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Dual Scattering Channel Schemes Extending the Johns Algorithm

Steffen Hein

TL;DR

The paper addresses limitations of Johns' Transmission Line Matrix method by introducing Dual Scattering Channel schemes that replace transmission lines with abstract scattering channels and permit non-trivial cell boundary scattering, producing a two-step update cycle with inherent duality. It develops a rigorous DSC framework, including node-boundary duality, near-field interaction, and stability guarantees, and demonstrates a heat diffusion model on a non-orthogonal mesh coupled to a lossy Maxwell field. A deflection formula and explicit conditions recover familiar TLM behavior as a special case while enabling broader, cross-transport modeling. The work broadens time-domain simulation capabilities, enabling stable, parallelizable computation on non-orthogonal grids and signaling potential applications to fluids and other transport phenomena beyond traditional Yee/FDTD methods.

Abstract

Dual scattering channel schemes extend the transmission line matrix numerical method (JOHNS' TLM algorithm) in two directions. For one point, transmission line links are replaced by abstract scattering channels in terms of paired distributions (characteristic impedances are thus neither needed, nor in general defined, e.g.). In the second place, non-trivial cell interface scattering is admitted during the connection cycle. Both extensions open a wide field of application beyond the range of classical time domain schemes, such as YEE's FDTD method and TLM. A DSC heat propagation [diffusion] scheme in non-orthogonal mesh, wherein heat sources are coupled to a lossy Maxwell field, illustrates the approach.

Dual Scattering Channel Schemes Extending the Johns Algorithm

TL;DR

The paper addresses limitations of Johns' Transmission Line Matrix method by introducing Dual Scattering Channel schemes that replace transmission lines with abstract scattering channels and permit non-trivial cell boundary scattering, producing a two-step update cycle with inherent duality. It develops a rigorous DSC framework, including node-boundary duality, near-field interaction, and stability guarantees, and demonstrates a heat diffusion model on a non-orthogonal mesh coupled to a lossy Maxwell field. A deflection formula and explicit conditions recover familiar TLM behavior as a special case while enabling broader, cross-transport modeling. The work broadens time-domain simulation capabilities, enabling stable, parallelizable computation on non-orthogonal grids and signaling potential applications to fluids and other transport phenomena beyond traditional Yee/FDTD methods.

Abstract

Dual scattering channel schemes extend the transmission line matrix numerical method (JOHNS' TLM algorithm) in two directions. For one point, transmission line links are replaced by abstract scattering channels in terms of paired distributions (characteristic impedances are thus neither needed, nor in general defined, e.g.). In the second place, non-trivial cell interface scattering is admitted during the connection cycle. Both extensions open a wide field of application beyond the range of classical time domain schemes, such as YEE's FDTD method and TLM. A DSC heat propagation [diffusion] scheme in non-orthogonal mesh, wherein heat sources are coupled to a lossy Maxwell field, illustrates the approach.

Paper Structure

This paper contains 6 sections, 6 theorems, 63 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Scattering channels
  3. The elements characterizing DSC schemes
  4. The model equations
  5. A heat propagation scheme in non-orthogonal mesh
  6. Conclusions

Key Result

Proposition 1

For every $I_{\zeta}^{\iota} \in I$ there exists exactly one function ${I_{\zeta}^{\iota \sptilde}:\mathcal{E}_{\zeta}^{\iota}\rightarrow} \\ {\mathcal{P}_{\zeta}^{n}} \,\, ( \,$note $I_{\zeta}^{\iota \sptilde} \notin I \,$ ), such that the following diagram is commutative \begin{CD} \mathcal{P}_{

Figures (4)

  • Figure 1: Ports on a cell face with their nodal images.
  • Figure 2: Non-orthogonal hexahedral mesh cell. (a) Edge vectors. (b) Node vectors.
  • Figure 3: Face vectors and temperature points (nodal section).
  • Figure 4: Transverse heat conduction over a square mesh using non-orthogonal cells. A Heaviside temperature step is imposed on one side and the transient temperature computed at the opposite side, assuming adiabatic boundary conditions on all but the heated sides. DSC results (+) are plotted over analytical solution (curve). (a) The mesh. (b) Horizontal (c) vertical propagation.

Theorems & Definitions (19)

  • definition thmcounterdefinition
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  • definition thmcounterdefinition
  • Proposition 1
  • proof
  • definition thmcounterdefinition
  • remark thmcounterremark
  • Theorem 1
  • proof
  • remark thmcounterremark
  • ...and 9 more