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 - <p style="text-align: justify;">We consider time domain formulations of Maxwell&#39;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&#39;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.</p>