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On energy preserving consistent boundary conditions for the Yee scheme in 2D. (English) Zbl 1257.78021

In 1966 K. S. Yee [IEEE Trans. Antennas Propag. 14, No. 3, 302–307 (1966; Zbl 1155.78304)] introduced the staggered grids in computational electromagnetics. The electric field components form a primary grid and the magnetic flux density components, normal to the electric field, a secondary grid in the successful Yee scheme, also called FDTD method, with the advantage that the corresponding Maxwellian grid equations are a consistent representation of the analytical equations in the sense that basic properties of the analytical fields (Gauss’ flux laws) are maintained independent of the discretization size. A drawback of the method consists in the demand to get a more accurate allowance of oblique boundary conditions.
Thus, it is the aim of the authors to improve the treatment of such boundary conditions by the use of a staircase approximation preserving good properties of the original Yee scheme, such as the energy conservation and the optimal CFL-condition. That is done by a modification of the coefficients of the update stencil near the boundary avoiding the change of the structure of the Yee scheme. The result is an extension of the consistent boundary treatment in [A.-K. Tornberg and B. Engquist, J. Comput. Phys. 227, No. 14, 6922–6943 (2008; Zbl 1154.65354)] with the additional property that now also the time stability (also called asymptotic stability), i. e. the energy conservation, and the CFL-condition are preserved modifying the coefficients in a new way. The findings are proved and validated by numerical examples for two dimensions. The authors announce an extension of the achievements to the three-dimensional case.

MSC:

78M20 Finite difference methods applied to problems in optics and electromagnetic theory
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L05 Wave equation
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
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[1] Cangellaris, A., Wright, D.: Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena. IEEE Trans. Antennas Propag. 39(10), 1518–1525 (1991)
[2] Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 220–236 (1994) · Zbl 0832.65098
[3] Dey, S., Mittra, R.: A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects. IEEE Microw. Guided Wave Lett. 7(9), 273–275 (1997)
[4] Dey, S., Mittra, R.: A modified locally conformal finite-difference time-domain algorithm for modeling three-dimensional perfectly conducting objects. Microw. Opt. Technol. Lett. 17(6), 349–352 (1998)
[5] Ditkowski, A., Dridi, K., Hesthaven, J.S.: Convergent Cartesian grid methods for Maxwell’s equations in complex geometries. J. Comput. Phys. 170(1), 39–80 (2001) · Zbl 1053.78021
[6] Gottlieb, D., Gustafsson, B., Olsson, P., Strand, B.: On the superconvergence of Galerkin methods for hyperbolic IBVP. SIAM J. Numer. Anal. 33(5), 1778–1796 (1996) · Zbl 0858.65094
[7] Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29(130), 396–406 (1975) · Zbl 0313.65085
[8] Gustafsson, B.: High Order Difference Methods for Time Dependent PDE. Springer, Berlin (2008) · Zbl 1146.65064
[9] Gustafsson, B., Kreiss, H.O., Sundström, A.: Stability theory of difference approximations for mixed initial boundary value problems. ii. Math. Comput. 26(119), 649–686 (1972) · Zbl 0293.65076
[10] Hao, Y., Railton, C.: Analyzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes. IEEE Trans. Microw. Theory Tech. 46(1), 82–88 (1998)
[11] Hesthaven, J.: High-order accurate methods in time-domain computational electromagnetics: A review. Adv. Imaging Electron Phys. 127, 59–123 (2003)
[12] Holland, R.: Pitfalls of staircase meshing. IEEE Trans. Electromagn. Compat. 35(4), 434–439 (1993)
[13] Jurgens, T., Taflove, A.: Three-dimensional contour FDTD modeling of scattering from single and multiple bodies. IEEE Trans. Antennas Propag. 41(12), 1703–1708 (1993)
[14] Jurgens, T., Taflove, A., Umashankar, K., Moore, T.: Finite-difference time-domain modeling of curved surfaces [EM scattering]. IEEE Trans. Antennas Propag. 40(4), 357–366 (1992)
[15] Mattsson, K.: Boundary procedures for summation-by-parts operators. J. Sci. Comput. 18, 133–153 (2003) · Zbl 1024.76031
[16] Monorchio, A., Mittra, R.: A hybrid finite-element finite-difference time-domain (FE/FDTD) technique for solving complex electromagnetic problems. IEEE Microw. Guided Wave Lett. 8(2), 93–95 (1998)
[17] Nieter, C., Cary, J.R., Werner, G.R., Smithe, D.N., Stoltz, P.H.: Application of Dey-Mittra conformal boundary algorithm to 3d electromagnetic modeling. J. Comput. Phys. 228, 7902–7916 (2009) · Zbl 1184.78080
[18] Nordström, J., Forsberg, K., Adamsson, C., Eliasson, P.: Finite volume methods, unstructured meshes and strict stability for hyperbolic problems. Appl. Numer. Math. 45(4), 453–473 (2003) · Zbl 1019.65066
[19] Railton, C., Craddock, I.: Stabilised CPFDTD algorithm for the analysis of arbitrary 3D PEC structures. IEE Proc., Microw. Antennas Propag. 143(5), 367–372 (1996)
[20] Railton, C., Schneider, J.: An analytical and numerical analysis of several locally conformal FDTD schemes. IEEE Trans. Microw. Theory Tech. 47(1), 56–66 (1999)
[21] Rylander, T., Bondeson, A.: Stable FEM-FDTD hybrid method for Maxwell’s equations. Comput. Phys. Commun. 125(1–3), 75–82 (2000) · Zbl 1003.78009
[22] Shang, J.S.: High-order compact-difference schemes for time-dependent Maxwell equations. J. Comput. Phys. 153(2), 312–333 (1999) · Zbl 0956.78018
[23] Tolan, J.G., Schneider, J.B.: Locally conformal method for acoustic finite-difference time-domain modeling of rigid surfaces. J. Acoust. Soc. Am. 114(5), 2575–2581 (2003)
[24] Tornberg, A.K.: Regularization techniques for singular source terms in differential equations. In: Laptev, A. (ed.) European Congress of Mathematics (ECM), Stockholm, Sweden, June 27–July 2 2004. European Mathematical Society, Zurich (2005) · Zbl 1079.65110
[25] Tornberg, A.K., Engquist, B.: Regularization techniques for numerical approximation of PDEs with singularities. J. Sci. Comput. 19, 527–552 (2003) · Zbl 1035.65085
[26] Tornberg, A.K., Engquist, B.: Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200, 462–488 (2004) · Zbl 1115.76392
[27] Tornberg, A.K., Engquist, B.: Regularization for accurate numerical wave propagation in discontinuous media. Methods Appl. Anal. 13, 247–274 (2006) · Zbl 1136.65084
[28] Tornberg, A.K., Engquist, B.: Consistent boundary conditions for the Yee scheme. J. Comput. Phys. 227(14), 6922–6943 (2008) · Zbl 1154.65354
[29] Tornberg, A.K., Engquist, B., Gustafsson, B., Wahlund, P.: A new type of boundary treatment for wave propagation. BIT Numer. Math. 46 (supplement), 145–170 (2006) · Zbl 1106.65078
[30] Trefethen, L.N.: Stability of finite-difference models containing two boundaries or interfaces. Math. Comput. 45(172), 279–300 (1985) · Zbl 0627.65099
[31] Turkel, E.: Progress in computational physics. Comput. Fluids 11(2), 121–144 (1983) · Zbl 0511.76002
[32] Wu, R.B., Itoh, T.: Hybrid finite-difference time-domain modeling of curved surfaces using tetrahedral edge elements. IEEE Trans. Antennas Propag. 45(8), 1302–1309 (1997)
[33] Yee, K.S.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14, 302–307 (1966) · Zbl 1155.78304
[34] Young, J., Gaitonde, D., Shang, J.: Toward the construction of a fourth-order difference scheme for transient em wave simulation: staggered grid approach. IEEE Trans. Antennas Propag. 45(11), 1573–1580 (1997) · Zbl 0947.78612
[35] Yu, W., Mittra, R.: A conformal FDTD algorithm for modeling perfectly conducting objects with curve-shaped surfaces and edges. Microw. Opt. Technol. Lett. 27(2), 136–138 (2000)
[36] Zagorodnov, I., Schuhmann, R., Weiland, T.: A uniformly stable conformal FDTD-method in Cartesian grids. Int. J. Numer. Model. 16(2), 127–141 (2003) · Zbl 1014.78014
[37] Zagorodnov, I., Schuhmann, R., Weiland, T.: Conformal FDTD-methods to avoid time step reduction with and without cell enlargement. J. Comput. Phys. 225(2), 1493–1507 (2007) · Zbl 1135.78015
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