The efficiency of transmission‐line matrix modelling—a rigorous viewpoint

Abstract

SUMMARYThe difference equations of the scalar linear transmission-line matrix (TLM) routine as introduced by Johns for numerically solving the diffusion equation are shown to be isomorphic to Goldstein's correlated random walk model of diffusion. For the infinite homogeneous bar their exact solution is derived algebraically and given in the form of Jacobi polynomials. This puts the TLM algorithm on a sounder mathematical and physical basis. The accuracy in solving the diffusion equation is investigated in general form and thus its astonishing efficiency explained. Several other basic questions of this numerical technique are also discussed.

A transmission line modelling (TLM) method for steady‐state convection–diffusion

“…This is equivalent to the one-dimensional diffusion equation with one extra term (the third term in (1)). If this (wave propagation) term is negligible then solving for the voltage on a lossy TL is equivalent to solving the diffusion equation [6][7][8]. This is the basis for the TLM method for modelling diffusion.…”

Transition probability coefficients and the stability of finite difference schemes for the diffusion and Telegraphers' equations

“…The computational space is represented by a discretized network of transmission lines (impedance, Z) and series resistors (magnitude 2R). It can be shown that the impedance Z ¼ Dt=C ¼ L=Dt (where C is the capacitance given by C d Dx and L is the This is a two-step Markov process [7] with a correlation coefficient ðr 2 À G 2 Þ which, although it can range in value between þ1 and À1; is unconditionally stable [8]. Since ðr þ GÞ ¼ 1 we can rewrite Equation (12) We can now return to the finite difference discretizations of the Telegraphers' equation and note the points of similarity with TLM.…”

TLM for diffusion: the artefact of the standard initial conditions and its elimination within an abstract TLM suite

“…Unconditional stability has been established as well as the relationship to previous work on correlated random walks and to the DuFort}Frankel scheme [2]. Many real-world applications have been performed successfully.…”