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Volume 12, Issue 2
Schemes and Estimates for the Long-Time Numerical Solution of Maxwell's Equations for Lorentz Metamaterials

Jichun Li & Simon Shaw

Int. J. Numer. Anal. Mod., 12 (2015), pp. 343-365.

Published online: 2015-12

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  • Abstract

We consider time domain formulations of Maxwell's equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall's inequality. The resulting estimates are, in this sense, sharp. We also give fully discrete formulations for each case and extend the sharp data-stability to these. Since the physical problem is linear it follows (and we show this with examples) that this stability property is also reflected in the constants appearing in the a priori error bounds. By removing the exponential growth in time from these estimates we conclude that these schemes can be used with confidence for the long-time numerical simulation of Lorentz metamaterials.

  • Keywords

Maxwell's equations, Lorentz model, metamaterial, Galerkin and mixed finite element method, long-time integration, time stepping.

  • AMS Subject Headings

65N30, 35L15, 78-08

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-12-343, author = {Jichun Li and Simon Shaw}, title = {Schemes and Estimates for the Long-Time Numerical Solution of Maxwell's Equations for Lorentz Metamaterials}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2015}, volume = {12}, number = {2}, pages = {343--365}, abstract = {

We consider time domain formulations of Maxwell's equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall's inequality. The resulting estimates are, in this sense, sharp. We also give fully discrete formulations for each case and extend the sharp data-stability to these. Since the physical problem is linear it follows (and we show this with examples) that this stability property is also reflected in the constants appearing in the a priori error bounds. By removing the exponential growth in time from these estimates we conclude that these schemes can be used with confidence for the long-time numerical simulation of Lorentz metamaterials.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/493.html} }
TY - JOUR T1 - Schemes and Estimates for the Long-Time Numerical Solution of Maxwell's Equations for Lorentz Metamaterials AU - Jichun Li & Simon Shaw JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 343 EP - 365 PY - 2015 DA - 2015/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/493.html KW - Maxwell's equations, Lorentz model, metamaterial, Galerkin and mixed finite element method, long-time integration, time stepping. AB -

We consider time domain formulations of Maxwell's equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall's inequality. The resulting estimates are, in this sense, sharp. We also give fully discrete formulations for each case and extend the sharp data-stability to these. Since the physical problem is linear it follows (and we show this with examples) that this stability property is also reflected in the constants appearing in the a priori error bounds. By removing the exponential growth in time from these estimates we conclude that these schemes can be used with confidence for the long-time numerical simulation of Lorentz metamaterials.

Jichun Li and Simon Shaw. (2015). Schemes and Estimates for the Long-Time Numerical Solution of Maxwell's Equations for Lorentz Metamaterials. International Journal of Numerical Analysis and Modeling. 12 (2). 343-365. doi:
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