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

Abstract

SUMMARYThis paper describes how the lossy transmission line modelling (TLM) method for diffusion can be extended to solve the convection-diffusion equation. The method is based on the correspondence between the convection-diffusion equation and the equation for the voltage on a lossy transmission line with properties varying exponentially over space. It is unconditionally stable and converges rapidly to highly accurate steady-state solutions for a wide range of Peclet numbers from low to high. The method solves the non-conservative form of the convection-diffusion equation but it is shown how it can be modified to solve the conservative form. Under transient conditions the TLM scheme exhibits significant numerical diffusion and numerical convection leading to poor accuracy, but both these errors go to zero as a solution approaches steady state.

An alternative TLM method for steady‐state convection–diffusion

“…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].…”

“…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.…”

“…To impose Dirichlet boundary conditions, nodes can be located at each boundary and the node voltages at those nodes simply fixed at the required values. It is not necessary to locate nodes at the boundaries but this does simplify the scheme [4]. The implementation of other types of boundary conditions is not considered here.…”

“…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].…”

“…Although TLM usually models in the time-domain, steady-state solutions can be calculated directly [5]. There is a rigorous procedure, described fully elsewhere [4,5], for determining the parameters of the TLM model from given coefficients of the CDE to be solved.…”