Journal of Computational Physics 258 (2014) 359–370 Contents lists available at ScienceDirect Journal of Computational Physics www.elsevier.com/locate/jcp Transformation optics based local mesh refinement for solving Maxwell’s equations Jinjie Liu a,∗ , Moysey Brio b , Jerome V. Moloney b a Department of Mathematical Sciences, Delaware State University, Dover, DE 19901, United States b Arizona Center for Mathematical Sciences at Department of Mathematics, The University of Arizona, Tucson, AZ 85721, United States article info abstract Article history: Received 21 August 2013 Received in revised form 21 October 2013 Accepted 26 October 2013 Available online 4 November 2013 Keywords: Maxwell’s equations FDTD Transformation optics Local mesh refinement Subgridding In this paper, a novel local mesh refinement algorithm based on transformation optics (TO) has been developed for solving the Maxwell’s equations of electrodynamics. The new algorithm applies transformation optics to enlarge a small region so that it can be resolved by larger grid cells. The transformed anisotropic Maxwell’s equations can be stably solved by an anisotropic FDTD method, while other subgridding or adaptive mesh refinement FDTD methods require time–space field interpolations and often suffer from the late-time instability problem. To avoid small time steps introduced by the transformation optics approach, an additional application of the mapping of the material matrix to a dispersive material model is employed. Numerical examples on scattering problems of dielectric and dispersive objects illustrate the performance and the efficiency of the transformation optics based FDTD method. © 2013 Elsevier Inc. All rights reserved. 1. Introduction The Finite-Difference Time-Domain (FDTD) method [1–4] is a very popular method of numerically solving Maxwell’s equation due to its simplicity, stability, minimal memory requirements, and ability to easily couple a large variety of material models. One of the major difficulties of the explicit finite-difference method is the computational cost of resolving small structures present in a large computational domain. One way to overcome this problem is to use small grid cells locally near the small object while using large grid cells elsewhere as in the subgridding or adaptive mesh refinement (AMR) methods [5–27]. However, as it has been pointed out in [24,25,27], current subgridding and adaptive mesh refinement FDTD methods often suffer from the late-time instability problems. Another drawback of the subgridding or AMR is the reduced accuracy when the ratio of the space steps is large. The Huygens subgridding method [25–27] can achieve very large ratio of spatial steps, but it also has the late-time instability problem. In contrast, all of the above methods, change only independent variables without corresponding field transformations and thus require temporal and/or spacial field interpolations that lead to late-time instabilities. Recently, transformation optics (TO) [28–32] has been used to design the metamaterials, with many interesting appli- cations, such as the metamaterial invisibility cloak [29], the space time cloak [33], shrinking an object [34], etc. A very important feature of the transformation optics is the invariance of the Maxwell’s equations after a coordinate transforma- tion. For example, the metamaterial cloak [29] is designed by transforming a disk to an annulus. As a result, the light is bent in the annulus and the region inside the inner circle of the annulus is cloaked. * Corresponding author. E-mail addresses: jliu@desu.edu (J. Liu), brio@math.arizona.edu (M. Brio), jml@acms.arizona.edu (J.V. Moloney). 0021-9991/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcp.2013.10.048