A TLM method for steady‐state convection–diffusion: Some additions and refinements

Abstract

SUMMARYA recent paper introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving steady-state convection-diffusion problems. This paper shows how this onedimensional scheme can be adapted to include reaction and source terms and how it can be implemented with non-equidistant nodes. It introduces new ways of calculating the necessary model parameters which can improve the accuracy of the scheme, shows how steady-state solutions can be obtained directly, and compares results with those from two finite difference (FD) methods. While the cost of implementation is higher than for the FD schemes, the new TLM scheme can be significantly more accurate, especially when convection dominates.

An alternative TLM method for steady‐state convection–diffusion

“…The one-dimensional steady-state convection-diffusion equation ( (5) Modelling such a transmission line is equivalent to solving the convection-diffusion equation. It should be noted that the distributed inductance does not appear in Equation (2).…”

“…It should be noted that the distributed inductance does not appear in Equation (2). The TLM method, however, used here to model the TL, requires a time step to be chosen, even if a steady-state solution is to be found directly, and this value determines the level of inductance [4,5].…”

“…the length of the TL) and time are divided into finite increments. Traditionally steady-state solutions have been found by running the scheme until transients reduce to an acceptable level [7,10,11], but a recent paper has shown that they can also be found directly [5]. The first step in modelling a transmission line using TLM is to choose the locations of the nodes at which the solution will be calculated and a time step length, Δt.…”

“…Once the values of the distributed resistance, impedance, and current source, are known, it is possible to determine the properties of the discrete components in the equivalent TLM network [4,5]. The pair of resistors between each pair of nodes must have the same total resistance as the equivalent section of the TL being modelled.…”

“…Recent papers have presented a novel Transmission Line Modelling (TLM) scheme, referred to here as the "varied impedance" (VI) method, which can solve the steady-state convection-diffusion equation in one dimension accurately and efficiently [4,5]. The method is particularly efficient when the convection term dominates, a situation in which most traditional schemes have difficulty producing accurate results [1][2][3]6].…”