Spatially weighted numerical models for the two‐dimensional wave equation: FD algorithm and synthesis of the equivalent TLM model

Abstract

AbstractIn this paper a new TLM model for the two‐dimensional wave equation is introduced. It is synthesized directly from a FD algorithm. The FD algorithm is second‐order‐accurate in both space and time, and is explicitly time‐stepped. The spatial derivatives in the FD algorithm are approximated by the weighted combination of two standard central difference stencils, one oriented as usual, the other rotated by 45° with its arms extended by a factor of (2)1/2. The TLM model is realized as the weighted connection of two original models (with the same geometrical configuration as the FD algorithm). The weighting in the TLM model is accomplished by using a variable intrinsic impedance for specific elemental transmission lines. The FD and TLM methods possess identical dispersion relations if the former is operated at its upper limit of stability. Therefore, under these conditions both represent identical models for the simulation of wave propagation. The propagation characteristics of the new model are investigated and the conditions for approximate numerical isotropy are provided. The numerical implementation (scattering matrix and transfer event) is described. To validate the new model, the calculation of cutoff frequencies of various modes in rectangular waveguide is performed. Comparison with analytical results (for an unfilled waveguide) and other numerical results (for a waveguide partially filled with a dielectric) validate the implementation of the model.