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

Abstract

SUMMARYWe demonstrate both analytically and numerically, that for the TLM algorithm for numerically solving the di!usion equation the standard initial conditions for the incident voltage pulses in terms of the initial concentration (or temperature) distribution introduce a signi"cant numerical error. We discover the origin of this error and present an alternative prescription for computing the node voltages at the "rst time step which removes this artefact and exhibits distinctly improved numerical accuracy. For this, we introduce the term abstract TLM suite (ATS). An ATS is a numerical scheme which formally parallels a TLM scheme, but the scattering coe$cients, and , may assume values outside the interval [!1, 1] or may not correspond to the values of a real TLM network.

Error analysis and reduction in lossy TLM

“…It may, however, be possible to temporarily adjust the transmission coefficient over one or more time steps so that the long-term variance error is zero even if the model is initialized in the standard way. Since the initial time step in a TLM model is not consistent with the other time steps [8], altering that time step in order to achieve more accurate results will not reduce the consistency of the method. Equation (46) can be rewritten to give the variance at time step k in terms of the node voltage variance and the incident pulse distribution mean positions at any previous time step j k s…”

“…The analytical solution of the diffusion equation for a one-dimensional system of length with boundary conditions (0, t) = ( , t) = 0 and initial condition (x, 0) = sin( x/ )+3 sin(2 x/ ) is [8] …”

“…This discrepancy leads to a significant error over these time steps and to a lesser long-term error that never reduces to zero. It has been shown that the first time step of the scheme is inconsistent with further time steps, but the method can be altered temporarily for that time step in such a way as to make it consistent [8]. An expression for the variance at each time step has been derived previously [9,10].…”

TLM for diffusion: consistent first time step. Two‐dimensional case

“…As in the 1D case [7,8], the error introduced by this inconsistency is of the same order in Dt and Dx 2 as the discretization error of the 2-step scheme itself. From the point of view of d'Alembert's solution to the wave equation, the deeper reason for this is the fact that the initial condition (13a) belongs to the case @f=@tðx; 0Þ ¼ 0: The contradiction is resolved for the case t ¼ 1=4; where (9) reduces to the standard explicit FD scheme and the wave properties are essentially lost.…”

“…However, it has recently been observed that the usual initial conditions applied to the TLM algorithm are not consistent with its own difference scheme [7,8]. In this paper, we investigate an improved model initialization and apply it to a two-dimensional (2D) case in a TLM link-line model without stubs.…”