Analysis of FDTD to UPML for Maxwell equations in polar coordinates

An FDTD system associated with uniaxial perfectly matched layer (UPML) for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered. Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic (TM) mode to Maxwell's equations is obtained by Yee's algorithm, and the open domain of the scattering problem is truncated by a circle with a UPML. Besides, an artificial boundary condition is imposed on the outer boundary of the UPML. Afterwards, stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality. Numerical experiments are designed to approve the theoretical analysis.