In this paper, a new method, called the lumped-component circuit method (LCM), is developed for one-dimensioal and two-dimensional convection–reaction–diffusion with low to moderate Peclet numbers, tested for modelling both steady-state and transient problems, and compared with standard finite volume method (FVM) schemes. The method has been developed principally for solving equations with piecewise-constant coefficients using nodes that are not positioned to correspond to the coefficient discontinuities. In such situations, the FVM solutions do not converge consistently as the node spacing is decreased, but LCM solutions do. In general, the LCM method is more accurate than the FVM schemes tested, and, while the computational cost of LCM is higher, results suggest that it can be more efficient. Like the transmission line method (TLM), it is an indirect scheme in which the problem to be solved is first represented by an analogous transmission line (TL). Unlike with TLM, however, the TL is then modelled using a lumped-component circuit, the voltages at nodes within that circuit being calculated.