Higher Order FDTD Formulation of Analytical ABCs

2 HIGHER ORDER FDTD FORMULATION OF ANALYTICAL ABCs 

Obviously, the combination of the aforementioned ABCs with higher order FDTD methods requires systematic modifications. This is mainly attributed to the dissimilar nodal 

Let us consider Lindman annihilators [3], which are constmcted through the use of projection operators incorporating past data at the boundary. Primarily, they involve the suitable field approximations by solving a system of partial differential equations in terms of certain correction functions. Focusing on the absorption of Ex electric component at the outer boundary, z = LAz, the higher order nonstandard FDTD form of its update expression for a lossy medium, in conjunction with (3.70) and (3.71), becomes 

Complementary operators, on the other hand, comprise an efficient boundary procedure that improves the absorption of numerical reflections by using previously developed ABCs [13]. Their competence is based on the implementation of two boundary operators that are complementary in their action. In this manner, new ABCs that produce prespecified reflection coefficients are derived. By solving the problem with each of the two operators and then averaging the two solutions, the technique annihilates the first-order artificial reflections of both obliquely propagating and evanescent waves, irrespective of their wave number. More specifically, absorption of the latter occurs even if the original ABC reflects them totally. The modified higher order development of COM starts from a well-posed and stable ABC that can be expressed by a single differential equation, defined as J-. If nonstandard operator [.] is 

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