Accelerated convergence in TLM algorithms for the Laplace equation

Transmission line matrix (TLM) schemes for the Laplace equation exhibit some curious features. There exist values of TLM parameters where convergence towards the analytical values is very rapid. This phenomenon is first examined using a binary scattering approach. A Fourier analysis of the equivalently bounded diffusion equation does not reveal any features that would account for these observations. However, a similar analysis using the Telegraphers' equation suggests that a TLM model under optimum conditions is operating at the transition between real and imaginary solutions. Small differences between the optimized parameters predicted by the two approaches are probably due to inaccuracies in the Fourier description of the heat-injecting boundary condition that is used in TLM models.