finite difference and finite element methods," SIAM J. Numer. Anal. 19, 671-681 (1982).
  44. T. Apel, S. Nicaise, "The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges," Math. Methods Appl. Sci. 21, 519-549 (1998).
  45. T. Apel, A.-M. Sandig, J. R. Whiteman, "Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains," Math. Methods Appl. Sci. 19, 63-85 (1996).
  46. K. Schmidt, P. Kauf, "Computation of the band structure of two-dimensional photonic crystals with HP finite elements," Comput. Methods Appl. Mech. Eng. 198, 1249-1259 (2009).

2011 (1)

R. Rodriguez-Oliveros, J. A. Sanchez-Gil, "Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surfaces," Opt. Exp. 19, 12208-12219 (2011).

2009 (1)

K. Schmidt, P. Kauf, "Computation of the band structure of two-dimensional photonic crystals with HP finite elements," Comput. Methods Appl. Mech. Eng. 198, 1249-1259 (2009).

2008 (3)

M. W. Chen, Y. F. Chau, D. P. Tsai, "Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres," Plasmonics 3, 157-164 (2008).

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, H. C. Chang, "Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations," IEEE J. Quantum Electron. 44, 56-66 (2008).

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, K. A. Feng, "A Legendre pseudospectral penalty scheme for solving time-domain Maxwell's equations," J. Sci. Comput. 36, 351-390 (2008).

2007 (4)

P. J. Chiang, C. P. Yu, H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, (2007) Art. ID 026703.

R. L. Chern, X. X. Liu, C. C. Chang, "Particle plasmons of metal nanospheres: Application of multiple scattering approach," Phys. Rev. E 76, (2007) Art. ID 016609.

Y. Liu, C. D. Sarris, G. V. Eleftheriades, "Triangular-mesh-based FDTD analysis of two-dimensional plasmonic structures supporting backward waves at optical frequencies," J. Lightw. Technol. 25, 938-946 (2007).

Y. Zhao, Y. Hao, "Finite-difference time-domain study of guided modes in nano-plasmonic waveguides," IEEE Trans. Antennas Propagat. 55, 3070-3077 (2007).

2006 (2)

R. Gomez-Medina, M. Laroche, J. J. Saenz, "Extraordinary optical reflection from sub-wavelength cylinder arrays," Opt. Exp. 14, 3730-3737 (2006).

M. Y. Ng, W. C. Liu, "Local-field confinement in three-pair arrays of metallic nanocylinders," Opt. Exp. 14, 4504-4513 (2006).

2005 (2)

S. A. Maier, H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, (2005) Art. ID 011101.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, D. W. Pohl, "Resonant optical antennas," Science 308, 1607-1609 (2005).

2004 (1)

G. Zhao, Q. H. Liu, "The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media," IEEE Trans. Antennas Propagat. 52, 742-749 (2004).

2003 (2)

E. Prodan, C. Radloff, N. J. Halas, P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419-422 (2003).

S. A. Maier, P. G. Kik, H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).

2002 (2)

Q. H. Liu, "A pseudospectral frequency-domain (PSFD) method for computational electromagnetics," IEEE Antennas Wireless Propagat. Lett. 1, 131-134 (2002).

J. S. Hesthaven, T. Warburton, "Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell's equations," J. Comput. Phys. 181, 186-221 (2002).

2001 (3)

J. P. Kottmann, O. J. F. Martin, D. R. Smith, S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, (2001) Art. ID 235402.

J. P. Kottmann, O. J. F. Martin, "Plasmon resonant coupling in metallic nanowires," Opt. Exp. 8, 655-663 (2001).

J. P. Kottmann, O. J. F. Martin, "Retardation-induced plasmon resonances in coupled nanoparticles," Opt. Lett. 26, 1096-1098 (2001).

1999 (2)

B. Yang, J. S. Hesthaven, "A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution," IEEE Trans. Antennas Propagat. 47, 132-141 (1999).

J. S. Hesthaven, P. G. Dinesen, J. P. Lynov, "Spectral collocation time-domain modeling of diffractive optical elements," J. Comput. Phys. 155, 287-306 (1999).

1998 (3)

T. Apel, S. Nicaise, "The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges," Math. Methods Appl. Sci. 21, 519-549 (1998).

S. Abarbanel, D. Gottlieb, "On the construction and analysis of absorbing layers in CEM," Appl. Numer. Math. 27, 331-340 (1998).

J. S. Hesthaven, "A stable penalty method for the compressible Navier-Stokes equations: III. Multidimensional domain decomposition schemes," SIAM J. Sci. Comput. 20, 62-93 (1998).

1997 (3)

S. Abarbanel, D. Gottlieb, "A mathematical analysis of the PML method," J. Comput. Phys. 134, 357-363 (1997).

S. Dey, R. Mittra, "A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects," IEEE Microw. Guided Wave Lett. 7, 273-275 (1997).

B. Yang, D. Gottlieb, J. S. Hesthaven, "Spectral simulations of electromagnetic wave scattering," J. Comput. Phys. 134, 216-230 (1997).

1996 (1)

T. Apel, A.-M. Sandig, J. R. Whiteman, "Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains," Math. Methods Appl. Sci. 19, 63-85 (1996).

1995 (1)

W. S. Don, A. Solomonoff, "Accuracy and speed in computing the Chebyshev collocation derivative," SIAM J. Sci. Comput. 16, 1253-1268 (1995).

1994 (1)

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).

1988 (1)

D. Funaro, D. Gottlieb, "A new method of imposing boundary-conditions in pseudospectral approximations of hyperbolic-equations," Math. Computation 51, 599-613 (1988).

1985 (1)

M. Moskovits, "Surface-enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985).

1982 (1)

D. Gottlieb, M. Gunzburger, E. Turkel, "On numerical boundary treatment of hyperbolic systems for finite difference and finite element methods," SIAM J. Numer. Anal. 19, 671-681 (1982).

1978 (1)

J. B. Andersen, V. V. Solodukhov, "Field behavior near a dielectric wedge," IEEE Trans. Antennas Propagat. 26, 598-602 (1978).

1973 (2)

W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).

W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending-function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).

1972 (1)

P. B. Johnson, R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).

1955 (1)

J. R. Wait, "Scattering of plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189-195 (1955).

1918 (1)

L. Rayleigh, "The dispersal of light by a dielectric cylinder," Philos. Mag. 36, 365-376 (1918).

Appl. Numer. Math. (1)

S. Abarbanel, D. Gottlieb, "On the construction and analysis of absorbing layers in CEM," Appl. Numer. Math. 27, 331-340 (1998).

Can. J. Phys. (1)

J. R. Wait, "Scattering of plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189-195 (1955).

Comput. Methods Appl. Mech. Eng. (1)

K. Schmidt, P. Kauf, "Computation of the band structure of two-dimensional photonic crystals with HP finite elements," Comput. Methods Appl. Mech. Eng. 198, 1249-1259 (2009).

IEEE Antennas Wireless Propagat. Lett. (1)

Q. H. Liu, "A pseudospectral frequency-domain (PSFD) method for computational electromagnetics," IEEE Antennas Wireless Propagat. Lett. 1, 131-134 (2002).

IEEE J. Quantum Electron. (1)

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, H. C. Chang, "Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations," IEEE J. Quantum Electron. 44, 56-66 (2008).

IEEE Microw. Guided Wave Lett. (1)

S. Dey, R. Mittra, "A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects," IEEE Microw. Guided Wave Lett. 7, 273-275 (1997).

IEEE Trans. Antennas Propagat. (4)

Y. Zhao, Y. Hao, "Finite-difference time-domain study of guided modes in nano-plasmonic waveguides," IEEE Trans. Antennas Propagat. 55, 3070-3077 (2007).

J. B. Andersen, V. V. Solodukhov, "Field behavior near a dielectric wedge," IEEE Trans. Antennas Propagat. 26, 598-602 (1978).

B. Yang, J. S. Hesthaven, "A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution," IEEE Trans. Antennas Propagat. 47, 132-141 (1999).

G. Zhao, Q. H. Liu, "The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media," IEEE Trans. Antennas Propagat. 52, 742-749 (2004).

J. Appl. Phys. (1)

S. A. Maier, H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, (2005) Art. ID 011101.

J. Comput. Phys. (5)

B. Yang, D. Gottlieb, J. S. Hesthaven, "Spectral simulations of electromagnetic wave scattering," J. Comput. Phys. 134, 216-230 (1997).

J. S. Hesthaven, P. G. Dinesen, J. P. Lynov, "Spectral collocation time-domain modeling of diffractive optical elements," J. Comput. Phys. 155, 287-306 (1999).

J. S. Hesthaven, T. Warburton, "Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell's equations," J. Comput. Phys. 181, 186-221 (2002).

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).

S. Abarbanel, D. Gottlieb, "A mathematical analysis of the PML method," J. Comput. Phys. 134, 357-363 (1997).

J. Lightw. Technol. (1)

Y. Liu, C. D. Sarris, G. V. Eleftheriades, "Triangular-mesh-based FDTD analysis of two-dimensional plasmonic structures supporting backward waves at optical frequencies," J. Lightw. Technol. 25, 938-946 (2007).

J. Sci. Comput. (1)

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, K. A. Feng, "A Legendre pseudospectral penalty scheme for solving time-domain Maxwell's equations," J. Sci. Comput. 36, 351-390 (2008).

Math. Computation (1)

D. Funaro, D. Gottlieb, "A new method of imposing boundary-conditions in pseudospectral approximations of hyperbolic-equations," Math. Computation 51, 599-613 (1988).

Math. Methods Appl. Sci. (2)

T. Apel, S. Nicaise, "The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges," Math. Methods Appl. Sci. 21, 519-549 (1998).

T. Apel, A.-M. Sandig, J. R. Whiteman, "Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains," Math. Methods Appl. Sci. 19, 63-85 (1996).

Numer. Math. (2)

W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending-function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).

W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).

Opt. Exp. (4)

R. Gomez-Medina, M. Laroche, J. J. Saenz, "Extraordinary optical reflection from sub-wavelength cylinder arrays," Opt. Exp. 14, 3730-3737 (2006).

M. Y. Ng, W. C. Liu, "Local-field confinement in three-pair arrays of metallic nanocylinders," Opt. Exp. 14, 4504-4513 (2006).

J. P. Kottmann, O. J. F. Martin, "Plasmon resonant coupling in metallic nanowires," Opt. Exp. 8, 655-663 (2001).

R. Rodriguez-Oliveros, J. A. Sanchez-Gil, "Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surfaces," Opt. Exp. 19, 12208-12219 (2011).

Opt. Lett. (1)

Philos. Mag. (1)

L. Rayleigh, "The dispersal of light by a dielectric cylinder," Philos. Mag. 36, 365-376 (1918).

Phys. Rev. B (3)

P. B. Johnson, R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).

J. P. Kottmann, O. J. F. Martin, D. R. Smith, S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, (2001) Art. ID 235402.

S. A. Maier, P. G. Kik, H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).

Phys. Rev. E (2)

R. L. Chern, X. X. Liu, C. C. Chang, "Particle plasmons of metal nanospheres: Application of multiple scattering approach," Phys. Rev. E 76, (2007) Art. ID 016609.

P. J. Chiang, C. P. Yu, H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, (2007) Art. ID 026703.

Plasmonics (1)

M. W. Chen, Y. F. Chau, D. P. Tsai, "Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres," Plasmonics 3, 157-164 (2008).

Rev. Mod. Phys. (1)

M. Moskovits, "Surface-enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985).

Science (2)

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, D. W. Pohl, "Resonant optical antennas," Science 308, 1607-1609 (2005).

E. Prodan, C. Radloff, N. J. Halas, P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419-422 (2003).

SIAM J. Numer. Anal. (1)

D. Gottlieb, M. Gunzburger, E. Turkel, "On numerical boundary treatment of hyperbolic systems for finite difference and finite element methods," SIAM J. Numer. Anal. 19, 671-681 (1982).

SIAM J. Sci. Comput. (2)

W. S. Don, A. Solomonoff, "Accuracy and speed in computing the Chebyshev collocation derivative," SIAM J. Sci. Comput. 16, 1253-1268 (1995).

J. S. Hesthaven, "A stable penalty method for the compressible Navier-Stokes equations: III. Multidimensional domain decomposition schemes," SIAM J. Sci. Comput. 20, 62-93 (1998).

Other (5)

J. S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems (Cambridge Univ., 2007).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

N. N. Rao, Elements of Engineering Electromagnetics (Prentice-Hall, 2004).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

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