On the stability of dual scattering channel schemes
Steffen Hein
TL;DR
This work addresses the stability of dual scattering channel (DSC) schemes, which generalize Johns' TLM by replacing transmission lines with abstract scattering channels. It introduces the $\alpha$-passivity framework, enabling a contraction-like, energy-like functional to characterize stability for causal maps, including nonlinear cases. The main result proves that finitely excited DSC processes generated by $\alpha$-passive reflection and connection maps are uniformly bounded, with a decomposition into two $\alpha$-passive sub-processes ensuring stability via a general theorem. Consequently, unconditional stability concepts known for linear TLM extend to DSC, providing robust stability guarantees for DSC-based simulations of wave propagation and related hyperbolic problems, even under nonlinear dynamics.
Abstract
Dual scattering channel (DSC) schemes generalize Johns' TLM algorithm in replacing transmission lines with abstract scattering channels in terms of paired distributions. A well known merit of TLM schemes is unconditional stability, a property that is commonly drawn upon the passivity of linear transmission line networks. So the question arises, if DSC algorithms remain stable in a neat sense. It is shown that a large class of alpha-passive processes are in fact unconditionally stable. The analysis applies to TLM and DSC schemes alike and includes non-linear situations.