×

Discretization of Maxwell’s equations for non-inertial observers using space-time algebra. (English) Zbl 1393.65033

Summary: We employ classical Maxwell’s equations formulated in space-time algebra to perform discretization of moving geometries directly in space-time. All the derivations are carried out without any non-relativistic assumptions, thus the application area of the scheme is not restricted to low velocities. The 4D mesh construction is based on a 3D mesh stemming from a conventional 3D mesh generator. The movement of the system is encoded in the 4D mesh geometry, enabling an easy extension of well-known 3D approaches to the space-time setting. As a research example, we study a manifestation of Sagnac’s effect in a rotating ring resonator. In case of constant rotation, the space-time approach enhances the efficiency of the scheme, as the material matrices are constant for every time step, without abandoning the relativistic framework.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
15A66 Clifford algebras, spinors
35Q61 Maxwell equations
78M12 Finite volume methods, finite integration techniques applied to problems in optics and electromagnetic theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Auchmann, B; Kurz, S, Observers and splitting structures in relativistic electrodynamics, J. Phys. A Math. Theor., 47, 435202, (2014) · Zbl 1302.78003
[2] Bossavit, A.: Computational Electromagnetism. Academic Press, San Diego (1998) · Zbl 0945.78001
[3] Bossavit, A, “generalized finite differences” in computational electromagnetics, PIER, 32, 45-64, (2001)
[4] Codecasa, L; Specogna, R; Trevisan, F, Symmetric positive-definite constitutive matrices for discrete eddy-current problems, IEEE Trans. Magn., 43, 510-515, (2007)
[5] Codecasa, L; Trevisan, F, Piecewise uniform bases and energetic approach for discrete constitutive matrices in electromagnetic problems, Int. J. Numer. Methods Eng., 65, 548-565, (2006) · Zbl 1241.78042
[6] Doran, C., Lasenby, A.: Geometric Algebra for Physicists, 2nd edn. Cambridge University Press, Cambridge (2003) · Zbl 1078.53001
[7] Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge Books Online. Cambridge University Press, Cambridge (2006) · Zbl 1121.53001
[8] Hehl, FW, History related maxwell’s equations in minkowski’s world: their premetric generalization and the electromagnetic energy-momentum tensor, Annalen der Physik, 17, 691-704, (2008) · Zbl 1207.78006
[9] Hestenes, D.: Space-Time Algebra. Documents on Modern Physics. Gordon and Breach, New York (1966)
[10] Hestenes, D; Brackx, F (ed.); Delanghe, R (ed.); Serras, H (ed.), Differential forms in geometric calculus, 269-285, (1993), Berlin
[11] Hestenes, D; Dorst, L (ed.); Lasenby, J (ed.), The shape of differential geometry in geometric calculus, 393-410, (2011), Berlin · Zbl 1292.53006
[12] Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. D. Reidel; Distributed in the USA and Canada by Kluwer Academic Publishers, Dordrecht; Boston; Hingham (1984) · Zbl 0541.53059
[13] Hiptmair, R, Finite elements in computational electromagnetism, Acta Numerica, 11, 237-339, (2002) · Zbl 1123.78320
[14] Jin, J.-M.: The Finite Element Method in Electromagnetics. Wiley, New York (2002) · Zbl 1001.78001
[15] Klimek, M., Roemer, U., Schoeps, S., Weiland, T.: Space-Time Discretization of Maxwell’s Equations in the Setting of Geometric Algebra. In: 2013 International Symposium on Electromagnetic Theory, editor, IEEE Xplore (2013)
[16] Mattiussi, C, The geometry of time-stepping, Prog. Electromagn. Res., 32, 123-149, (2001)
[17] Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman and Company, New York (1973)
[18] Mur, G, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations, IEEE Trans. Electromagn. Compat. EMC, 23, 377-382, (1981)
[19] Novitski, R; Scheuer, J; Steinberg, BZ, Unconditionally stable finite-difference time-domain methods for modeling the Sagnac effect, Phys. Rev. E, 87, 023303, (2013)
[20] Peng, C; Hui, R; Luo, X; Li, Z; Xu, A, Finite-difference time-domain algorithm for modeling Sagnac effect in rotating optical elements, Opt. Express, 16, 5227-5240, (2008)
[21] Salamon, J., Moody, J., Leok M.: Geometric representations of whitney forms and their generalization to Minkowski spacetime. arXiv:1402.7109v1 (2014)
[22] Schuhmann, R; Weiland, T, A stable interpolation technique for FDTD on nonorthogonal grids, Int. J. Numer. Model. Electron. Netw. Devices Fields, 11, 299-306, (1998) · Zbl 0926.65092
[23] Sobczyk, G.: Simplicial calculus with geometric algebra. In: Micali, A., Boudet, R., Helmstetter, J. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 279-292. Springer, Berlin (2011) · Zbl 0783.58005
[24] Steinberg, BZ; Shamir, A; Boag, A, Two-dimensional green’s function theory for the electrodynamics of a rotating medium, Phys. Rev. E, 74, 016608, (2006)
[25] Stern, A., Tong, Y., Desbrun, M., Marsden, J.E.: Geometric computational electrodynamics with variational integrators and discrete differential forms. arXiv:0707.4470 [v2] (2009) · Zbl 1337.78014
[26] Weiland, T, Time domain electromagnetic field computation with finite difference methods, Int. J. Numer. Model., 9, 295-319, (1996)
[27] Yang, Y., Pesavento, M.: A unified successive pseudoconvex approximation framework. IEEE. Trans. Sig Process. 65(13), 3313-3328 (2017). https://doi.org/10.1109/TSP.2017.2684748
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.