Gauge techniques in time and frequency domain TLM
Steffen Hein
TL;DR
The paper formalizes gauge-like internal degrees of freedom in the Transmission Line Matrix (TLM) method using a propagator framework, showing that stub loading embodies perturbative gauge interactions linked to gauge-field concepts. It introduces the deflection lemma as a canonical tool for algorithm synthesis and extends gauge-like perturbations to nonlinear settings. In the frequency domain, the work separates gauge contributions from conventional scattering in the system's S-parameters, enabling direct interpretation of mesh properties. A prototype coupling of a relativistic charged particle current to Maxwell fields demonstrates the method on a physically meaningful nonlinear problem and illustrates stability and convergence properties. Overall, the results provide a principled basis for extending TLM with gauge-inspired transformations and for bridging TLM with classical network and gauge-field concepts in computational plasma physics.
Abstract
Typical features of the Transmission Line Matrix (TLM) algorithm in connection with stub loading techniques and prone to be hidden in common frequency domain formulations are elucidated within the propagator approach to TLM. In particular, the latter reflects properly the perturbative character of the TLM scheme and its relation to gauge field models. Internal 'gauge' degrees of freedom are made explicit in the frequency domain by introducing the complex nodal S-matrix as a function of operators that act on external or internal fields or virtually couple the two. As a main benefit, many techniques and results gained in the time domain thus generalize straight away. The recently developed deflection method for algorithm synthesis, which is extended in this paper, or the non-orthogonal node approximating Maxwell's equations, for instance, become so at once available in the frequency domain. In view of applications in computational plasma physics, the TLM model of a relativistic charged particle current coupled to the Maxwell field is treated as a prototype.