Abstract

The discontinuous Galerkin time-domain method (DGTD) is an emerging technique for the numerical simulation of time-dependent electromagnetic phenomena. For many applications it is necessary to model the infinite space which surrounds scatterers and sources. As a result, absorbing boundaries which mimic its properties play a key role in making DGTD a versatile tool for various kinds of systems. Popular techniques include the Silver-Müller boundary condition and uniaxial perfectly matched layers (UPMLs). We provide novel instructions for the implementation of stretched-coordinate perfectly matched layers in a discontinuous Galerkin framework and compare the performance of the three absorbers for a three-dimensional test system.

© 2011 Optical Society of America

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References

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  1. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  2. J. S. Hesthaven, and T. Warburton, “Nodal high-order methods on unstructured grids–I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
    [CrossRef]
  3. J. S. Hesthaven, and T. Warburton, Nodal Discontinuous Galerkin Methods—Algorithms, Analysis, and Applications (Springer, 2007).
    [PubMed]
  4. T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
    [CrossRef]
  5. J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
    [CrossRef]
  6. M. König, K. Busch, and J. Niegemann, “The discontinuous Galerkin time-domain method for Maxwell’s equations with anisotropic materials,” Photonics Nanostruct. Fundam. Appl. 8, 303–309 (2010).
    [CrossRef]
  7. A. Hille, R. Kullock, S. Grafström, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method computational,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
    [CrossRef]
  8. N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: the role of separation and relative orientation,” Opt. Express 18, 6545–6554 (2010).
    [CrossRef] [PubMed]
  9. J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
    [CrossRef]
  10. T. Hagstrom, and S. Lau, “Radiation boundary conditions for Maxwell’s equations: A review of accurate timedomain formulations,” J. Comput. Math. 25, 305–336 (2007).
  11. J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  12. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. 43, 1460–1463 (1995).
    [CrossRef]
  13. W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
    [CrossRef]
  14. J.-P. Bérenger, Perfectly Matched Layer (PML) for Computational Electromagnetics (Morgan & Claypool Publishers, 2007).
  15. P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
    [CrossRef] [PubMed]
  16. M. H. Carpenter, and C. A. Kennedy, “Fourth-order 2N-storage Runge–Kutta schemes,” NASA Tech. Memo. 109112 (1994).
  17. R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
    [CrossRef]
  18. R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, 2002).
    [CrossRef]

2010 (4)

M. König, K. Busch, and J. Niegemann, “The discontinuous Galerkin time-domain method for Maxwell’s equations with anisotropic materials,” Photonics Nanostruct. Fundam. Appl. 8, 303–309 (2010).
[CrossRef]

A. Hille, R. Kullock, S. Grafström, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method computational,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: the role of separation and relative orientation,” Opt. Express 18, 6545–6554 (2010).
[CrossRef] [PubMed]

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

2009 (2)

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

2007 (1)

T. Hagstrom, and S. Lau, “Radiation boundary conditions for Maxwell’s equations: A review of accurate timedomain formulations,” J. Comput. Math. 25, 305–336 (2007).

2004 (1)

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

2002 (1)

J. S. Hesthaven, and T. Warburton, “Nodal high-order methods on unstructured grids–I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

1995 (1)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

1994 (2)

W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bérenger, J. P.

J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Busch, K.

N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: the role of separation and relative orientation,” Opt. Express 18, 6545–6554 (2010).
[CrossRef] [PubMed]

M. König, K. Busch, and J. Niegemann, “The discontinuous Galerkin time-domain method for Maxwell’s equations with anisotropic materials,” Photonics Nanostruct. Fundam. Appl. 8, 303–309 (2010).
[CrossRef]

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

Cai, W.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

Chew, W. C.

W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Diehl, R.

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

Eng, L. M.

A. Hille, R. Kullock, S. Grafström, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method computational,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

Feth, N.

Grafström, S.

A. Hille, R. Kullock, S. Grafström, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method computational,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

Hagstrom, T.

T. Hagstrom, and S. Lau, “Radiation boundary conditions for Maxwell’s equations: A review of accurate timedomain formulations,” J. Comput. Math. 25, 305–336 (2007).

Hesthaven, J. S.

J. S. Hesthaven, and T. Warburton, “Nodal high-order methods on unstructured grids–I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

Hille, A.

A. Hille, R. Kullock, S. Grafström, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method computational,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

Husnik, M.

Kingsland, D. M.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

König, M.

N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: the role of separation and relative orientation,” Opt. Express 18, 6545–6554 (2010).
[CrossRef] [PubMed]

M. König, K. Busch, and J. Niegemann, “The discontinuous Galerkin time-domain method for Maxwell’s equations with anisotropic materials,” Photonics Nanostruct. Fundam. Appl. 8, 303–309 (2010).
[CrossRef]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

Kullock, R.

A. Hille, R. Kullock, S. Grafström, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method computational,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

Lau, S.

T. Hagstrom, and S. Lau, “Radiation boundary conditions for Maxwell’s equations: A review of accurate timedomain formulations,” J. Comput. Math. 25, 305–336 (2007).

Lee, J.-F.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

Lee, R.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

Linden, S.

Lu, T.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

Niegemann, J.

M. König, K. Busch, and J. Niegemann, “The discontinuous Galerkin time-domain method for Maxwell’s equations with anisotropic materials,” Photonics Nanostruct. Fundam. Appl. 8, 303–309 (2010).
[CrossRef]

N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: the role of separation and relative orientation,” Opt. Express 18, 6545–6554 (2010).
[CrossRef] [PubMed]

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

Pernice, W.

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

Sacks, Z. S.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

Stannigel, K.

N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: the role of separation and relative orientation,” Opt. Express 18, 6545–6554 (2010).
[CrossRef] [PubMed]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

Warburton, T.

J. S. Hesthaven, and T. Warburton, “Nodal high-order methods on unstructured grids–I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

Weedon, W. H.

W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Wegener, M.

Zhang, P.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

J. Comput. Math. (1)

T. Hagstrom, and S. Lau, “Radiation boundary conditions for Maxwell’s equations: A review of accurate timedomain formulations,” J. Comput. Math. 25, 305–336 (2007).

J. Comput. Phys. (3)

J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. S. Hesthaven, and T. Warburton, “Nodal high-order methods on unstructured grids–I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

J. Comput. Theor. Nanosci. (2)

A. Hille, R. Kullock, S. Grafström, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method computational,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Opt. Express (1)

Photonics Nanostruct. Fundam. Appl. (2)

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

M. König, K. Busch, and J. Niegemann, “The discontinuous Galerkin time-domain method for Maxwell’s equations with anisotropic materials,” Photonics Nanostruct. Fundam. Appl. 8, 303–309 (2010).
[CrossRef]

Other (6)

J. S. Hesthaven, and T. Warburton, Nodal Discontinuous Galerkin Methods—Algorithms, Analysis, and Applications (Springer, 2007).
[PubMed]

A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

J.-P. Bérenger, Perfectly Matched Layer (PML) for Computational Electromagnetics (Morgan & Claypool Publishers, 2007).

P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
[CrossRef] [PubMed]

M. H. Carpenter, and C. A. Kennedy, “Fourth-order 2N-storage Runge–Kutta schemes,” NASA Tech. Memo. 109112 (1994).

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, 2002).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Working principle of perfectly matched layers. Incident radiation is attenuated in the PML region. At the boundary of the computational domain the light wave is reflected and undergoes continued attenuation. Once it reenters the physical region, its amplitude (typically suppressed by multiple orders of magnitude) no longer presents a significant perturbation to the physical fields. Please note the absence of reflections at the interface between the physical and the PML region.

Fig. 2
Fig. 2

Setup of the test system. The left panel shows a surface mesh of the computational domain. The red triangles indicate the extent of the boundary layer. The blue surface is used for the injection of radiation as generated by an oscillating dipole located at the center of the system. The volume mesh consists of a number of tetrahedrons, a few of which are depicted in the right panel. As outlined in the text, five tetrahedrons make up a cube. For better visibility, all tetrahedrons have been colored and shrunken.

Fig. 3
Fig. 3

Performance of the stretched-coordinate formulation in comparison to UPMLs and Silver-Müller boundary conditions for p = 3 and one layer of PMLs. The left panel shows the average error ℰ for a test system terminated using the PEC boundary condition. The right-hand side shows corresponding results with the PEC replaced by a Silver-Müller boundary condition. Please note the logarithmic scales of the false color plots. For a description of the isocontours please refer to section 7.2.

Fig. 4
Fig. 4

Performance of the stretched-coordinate formulation in comparison to UPMLs and Silver-Müller boundary conditions for p = 4 and one layer of PMLs. For a more detailed explanation please refer to the caption of Fig. 3.

Fig. 5
Fig. 5

Performance of the stretched-coordinate formulation in comparison to UPMLs and Silver-Müller boundary conditions for p = 5 and one layer of PMLs. For a more detailed explanation please refer to the caption of Fig. 3.

Fig. 6
Fig. 6

Comparison of the PML performance for two and three layers and p = 3. For a more detailed explanation please refer to the caption of Fig. 3.

Tables (2)

Tables Icon

Table 1 Optimal R-parameters for one layer of stretched-coordinate PMLs for various orders p. The computational domain is terminated using a Silver-Müller boundary condition. Please note that the optimal value of the complex frequency-shift α depends on the excitation and, thus, cannot be easily tabularized.

Tables Icon

Table 2 Computational effort required for the simulation of various systems. The table features a short system description (all systems terminated by SMBCs), the total number of elements Ntot in the mesh, the number of PML elements NPML in the mesh, and the CPU time required to evolve the respective system for 15 time units (see section 7.2). The next column compares the CPU time against the system without PMLs and against the system with one layer of UPMLs. Finally, the last column provides an error comparison for the different boundary conditions.

Equations (29)

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x 1 s x x , y 1 s y y , z 1 s z z ,
s i ( ω ) κ i σ i i ω α i .
i ω ɛ E x ( ω ) = 1 s y y H z ( ω ) 1 s z z H y ( ω ) .
E x ( t ) E x ( ω ) t E x ( t ) i ω E x ( ω )
1 s i = 1 + ( 1 s i 1 )
i ω ɛ E x ( ω ) = y H z ( ω ) z H y ( ω ) G x y E ( ω ) G x z E ( ω ) , G x y E ( ω ) = ( 1 s y 1 ) y H z ( ω ) , G x z E ( ω ) = ( 1 s z 1 ) z H y ( ω ) .
1 s i 1 = σ i i ω ( σ i + α i )
i ω G x y E ( ω ) = σ y y H z ( ω ) ( α y + σ y ) G x y E ( ω ) , i ω G x z E ( ω ) = σ z z H y ( ω ) ( α z + σ z ) G x z E ( ω ) .
ɛ t E x ( t ) = y H z ( t ) z H y ( t ) G x y E ( t ) G x z E ( t ) , 1 σ y t G x y E ( t ) = y H z ( t ) α y + σ y σ y G x y E ( t ) , 1 σ z t G x z E ( t ) = z H y ( t ) α z + σ z σ z G x z E ( t ) .
𝒬 t q + F + S = 0 .
q = ( E x , E y , E z , H x , H y , H z , G x y E , G x z E , G y x E , G y z E , G z x E , G z y E , G x y H , G x z H , G y x H , G y z H , G z x H , G z y H ) T .
𝒬 = diag ( ɛ , ɛ , ɛ , μ , μ , μ , 1 σ y , 1 σ z , 1 σ x , , 1 σ y ) .
F x ( q ) = ( 0 , H z , H y , 0 , E z , E y , 0 , 0 , H z , 0 , 0 , H y , 0 , 0 , E z , 0 , 0 , E y ) T F y ( q ) = ( H z , 0 , H x , E z , 0 , E x , H z , 0 , 0 , 0 , H x , 0 , E z , 0 , 0 , 0 , E x , 0 ) T F z ( q ) = ( H y , H x , 0 , E y , E x , 0 , 0 , H y , 0 , H x , 0 , 0 , 0 , E y , 0 , E x , 0 , 0 ) T
S = ( G x y E + G x z E , , G z x H + G z y H , α y + σ y σ y G x y E , , α y + σ y σ y G z y H ) T .
V ( 𝒬 t q L i F L i + S L i ) d 3 r = V ( n ^ F ) L i d 2 r 0 ,
V ( 𝒬 t q + F + S ) L i d 3 r = V n ^ ( F F * ) L i d 2 r 0 ,
ɛ t E ˜ x = 𝒟 y H ˜ z 𝒟 z H ˜ y ( G ˜ x y + G ˜ x z ) + 1 [ n ^ ( F F * ) ] E x , 1 σ y t G ˜ x y = 𝒟 y H ˜ z ( α y + σ y ) G x y + 1 [ n ^ ( F F * ) ] G x z , 1 σ z t G ˜ x z = 𝒟 z H ˜ y ( α z + σ z ) G x z + 1 [ n ^ ( F F * ) ] G x z .
[ n ^ ( F F * ) ] E = n ^ × Z + Δ H n ^ × Δ E Z + + Z
[ n ^ ( F F * ) ] E = n ^ × Z + Δ H n ^ × Δ E Z + + Z , [ n ^ ( F F * ) ] G i j E = k = 1 3 ɛ i j k n j ( Z + Δ H n ^ × Δ E Z + + Z ) k
[ n ^ ( F F * ) ] E x [ n ^ ( F F * ) ] G x y + [ n ^ ( F F * ) ] G x z .
ɛ _ = Λ _ ɛ , μ _ = Λ _ μ , Λ _ = diag ( s y s z s x , s x s z s y , s x s y s z ) .
s i = 1 σ i i ω ,
ɛ t E ˜ x = 𝒟 y H ˜ z 𝒟 z H ˜ y + ɛ ( σ x σ y σ z ) E ˜ x P ˜ x + 1 [ n ^ ( F F * ) ] E x , t P ˜ x = ( σ x 2 σ x σ y σ x σ z + σ y σ z ) ɛ E ˜ x σ x P ˜ x .
j ( t ) = { e ^ z sin ( ω 0 t ) exp ( ( t t 0 ) 2 2 w 2 ) , 0 t 2 t 0 0 , otherwise
r A = ( 1.8 , 0.0 , 0.0 ) , r B = ( 1.8 , 1.8 , 0.0 ) , r C = ( 1.8 , 1.8 , 1.8 ) .
Error ( r p ) = max t i 15 | E z abs ( r p , t i ) E z ref ( r p , t i ) | max t i 15 | E z ref ( r p , t i ) | , p = A , B , C .
= 1 3 [ Error ( r A ) + Error ( r B ) + Error ( r C ) ] .
σ = R 2 d ,
log 10 ( R , α ) 0.1 n ,

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