Transmission‐line matrix models for solving transient problems of diffusion with recombination

Abstract

Starting from the general telegrapher's equation, we investigate two nodal network constructions for modelling diffusion with recombination by means of the transmission-line matrix (TLM) method. The diffusion effect is modelled by the series and shunt capacitance in one approach, and by the series inductance and shunt resistance in the other. Both approaches use the series and shunt resistances to model the recombination effect. The constraint of using both TLM networks for solving transient problems of diffusion with recombination is found to be identical in terms of the physics behind the numerical routines. A practical way of determining the spatial resolution and iteration time step for accurate TLM numerical computations is suggested based on a simple frequency analysis

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

“…Reaction-diffusion can be modelled using the TLM method by including leakage resistors in the TL network connecting each node to ground [10,17]. There would appear to no reason why the convection-diffusion method described in this paper could not be extended to solve reaction-convection-diffusion equations in a similar manner.…”

Use of lossless transmission‐line segments and shunt resistors for TLM diffusion modelling

“…When this value is specified, however, the value of the transmission-line characteristic impedance is also uniquely given for any predetermined spatial resolution and time step. In brief, the modelling parameters including Dl; Dt; R L and Z must be appropriately correlated to the physical parameter characterizing the diffusion process, namely the diffusion coefficient D: This relationship is dictated by Equation (13), which ensures the solution of the correct diffusion equation by the TLM modelling network.…”

“…However, (2) or (A10) tells us that it is necessary to have both G d and R d for describing the absorption effect. Detailed discussions regarding the TLM models of diffusion with absorption, including the constraints and the determination of spatial resolution and iteration time step, are shown in Reference [13].…”

Boundary conditions in TLM diffusion modelling

“…On the one hand, the incorporation of the inductive component in the transmission-line analogue is necessary for introducing signal delay and naturally enabling time discretization. On the other hand, the inductive error or the second derivative term with respect to time in Equation (5) can be made negligibly small by choosing a suitable time iteration interval [2,6]. Based on this assumption, the comparison of (5) with (1) yields the following equivalence:…”