Exact 'absorbing' conditions in initial boundary-value problems of the theory of pulse wave radiation

Abstract

The outcomes of a search for efficient and correct ways of the truncation of computational domain in finite-difference methods are presented. The relevant mathematical problem is resolved rigorously both for two-dimensional scalar model problems and for three-dimensional vector problems. In the framework of this abstract, we describe briefly only two typical two-dimensional situations. The peculiarities associated with a change to the analysis of three-dimensional vector problems are to be covered in the report.

An FFT-Accelerated FDTD Scheme With Exact Absorbing Conditions for Characterizing Axially Symmetric Resonant Structures

“…The discretization of the EAC (18) on L 2 can be done in a similar way. In a conventional FDTD scheme, it is more convenient to discretize (15) for the amplitudes u n1 (z, t) and then use (2) to obtain boundary values of the total field U (ρ, −L 1 , t) rather than to discretize (16) directly. On L 1 the total field is a sum of the scattered field and the incident wave: U (ρ, z, t) = U s (ρ, z, t) + U i (ρ, z, t), and one should remember that (16) is formulated for the scattered field and the problem (45) is being solved for the total field.…”

“…T E 0n and T M 0n waves propagating in the axially symmetric waveguide unit and radiating to the outside satisfy [14,15]: Figure 1. Geometry of the model problems: (a) a waveguide cavity resonator fed and terminated by coaxial waveguides; (b) a parabolic radiator with elliptical subreflector and infinite flange.…”

“…Inserting (2b) into the homogenous version of (1) (i.e., assuming ε r (g) = 1, σ (g, t) = 0, ϕ (g) = 0 and ψ (g) = 0), equations satisfied by µ n1 (ρ) and u n1 (z, t) are obtained (see (4) and (5) below). The transverse functions µ n1 (ρ) and the corresponding eigenvalues λ n1 are specified by nontrivial solutions of the homogeneous problem [14,15] …”

“…In addition, the same FFT-based acceleration method is used for computing the convolutions of nonlocal EACs enforced on spherical boundaries, which are located external to the structures only on their radiating ends. To briefly summarize, the contributions of this paper are threefold: (i) A detailed derivation of the (planar and spherical) EACs for axially symmetric waveguide and radiating structures, which follows the approach developed in [14,15], is presented. (ii) Implementation and discretization of these (local and nonlocal) EACs in an FDTD method are discussed.…”

“…The computation domain is truncated using the EACs derived from the radiation conditions for the outgoing modes' spatio-temporal amplitudes [14,15]. The nonlocal EACs on the planar boundaries located inside the waveguides are localized without the loss of mathematical exactness.…”