Applied Numerical Mathematics 33 (2000) 517–524 Reflectionless sponge layers for the numerical solution of Maxwell’s equations in cylindrical and spherical coordinates P.G. Petropoulos 1,2 Department of Mathematics, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA Abstract We review the scaling argument used to derive reflectionless wave absorbing layers for use as Absorbing Boundary Conditions (ABC) in numerical solutions of the elliptic and hyperbolic Maxwell equations in cylindrical and spherical coordinates, and show that thus obtained absorbing layers are described in the time-domain by causal, strongly well-posed hyperbolic systems. Representative results are given for scattering by cylinders. Also, we study the reflection of local ABC’s in discrete space.  2000 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Absorbing boundary conditions; Maxwell’s equations We wish to solve numerically the elliptic (frequency-domain) and hyperbolic (time-domain) Maxwell equations in a domain Ω c ⊂ R 3 embedded in an infinite dielectric background medium Ω m of constant permittivity ε and permeability µ. On the computational domain boundary, ∂Ω c , an absorbing boundary condition must be imposed to provide field values for the interior solution algorithm. An alternative to ABC’s is to surround Ω c with a wave absorbing layer Ω m of thickness d that possesses the following properties: the transition from Ω c to Ω m should not produce wave reflection, while the fields that have penetrated into Ω m should attenuate as they propagate outward. The existence of such layers was shown by Berenger [2], who produced the first split-field PML analyzed in [1]. Subsequently, the unsplit PML [6] has become popular and it has also been applied to high-order finite difference schemes [5]. Also, it is simple to apply such layers to pseudo-spectral codes for the Maxwell equations. We present here an extension of [6] to cylindrical and spherical coordinates. Section 1 contains our general approach. The discussion of the time-domain (hyperbolic) formulation is in Section 3, and Section 4 presents a representative result. All the details can be found in [3] (which is available upon request). 1 E-mail: peterp@m.njit.edu 2 Supported in part by AFOSR Grant F49620-99-1-0072. 0168-9274/00/$20.00  2000 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0168-9274(99)00120-8