A 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 one-dimensional 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.